Design and Analysis of Matching and Auction Markets · Design and Analysis of Matching and Auction ... Design and Analysis of Matching and Auction Markets Daniela Saban ... The outcome

Design and Analysis of Matching and AuctionMarkets

Daniela Saban

Submitted in partial fulfillment of the

requirements for the degree

of Doctor of Philosophy

under the Executive Committee

of the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2015

c©2015

Daniela Saban

All Rights Reserved

ABSTRACT

Design and Analysis of Matching and AuctionMarkets

Daniela Saban

Auctions and matching mechanisms have become an increasingly important tool to allocate

scarce resources among competing individuals or firms. Every day, millions of auctions are

run for a variety of purposes, ranging from selling valuable art or advertisement space in

websites to acquiring goods for government use. Every year matching mechanisms are used

to decide the public school assignments of thousands of incoming high school students, who

are competing to obtain a seat in their most preferred school. This thesis addresses several

questions that arise when designing and analyzing matching and auction markets.

The first part of the dissertation is devoted to matching markets. In Chapter 2, we

study markets with indivisible goods where monetary compensations are not possible. Each

individual is endowed with an object and has ordinal preferences over all objects. When

preferences are strict, the Top-Trading Cycles (TTC) mechanism invented by Gale is Pareto

efficient, strategy-proof, and finds a core allocation, and is the only mechanism satisfying

these properties. In the extensive literature on this problem since then, the TTC mechanism

has been characterized in multiple ways, establishing its central role within the class of

all allocation mechanisms. In many real applications, however, the individual preferences

have subjective indifferences; in this case, no simple adaptation of the TTC mechanism

is Pareto efficient and strategy-proof. We provide a foundation for extending the TTC

mechanism to the preference domain with indifferences while guaranteeing Pareto efficiency

and strategy-proofness. As a by-product, we establish sufficient conditions for a mechanism

(within a broad class of mechanisms) to be strategy-proof and use these conditions to design

computationally efficient mechanisms.

In Chapter 3, we study several questions associated to the Random Priority (RP) mech-

anism from a computational perspective. The RP mechanism is a popular way to allocate

objects to agents with strict ordinal preferences over the objects. In this mechanism, an

ordering over the agents is selected uniformly at random; the first agent is then allocated

his most-preferred object, the second agent is allocated his most-preferred object among

the remaining ones, and so on. The outcome of the mechanism is a bi-stochastic matrix

in which entry (i, a) represents the probability that agent i is given object a. It is shown

that the problem of computing the RP allocation matrix is #P-complete. Furthermore, it

is NP-complete to decide if a given agent i receives a given object a with positive probabil-

ity under the RP mechanism, whereas it is possible to decide in polynomial time whether

or not agent i receives object a with probability 1. The implications of these results for

approximating the RP allocation matrix as well as on finding constrained Pareto optimal

matchings are discussed.

Chapter 4 focuses on assignment markets (matching markets with transferable utilities),

such as labor and housing markets. We consider a two-sided assignment market with agent

types and stochastic structure similar to models used in empirical studies, and characterize

the size of the core in such markets. We allow the number of agents to grow, keeping the

number of agent types fixed. Let n be the number of agents and K be the number of types

on the side of the market with more types. We find, under reasonable assumptions, that the

relative variation in utility per agent over core outcomes is bounded as O∗(1/n1/K), where

polylogarithmic factors have been suppressed. Further, we show that this bound is tight in

worst case, and provide a tighter bound under more restrictive assumptions.

In the second part of the dissertation, we study auction markets. Chapter 5 considers

the problem faced by a procurement agency that runs an auction-type mechanism to con-

struct an assortment of products with posted prices, from a set of differentiated products

offered by strategic suppliers. Heterogeneous consumers then buy their most preferred al-

ternative from the assortment as needed. Framework agreements (FAs), widely used in the

public sector, take this form; this type of mechanism is also relevant in other contexts, such

as the design of medical formularies and group buying. When evaluating the bids, the pro-

curement agency must consider the trade-off between offering a richer menu of products for

consumers, versus offering less variety, hoping to engage the suppliers in a more aggressive

price competition. We develop a mechanism design approach to study this problem, and

provide a characterization of the optimal mechanisms. This characterization allows us to

quantify the optimal trade-off between product variety and price competition, in terms of

suppliers’ costs, products’ characteristics, and consumers’ characteristics. We then use the

optimal mechanism as a benchmark to evaluate the performance of the Chilean government

procurement agency’s current implementation of FAs, used to acquire US$2 billion worth

of goods per year. We show how simple modifications to the current mechanism, which in-

crease price competition among close substitutes, can considerably improve performance.

Table of Contents

List of Figures v

List of Tables vi

1 Introduction 1

1.1 Matching Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Auction Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

I Matching Markets 8

2 House Allocation with Indifferences 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 A brief overview of the TTC Algorithm . . . . . . . . . . . . . . . . 15

2.3 The Trading Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Pareto efficiency, weak-core and generality of the Trading mechanisms 19

2.3.2 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Strategy-proofness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1 An alternative characterization of strategy-proofness . . . . . . . . . 25

2.4.2 Sufficient conditions for local invariance . . . . . . . . . . . . . . . . 28

2.5 Selection Rules: Old and New . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5.1 Improving-cycles-only rules . . . . . . . . . . . . . . . . . . . . . . . 34

i

2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Complexity of Computing the RP Matrix 46

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 The complexity of Random Priority . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Decision problems associated with Random Priority . . . . . . . . . . . . . 53

3.4.1 The SD Feasibility problem . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.2 The SD Unique Assignment problem . . . . . . . . . . . . . . . . . . 61

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4 The Size of the Core in Assignment Markets 65

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.1 Structure of Φ(i, j) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4 Overview of the proof of the main result . . . . . . . . . . . . . . . . . . . . 75

4.4.1 Overview of the upper bound proof . . . . . . . . . . . . . . . . . . . 75

4.4.2 Hypercube definitions and key lemmas . . . . . . . . . . . . . . . . . 77

4.4.3 Proof of the lower bound . . . . . . . . . . . . . . . . . . . . . . . . 80

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

II Auction Markets 83

5 Procurement Mech. for Differentiated Products 84

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2 Related literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3 Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3.2 Mechanism Design Problem Formulation . . . . . . . . . . . . . . . . 95

ii

5.4 General Solution Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.5 Affine Demand Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.5.1 Applying the Solution Approach to Affine Demand Models . . . . . 101

5.5.2 Optimal Mechanism for Hotelling Demand Model . . . . . . . . . . . 103

5.5.3 Optimal mechanisms for general Affine Demand models . . . . . . . 109

5.6 Case Study: ChileCompra-Style Framework Agreements . . . . . . . . . . . 111

5.6.1 Competition For the Market and Competition In the Market . . . . 112

5.6.2 ChileCompra’s Framework Agreements . . . . . . . . . . . . . . . . . 113

5.6.3 Analytical Evaluation of ChileCompra-Style FAs in Simple Model . 115

5.6.4 Robustness Results: Numerical Experiments . . . . . . . . . . . . . 123

5.7 Conclusions and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

III Bibliography 127

Bibliography 128

IV Appendices 137

A House Allocation with Indifferences 138

B The Size of the Core in Assignment Markets 142

B.1 Results on point processes in the unit hypercube . . . . . . . . . . . . . . . 142

B.2 Proof of Theorem 9 upper bound . . . . . . . . . . . . . . . . . . . . . . . . 149

B.3 Theorem 9 lower bound: Proof of Proposition 4 . . . . . . . . . . . . . . . . 156

B.4 Proof of Theorem 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

C Procurement Mech. for Differentiated Products 170

C.1 Proof of Proposition 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

C.2 Hotelling GAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

C.3 Optimal mechanisms for Vertical Demand Model . . . . . . . . . . . . . . . 174

C.4 Extensions to our model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

C.4.1 Extension to multiple products per agents . . . . . . . . . . . . . . . 176

iii

C.4.2 Demand Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

C.5 Proof of Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

C.5.1 The coefficient matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 181

C.5.2 Definitions and notation . . . . . . . . . . . . . . . . . . . . . . . . . 185

C.5.3 Auxiliary Lemmas and Properties . . . . . . . . . . . . . . . . . . . 187

C.5.4 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

C.6 Supplement to Section 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

C.7 Proofs of Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

iv

List of Figures

2.1 Algorithm 1 might fail to terminate. . . . . . . . . . . . . . . . . . . . . . . 20

2.2 The trading algorithms may not find some efficient and weak-core allocations. 22

2.3 A selection rule that is not strategy-proof . . . . . . . . . . . . . . . . . . . 43

2.4 Example illustrating the steps of the mechanism induced by the Highest

Priority Object rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.5 Examples illustrating that the Highest Priority Object rule is different from

the TCRP and TTAS rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1 Expected total costs as a function of the transportation cost for optimal,

ChileCompra and BRE mechanisms . . . . . . . . . . . . . . . . . . . . . . 121

5.2 When it is profitable to restrict the entry as a function of the differentiation

cost and fL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

A.1 Example showing that “Common ordering on agents, individual ordering on

agents” is not strategy-proof. . . . . . . . . . . . . . . . . . . . . . . . . . . 141

C.1 When it is profitable to restrict the entry using a FPA as a function of fL

and δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

v

List of Tables

5.1 Optimality gaps as a function of both the differentiation cost δ and fL. . . 117

C.1 Comparison betweent te optimal mechanism and ChileCompra mechanism

with reserve price θH . In all cases, the expected price for an item of cost θH

is θH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

vi

Acknowledgments

I would like to start by thanking my advisors, Prof. Jay Sethuraman and Prof. Gabriel

Weintraub, for the countless hours they both spent meeting with me, providing professional

and personal advice. It is impossible to quantify how much I have learned from them.

Chapter 4 in this thesis is a collaboration with Prof. Yash Kanoria; this chapter would

never have existed without his generosity, his guidance, and his extreme patience.

I am greatly indebted to Professors Awi Federgruen, Omar Besbes, Jacob Leshno, Nelson

Fraiman and Carri Chan for their contribution to my personal and professional development;

they have always had the right word of advice. I am also grateful to the DRO and doctoral

office staff, Clara, Winnie, Joyce, Cristina, Liz and Dan; they have been so helpful!

I am thankful to Prof. Nicolas Stier, for inspiring me to pursue an academic career; his

support during all these years has been invaluable. I would also like to thank Professors

Flavia Bonomo, Javier Marenco and Willy Duran; they taught me the value of patience

and persistence.

Among the things I cherish the most from these past five years pursuing my PhD are

all the good friends I have made. My special thanks go to Juan, Daniel, Nikhil, and Peter;

all the meals, trips, game-nights, chats, uncountable tea-breaks, and ridiculous activities

we shared were an essential part of my life in New York.

Finally, this thesis would have never been possible without Carlos. Many attempts to

prove the results in this dissertation were made at times when I had promised to walk the

dog (and didn’t), or had promised to cook (and didn’t). I owe him more than any words

can describe.

vii

To my parents

viii

CHAPTER 1. INTRODUCTION 1

Chapter 1

Introduction

Auctions and matching mechanisms have become an increasingly important tool to allocate

scarce resources among competing individuals or firms. Every day, millions of auctions

are run for a variety of purposes, ranging from selling valuable art or advertisement space

in websites to acquiring goods for government use. Every year matching mechanisms are

used to decide the public school assignments of thousands of incoming high school students,

who are competing to obtain a seat in their most preferred school. Though different in

appearance, these settings require understanding how to allocate scarce resources (ad space,

school seats) among competing participants (bidders, students) to achieve a certain goal,

such as maximizing revenue or social welfare. This dissertation addresses several such

questions arising in matching and auction markets.

Broadly speaking, two natural sets of questions arise when studying matching and auc-

tion markets. The first set is related to design of new mechanisms (or the analysis of

existing ones) to procure or allocate resources. An important challenge when addressing

these questions is how to account for operational constraints. For example, a publisher

may have limited ad space to sell, an education department must guarantee that payments

cannot be made in order to obtain a seat in a better school, or a government might want

to design procurement auctions that favor domestic producers or small businesses. As a

consequence, a rich set of strategic and tactical questions emerges: What is the optimal

mechanism to reach a certain goal in the presence of these constraints? How well do em-

pirically relevant mechanisms perform relative to the optimum? Several examples over the


last few decades, such as spectrum auctions in Europe and the National Resident Matching

Program in the US, have illustrated how different outcomes can arise in similar markets, as a

consequence of the mechanisms used. Which mechanism performs better generally depends

on the specific application context. Therefore, it becomes crucial to explicitly incorporate

operational constraints and application details into the models used to guide the design of

practical mechanisms. In Chapters 2 and 5 we study how to design new matching and

auction mechanisms for given application contexts. In Chapter 3, we analyze a well-known

mechanism from a computational point of view.

The other main set of questions seeks to improve our understanding of the outcomes

arising in decentralized markets, i.e., markets where there is no central-planner or clear-

inghouse who finds the outcome based on the reported preferences. Understanding market

equilibria in decentralized settings is crucial, as these equilibria play a prominent role in

market predictions; most of the literature, empirical and theoretical, assumes that the out-

come of markets coincides with one of their equilibria. Chapter 4 aims to understand the

set of equilibria that arise in a decentralized matching market.

This dissertation is divided in two parts: in the first part, we focus on matching markets;

the second part is devoted to auction markets. In the remainder of this chapter, we briefly

discuss the main contributions in each of the parts; more details can be found in the specific

chapters.

1.1 Matching Markets

In the traditional one-sided matching market (or resource allocation) problem, a central

planner must efficiently allocate (usually indivisible) objects to agents. Each agent has pri-

vate preferences over these objects, and submits a preference list or ranking to the planner.

Using the reported preferences, the planner then runs a mechanism to decide the final allo-

cation. In recent years, matching mechanisms have been used in a variety of non-standard

applied settings, such as the student assignment process in several US cities, and regional

kidney exchange programs. In such markets, where monetary transfers are not permitted,

the goal is to find a mechanism that satisfies some desirable properties.


For example, as preferences are private, it is usually desirable to find allocation mech-

anisms that are truthful (or strategyproof ), where agents do not have an incentive to mis-

represent their preferences. In addition, one would like the final allocation to satisfy Pareto

efficiency, i.e., no agent can obtain a better object without making at least one agent

worse off. Moreover, sometimes imposing some type of “fairness” in the way agents are

treated by the mechanism is desirable. However, satisfying all these properties at once is

rarely possible, as they usually conflict with each other. Hence, designing a mechanism

involves a series of trade-offs, which raise new theoretical questions. Chapters 2 and 3 aim

to explore the boundary between what can and cannot be achieved in terms of design and

implementability.

In Chapter 2, we consider the problem of allocating a number of indivisible objects to

a group of individuals (also called agents) when monetary compensations are not possible.

Each individual is endowed with at most object, has preferences over all objects and wishes

to be allocated exactly one object. Examples of this setting include the allocation of public

school seats to students, or of landing slots at airports to airlines, or of kidneys to patients,

or of time sharing slots at a vacation home among its owners [Roth et al., 2004; Sonmez

and Unver, 2011; Wang and Krishna, 2006; Papai, 2000].

When preferences are strict, the Top-Trading Cycles (TTC) mechanism invented by

Gale and introduced by Shapley and Scarf [Shapley and Scarf, 1974] is Pareto efficient,

strategy-proof, and finds a core allocation (i.e., no coalition of individuals can (weakly)

improve their current allocations). Furthermore, it is the only mechanism simultaneously

satisfying these properties. In the extensive literature on this problem since then, the TTC

mechanism has been characterized in multiple ways and has been used in a variety of applied

settings, establishing its central role within the class of all allocation mechanisms.

An important limitation of the original Shapley-Scarf model is that agents are assumed

to have strict preferences. In many applications, however, it is not realistic to rule out

indifferences in the agents’ preferences. For example, the agents may not have enough

information about all the objects, and so it is reasonable to expect that their limited knowl-

edge is only sufficient to place the objects in different indifference classes. It has already

been shown in the literature (e.g. in the school choice setting) that ignoring indifferences


could lead to significant losses in the overall social welfare. Therefore, we aim to under-

stand how the TTC mechanism can be extended to allow agents to report indifferences

between objects, while still maintaining some of the desired properties such as truthfulness

and efficiency.

Unfortuntaley, when indifferences are present, no simple adaptation of the TTC mecha-

nism is Pareto efficient and strategy-proof. Chapter 2 provides a foundation for extending

the TTC mechanism to the preference domain with indifferences. We unify and generalize

earlier results by describing a family of strategy-proof mechanisms that always find alloca-

tions in the weak-core (the set of allocations such that no coalition of agents can strictly

improve upon it) that are also Pareto efficient. As a by-product, we establish sufficient con-

ditions for a mechanism (within a broad class of mechanisms) to be strategy-proof. Finally,

we use these conditions to design computationally efficient mechanisms.

In Chapter 3, we study several problems associated with the efficient allocation of ob-

jects from a computational perspective. We start by studying the Random Priority (RP)

mechanism, which is a popular way to allocate objects to agents who have strict ordinal

preferences over the objects. In the RP mechanism, an ordering over the agents is selected

uniformly at random; the first agent is then allocated his most-preferred object, the second

agent is allocated his most-preferred object among the remaining ones, and so on. The

outcome of the mechanism is a bi-stochastic matrix in which entry (i, a) represents the

probability that agent i is given object a. We show that the problem of computing the RP

allocation matrix is #P-complete, and thus suspected to be computationally intractable.1

It is worth noting that there is a close relationship between the potential outcomes of

the RP mechanism and the (Pareto) efficient allocation of goods —every outcome of the RP

mechanism is efficient, and every efficient allocation can be obtained by the RP mechanism

under some ordering of agents. Therefore, we study two decision problems associated with

the efficient allocation of objects. First, we show that it is possible to decide in polynomial

time whether or not a given agent i receives a given object a with probability 1 in the RP

mechanism; this is equivalent to showing that agent i will get the same object (object a) in

1The formal definition of the NP and #P complexity classes as well as the discussion on computational

tractability is deferred to Chapter 3.


every efficient matching. Second, we show that it is computationally hard (NP-complete)

to decide if a given agent i receives a given object a with positive probability under the

RP mechanism —equivalently, if agent i is assigned object a in some efficient matching.

The second result has important implications. First, as a corollary of the NP-completeness

result, we establish that the RP allocation matrix is even hard to approximate. In addition,

the result establishes that computing efficient matchings with constraints, which naturally

arise in social choice applications where affirmative action is imposed, is computationally

intractable. This raises questions as to whether (Pareto) efficiency is an appropriate goal

in this setting. Finally, it allows us to show that whenever a market is assumed to be

efficient, deriving reported preferences from just observing the final allocation is in general

computationally hard. This is related to the more general question of identifying properties

of the preference orderings that can be inferred by simply observing the outcomes of the

market.

Chapter 4 focuses on a different kind of matching markets: two-sided matching markets

with transferable utility. Agents are divided in two sides (e.g., sellers-buyers, workers-firms).

Each agent can partner with at most one other agent from the opposite side, generating a

certain value. In this model, transfers (or payments) are allowed between pairs of agents who

form a match. Examples of such markets include labor, housing and marriage markets. As

opposed to the previous chapters, where a central-planner chooses a mechanism to match

agents and objects, here we assume a decentralized market. Therefore, the objective of

this chapter is to understand the size and structure of the set of competitive equilibria in

matching markets with transfers.

In two-sided matching markets with transfers, it has been shown that competitive equi-

libria agree with stable outcomes, i.e., an outcome in which there is no pair of agents who

would be happier with each other than with their current match. From this observation, we

know that equilibria exist but are seldom unique. Despite this fact, most theoretical and

empirical studies in matchings assume a nearly unique stable outcome in order to facilitate

predictions, but there is little theoretical understanding about when this occurs.

To that end, we consider the classical Shapley-Shubik-Becker model for two-sided as-

signment markets with agent types [Shapley and Shubik, 1971; Becker, 1973]. We use a


generative model for the value of a match (similar models have been used in empirical

studies), where each agent has a randomly drawn productivity with respect to each type

of agent on the other side. The value generated from a match between a pair of agents

is the sum of the two productivity terms, each of which depends on the identity of one

agent but only on the type (not the identity) of the other agent, and a third term driven by

the types of both agents. In this setting, we study how the size and structure of the core

(i.e., the set of stable outcomes) is determined by market characteristics. We prove that an

approximately unique stable outcome emerges when a constant number of types is assumed.

Specifically, let n be the number of agents and K be the number of types on the side of the

market with more types. We find, under reasonable assumptions, that the relative variation

in utility per agent over core outcomes is bounded as O∗(1/n1/K), where the star notation

indicates that polylogarithmic factors have been suppressed. As a corollary, we obtain that

the expected size of the core is bounded as O∗(1/n1/K). Further, we show the tightness of

the result in the worst case, by providing a family of instances for which the expected size

of the core is Ω∗(1/n1/K). Finally, under more restrictive assumptions, we are able to show

tighter bounds on the size of the core.

1.2 Auction Markets

The second part of the dissertation centers on auction markets. Chapter 5 focuses on the

challenges that arise in the design of framework agreements (FAs), a popular procurement

mechanism used by public procurement agencies all around the world. In particular, FAs

play a central role in the procurement strategy of the Chilean government, our collaborator

in this work: every year, US$2 billion worth of products and services ranging from food to

office supplies, dialysis services and medicines, are acquired through FAs.

In a FA, the procurement agency uses an auction mechanism to select a menu (assort-

ment of products with posted prices), from a set of potential suppliers offering differen-

tiated products. Then, heterogeneous public organizations (hospitals, schools, etc.) buy

their most preferred option from the menu according to their needs. In these agreements

the government typically requests products within a certain category, without specifying


brands and other characteristics. As a result, suppliers submit bids for imperfect substitute

products that cannot be directly compared. This motivates the main question: how should

bids for differentiated products be evaluated, taking into account the heterogeneous prefer-

ences of the public organizations? From a theoretical perspective, we want to understand

how optimal menus should be designed. Based on the theoretical insights, we then eval-

uate the agency current FAs’ design and propose practical modifications to improve their

performance.

When evaluating the bids for a FA, the agency must consider the trade-off between

offering a richer menu of products for the organizations, versus offering less variety, hoping

to engage the suppliers in a more aggressive price competition. We develop a mechanism

design approach to understand how menus should be decided and characterize the optimal

direct-revelation posted price mechanism. Typically, the optimal mechanism restricts the

entry of close-substitute products to the assortment by selecting only one supplier from

that set; this induces more price competition without damaging much variety. On the other

hand, if a product is perceived by consumers as “unique” (not easily substitutable by other

product), then such a product will typically be added to the optimal assortment even if it

does not have a competitive cost, so as to improve market coverage. The characterization

of the optimal mechanism allows us to formalize these ideas, by describing the optimal

menus in terms of suppliers’ costs, product characteristics, and substitution patterns. More

broadly, the theoretical results in this chapter also shed light on how optimal menus of

differentiated products should be constructed in the presence of strategic suppliers and

heterogeneous consumers, a problem that also arises in other contexts such as the design of

medical formularies and group buying.

In the second part of the chapter, we use our theoretical framework to study the type

of FAs currently used by our collaborator, the Chilean government. Using our theoretical

results, we show how the current implementation fails to generate sufficient price compe-

tition among suppliers. We propose a simple modification to the rule currently used to

decide which products to include in the assortment. Our suggested modification signifi-

cantly increases incentives to compete in prices, yielding menus that are similar to those in

the optimal mechanism. As a result, the overall performance of the mechanism is improved.

8

Part I

Matching Markets

CHAPTER 2. HOUSE ALLOCATION WITH INDIFFERENCES 9

Chapter 2

House Allocation with

Indifferences: A Generalization

and a Unified View

2.1 Introduction

We consider the problem of allocating a number of indivisible objects to a group of indi-

viduals (also called agents) when monetary compensations are not possible. Agents have

preferences over the objects, and wish to be allocated exactly one object. Moreover, each

individual is endowed with at most one object. This fundamental allocation problem arises

in many settings such as the allocation of public school seats to students, or of landing slots

at airports to airlines, or of kidneys to patients, or of time sharing slots at a vacation home

among its owners [Roth et al., 2004; Sonmez and Unver, 2011; Wang and Krishna, 2006;

Papai, 2000].

The classic paper of Shapley and Scarf [Shapley and Scarf, 1974] considers the special

case of this model when the individual preference orderings over the objects are strict, and

when each object is endowed to exactly one individual, and each individual is endowed with

exactly one object. They propose the Top Trading Cycles (TTC) algorithm1, attributed

1The TTC algorithm is described in detail in Section 2.2.2


to David Gale, for the re-allocation of objects in such markets, and show that the TTC

allocation is in the strict core: no coalition of individuals can (weakly) improve their allo-

cation using only their endowments. This implies, in particular, that the TTC allocation

has two very desirable properties: it is Pareto efficient, i.e., no agent can obtain a better

object without making at least one agent worse off, and individually rational, meaning that

every agent weakly prefers his allocation to his endowment. Shapley and Scarf also observe

that the TTC outcome can be viewed as a competitive equilibrium. Subsequent research

has established the central role of the TTC mechanism for this problem when preferences

are strict: There is a unique core allocation, and it is also the unique competitive equilib-

rium [Roth and Postlewaite, 1977]; the TTC mechanism is strategy-proof, that is, no agent

can do better by misreporting his preferences [Roth, 1982]; and the TTC mechanism is

group strategyproof, that is, no coalition of agents can do better by misreporting their pref-

erences [Bird, 1984]. Furthermore, it is the only mechanism satisfying individual rationality,

Pareto-efficiency and strategy-proofness on the strict preference domain [Ma, 1994].

An important limitation of the original Shapley-Scarf model is that agents are assumed

to have strict preferences. In many applications, however, it is not realistic to rule out

indifferences in the preferences of the agents. For example, the agents may not have enough

information about all the objects, and so it is reasonable to expect that their limited knowl-

edge is only sufficient to place the objects in different indifference classes. For instance,

when applying for university housing, students may care only about certain characteristics

of the houses (e.g., number of rooms and roommates), so all houses sharing these charac-

teristics may be in the same indifference class. Another class of examples are online trading

sites, which are popular for certain types of goods, such as video games and anime. A

widely known website among the gamer community is GameTZ, where users keep track of

their desired and available games in the site’s database, and GameTZ’s matching system

helps the users find mutually beneficial trades. In this application, users are allowed to

express indifferences in their “ranking” of their desired set of games. A final, and impor-

tant, motivation is the renewed interest in the TTC mechanism for allocating students to

schools. The New Orleans Recovery School District has recently adopted the TTC mech-

anism to assign more than 28,000 students to school seats, taking into account both the

http://gametz.com


preferences of students and the priorities of schools. In addition, the San Francisco School

Board approved in 2010 the use of a mechanism based on the TTC for the assignment of

school seats in their district. In these cases school priorities at schools are “coarse,” so

that many students have the same priority at a given school or set of schools, typically

based on criteria such as their residential address, whether or not the student has a sibling

attending the same school, etc. While the student preferences are required to be strict in all

school assignment mechanisms so far, one could envision mechanisms in which (families of)

students are allowed to express indifferences in preferences. Any mechanism that does not

directly account for the indifferences, such as the ones using a random tie-breaking rule to

create strict preferences may be extremely inefficient [Erdil and Ergin, 2008]. The extensive

literature on school choice documents inefficiencies caused by randomly breaking ties in

the priorities at schools [Abdulkadiroglu et al., 2009; Erdil and Ergin, 2006] when using the

deferred-acceptance mechanism because of the artificial stability constraints. While one can

show that similar inefficiencies do not arise when the TTC mechanism is used (as long as

students have strict preferences), it is useful to understand the efficiency cost of insisting on

a strict ranking of the schools by the students, and our work is a first step in this direction2.

The fundamental importance of the allocation problem with indifferences has motivated

a number of recent research papers, resulting in a better understanding of possibility and

impossibility results for such a model. Shapley and Scarf [Shapley and Scarf, 1974] observe

that the strict-core may be empty when indifferences are present. Therefore, the core

requirement is usually relaxed to finding allocations in the weak-core, which is the set

of allocations such that no coalition of agents can strictly improve upon it. Unfortunately,

membership in the weak-core does not guarantee Pareto efficiency. Moreover, many negative

results have been established on the full preference domain: Ehlers [Ehlers, 2002] shows that

no group strategyproof mechanism can be Pareto efficient when indifferences are permitted;

and Bogomolnaia, Deb, and Ehlers [Bogomolnaia et al., 2005] show that the only strategy-

2The idea of working with indifferences in student preferences is not new. Indeed, Pathak and Sethura-

man [Pathak and Sethuraman, 2014] introduce such a model as an approximation for the appeals process in

school admissions. In their model, agents are allowed to list a subset of schools that they strictly prefer to

their assigned school, but are not allowed to rank-order this subset of preferred schools.


proof mechanisms that are non-bossy3 and Pareto efficient are serial dictatorships. As a

consequence, the only strategyproof mechanisms that are individually rational and efficient

must be bossy mechanisms4. Contrast this to the strict preference domain, where the TTC

mechanism simultaneously achieves all these properties. Our main objective in this chapter

is to design “TTC-like” mechanisms that are strategyproof, and that guarantee individual

rationality and Pareto efficiency.

To the best of our knowledge, four recent papers deal with generalizations of the TTC

mechanism to the full preference domain. Simultaneously and independently, [Alcalde-Unzu

and Molis, 2011] and [Jaramillo and Manjunath, 2012] propose different generalizations of

the TTC algorithm, both of which are strategy-proof and find efficient allocations that are

also in the weak core. These mechanisms are discussed in more detail in Section 2.5. [Aziz

and de Keijzer, 2012] generalize the ideas of the previous mechanisms into a single family of

mechanisms. The show that every member of that family is Pareto-efficient and individually

rational, and provide an example to illustrate that strategy-proofness may fail to hold. Their

main contribution is to show that the family of mechanisms introduced by [Alcalde-Unzu

and Molis, 2011] has an exponential running time. Finally, in a recent paper that was

written at the same time as this work, [Plaxton, 2012] provides a different generalization of

the TTC mechanism with the same properties and proposes a O(n3) implementation.

This chapter unifies and generalizes these earlier results by describing a family of

strategy-proof mechanisms that always allocations in the weak-core that are also Pareto

efficient. The family of mechanisms we identify is a subfamily of the ones discussed by

[Aziz and de Keijzer, 2012], and includes the mechanisms of Jaramillo & Manjunath, and

Alcalde-Unzu & Molis. Our focus is on generalizing the TTC mechanism in the following

manner: We construct a graph in which each agent points to the owners of his most pre-

ferred objects and provide general conditions that ensure Pareto-efficiency and membership

3A mechanism is non-bossy if no agent can alter the allocation of another agent without changing his

own.

4In a recent paper, [Ehlers et al., 2011] imposes non-bossiness and relaxes Pareto efficiency, and finds

that TTC with a tie-breaking rule satisfies all these properties, and that the additional requirements of weak

Pareto efficiency and consistency, in a sense, characterizes that class of mechanisms.


in the weak core. The mechanisms in this family differ only in the rule used to select the

trading cycles that are implemented; thus each selection rule induces a mechanism. To

get insights on which selection rules induce strategy-proof mechanisms, we impose two ad-

ditional requirements and show that a property, named local invariance, characterizes all

strategy-proof mechanisms satisfying these requirements. One of our key contributions is

to derive sufficient conditions on the selection rules to guarantee local invariance (and thus

strategy-proofness) of the induced mechanisms. In addition, we present a family of selection

rules inducing strategy-proof mechanisms that run in polynomial-time. The mechanisms

in our family only solve improving cycles, meaning that in each cycle implemented by the

mechanism there is at least one agent who strictly prefers his new allocation to the old.

Finally, we show that a member of that family runs in O(n2 log n + n2γ) (where γ is the

maximum size of an equivalence class in any preference list), which is the fastest known

strategy-proof mechanism for this problem.

The rest of the chapter is organized as follows. We start with a formal definition of the

model in Section 2.2, which also includes some basic definitions, notation, and a very quick

overview of the TTC mechanism. In Section 2.3, we introduce our family of mechanisms

and show that every mechanism in the class finds an efficient allocation that is also in the

weak core. In Section 2.4, we discuss further conditions that the selection rules for trading

cycles must satisfy and show that, in this setting, proving strategy-proofness is equivalent

to showing a much simpler statement. We end Section 2.4 by providing sufficient conditions

on the rules for that statement to hold. In Section 2.5, we discuss the existing mechanisms

and show why they are members of our family of mechanisms. In addition, we present

a new class of selection rules and show that the mechanisms induced by these rules are

strategy-proof. We conclude with a brief discussion of some open problems in Section 2.6.

2.2 Preliminaries

2.2.1 Definitions and Notation

We consider a market consisting of a set of n agents N = 1, . . . , n. Each agent is initially

endowed with an object and has preferences, possibly non-strict, over all the objects, includ-


ing his endowment. We denote agent i’s endowment by ω(i) and his preferences by Pi. We

denote by ω = (ω(1), . . . , ω(n)) the vector of initial endowments and by P = (P1, . . . , Pn)

the preference profile for all agents. A problem is completely defined by the triple 〈N,ω, P 〉.For any two objects a and b we write a ≥Pi b (resp. a =Pi b) if agent i weakly prefers a

to b (resp. is indifferent between a and b). We write a >Pi b to indicate that agent i strictly

prefers object a to object b. An allocation is a re-distribution of the objects in which each

agent obtains exactly one object. For the allocation µ, let µ(i) be the object allocated to

agent i.

An allocation µ is Pareto-efficient if, for every allocation µ′ and every agent i, µ′(i) >Pi

µ(i) implies µ(j) >Pj µ′(j) for some j ∈ N ; that is, we cannot improve agent i’s allocation

without making someone worse-off. An allocation µ is in the weak core if it is not possible

for any subset S of agents to reallocate their endowments in such a way that every one of

them strictly prefers this new allocation to their allocation under µ. In particular, if S is a

singleton agent, this condition reduces to individual rationality. A mechanism is said to be

Pareto-efficient and in the weak-core if it always finds allocations that are Pareto efficient

and in the weak-core, respectively. A mechanism is strategy-proof if the allocation an agent

obtains when reporting his true preference ordering is weakly preferred to the allocation he

obtains by reporting any other preference ordering.

In the basic model we consider we assume that the number of agents is the same as

the number of objects, and that a bijective map between these sets specifies the initial

endowments. In many applications, however, there may be an unequal number of agents

and objects, and some of the objects may be the social endowment, instead of being endowed

to a single agent. While these models are distinct when preferences are required to be strict

(and indeed there is an extensive literature on such allocation problems, see [Abdulkadiroglu

and Sonmez, 2003; Papai, 2000]), these distinctions disappear when indifferences are allowed

in the preferences of the agents. This construction is standard, but we include it here for the

sake of completeness and because these hybrid models are important in some applications

(e.g., student housing). For each agent not endowed with an object, create a distinct dummy

object that is endowed to this agent; and each agent ranks the dummy objects strictly below

their true preference ordering over the real objects; similarly, for each object that is not


endowed to any agent, create a dummy agent who is the owner of this object, and who is

indifferent between all (both real and dummy) objects in the problem. It is easy to see that

Pareto efficiency, membership in the weak-core and strategy-proofness are preserved in this

transformation, so in the rest of the chapter we work with the basic model.

We define a few basic concepts from graph theory that we shall use heavily in the rest

of the chapter. A directed graph G = (V,E) consists of a set V of vertices (or nodes) and

a set E of ordered pairs of vertices called edges (or arcs). Given a graph G = (V,E), a

subgraph of G is a graph whose vertex set is a subset V ′ ⊆ V , and whose edges are a subset

of E restricted to this subset V ′. The subgraph of G that contains every edge of E with

endpoints in V ′ is the subgraph induced by V ′ and is denoted G[V ′]. A directed graph is

strongly connected if there is a path from each vertex in the graph to every other vertex.

The strongly connected components of a directed graph are its maximal strongly connected

subgraphs. A strongly connected component S is a sink if all the neighbors of each vertex

in S are also in S. It is easy to see that every directed graph has at least one sink: the

graph of its strongly connected components cannot have a directed cycle, so at least one

component has no outgoing edge, and must be a sink.

2.2.2 A brief overview of the TTC Algorithm

Shapley and Scarf [Shapley and Scarf, 1974] proposed the Top Trading Cycles (TTC) al-

gorithm (attributed to David Gale) for the re-allocation of objects when agents have strict

preferences. Agents perform trades according to the following rules, which are repeated

until no agent is left:

1. Construct a graph with one vertex per agent. Each agent points to the owner of his

top-ranked object among the remaining ones. As all vertices have out-degree 1, at

least one cycle must exist and no two cycles overlap. Select the cycles in this graph.

2. Permanently assign to each agent in a cycle the object owned by the agent he points

to. Remove all agents and objects involved in a cycle from the problem.

A straightforward approach to account for indifferences is to resolve the indifferences

using an arbitrary tie-breaking rule and then apply the TTC algorithm to the resulting


strict-preference instance. It is unfortunate, therefore, that this idea may lead to an allo-

cation that is not Pareto efficient. To illustrate, consider two agents, 1 and 2 endowed with

objects a and b respectively. Suppose agent 1 is indifferent between objects a and b, whereas

agent 2 strictly prefers a to b. To guarantee Pareto efficiency, the agents must trade their

endowments, but an arbitrary tie-breaking rule may rank a before b for agent 1, resulting in

the inefficient allocation where agents do not trade. This argues for a nuanced tie-breaking

rule that takes into account the reported preferences of the agents; but in such a case,

ensuring strategy-proofness is tricky, as agents may be able to manipulate the tie-breaking

rule (and hence the outcome) by changing their preferences.

The special case of strict preferences is exploited by the TTC algorithm in two (related)

ways: in any iteration of the TTC algorithm, each agent has a unique best object (because

of strict preferences), and so points to exactly one agent (the owner of the said best object);

furthermore, the cycles that are formed in this graph have no overlaps, as each agent

points to exactly one other agent. When indifferences are permitted, the natural extension

of the TTC algorithm is to let each agent point to all the owners of his most preferred

objects (among the remaining ones) and perform the trades as indicated by the cycles of

this graph. This is difficult because the cycles in the TTC graph may overlap, causing an

agent to be involved in multiple cycles. In the earlier example, agent 1 is a member of

two cycles: one involving only himself (a self-loop), and the other involving agents 1 and

2. Even if one uses some kind of a priority rule to resolve these cycles, ensuring Pareto

efficiency and strategy-proofness is likely to be non-trivial. Additionally, in the original

model, an agent can leave the problem as soon as he takes part in a trading cycle; in

the model with indifferences, however, it may be necessary for an agent to “stay” in the

problem to help execute trades that are Pareto-improving for others, but not for him. In the

earlier example with two agents, if agent 1 leaves because he has one of his best objects, the

resulting allocation will be inefficient. Thus, while we can think of a natural extension of the

TTC framework to accommodate indifferences in preferences, ensuring that the resulting

mechanism is Pareto-efficient, individually rational, and strategy-proof is non-trivial and

necessitates a more careful understanding of the interactions between these requirements.

The rest of the chapter is devoted to precisely this goal.


2.3 The Trading Algorithm

In this section, we present a family of mechanisms that finds allocations in the weak-

core satisfying Pareto-efficiency. The mechanisms in the family are iterative, and have a

structure very similar to the TTC family. Given any problem we construct the TTC graph

and identify when a subset of agents and objects can be removed safely from the problem

without violating any of the desired properties. The remaining agents and objects constitute

a reduced problem to which we apply the same idea. As mentioned earlier, extending the

TTC mechanism to the full-preference domain presents two main design challenges: the

departure condition for agents and objects, and the selection rule to decide which cycle(s)

to implement when trading cycles overlap.

We start with some useful terminology. Given a (reduced) problem, an agent is said

to be satisfied if he owns one of his most-preferred objects among the remaining ones and

unsatisfied otherwise. The TTC-graph associated with a problem is a graph containing one

vertex per agent and each agent points to the owners of his top-ranked objects among the

remaining ones. As each vertex has out-degree at least 1, there is at least one directed

cycle in the TTC graph; these cycles are also called “trading cycles” as they represent an

exchange of objects that is (weakly) Pareto improving for the agents involved in the cycle.

Given the TTC-graph associated with a (reduced) problem, a trading cycle is said to be

improving if it contains at least one unsatisfied agent, and non-improving otherwise. A

trading cycle is solved or implemented when we assign to each agent in the cycle the object

owned by the agent he points to. The TTC-graph is a directed graph and so has at least one

sink. A sink in the TTC-graph is terminal if every vertex in the sink has an edge pointing

to itself, i.e., if all the agents in the sink are satisfied.

Algorithm 1 describes the family of mechanisms, which follows the same basic setup

as in prior work. Every mechanism in this family consists of the same two phases: the

removal and update phase and the improvement phase. During the removal and update

phase, we address the departure problem by iteratively removing those agents (together

with the objects they own) who are satisfied and cannot be part of an improving cycle, and

updating the problem accordingly. Each time one of the outer loops is executed, i.e., both

the removal and update and improvement phases are executed, we say that a step of the


algorithm takes place. We refer to the executions of the loop in the removal and update

phase as iterations within a step.

During the improvement phase, we solve some top trading cycles that are specified by

the selection rule.

Algorithm 1 Trading Algorithms

Input: Agents’ non-strict preference lists, ownership list.

Output: A Pareto efficient and individually rational allocation. Strategy-proofness de-

pends on the choice of rule F .

Repeat until no agent is left:

1. (Removal and Update phase)

(a) Construct a TTC-graph G: there is a vertex per agent and each agent points

to the owners of his top choices.

(b) Repeat until G has no terminal sinks:

Analyze the strongly connected components of G. There must be at

least one sink component S. For every sink component S:

If S is a terminal sink (i.e., every agent in S owns one of his top

choices), permanently assign to each agent in S his own object

and remove all the agents and objects in S from the problem, as

well as from the preference lists of the remaining agents.

Update the graph G so that each agent points to the owners of his (new)

top-choices.

2. (Improvement phase) Apply the selection rule onG to obtain a set of disjoint trading

cycles. Solve the cycles.

In the rest of this chapter, we shall focus on selection rules satisfying the following

natural properties:


1. For each agent i the rule selects a unique vertex j to whom i will point. Vertex j

must own one of i’s most preferred objects among the remaining ones.

2. Algorithm 1 terminates.

The second condition needs to be explicitly enforced, since not every rule satisfying

the first condition guarantees termination.5 To illustrate, consider the example described

in Figure 2.1. The market consists of three agents (N = 1, 2, 3) endowed with objects

ω = (a, b, c). Agent 1 is indifferent between a, b and c, agent 2 is indifferent between a and

b and both are strictly preferred over c and agent 3 has object a as his unique top-choice.

Suppose that a common priority ordering over agents is given, where 1 has the highest

priority followed by 2 and finally by 3. Let the selection rule be: each agent points to the

highest priority agent different from himself that owns one of his top ranked objects. Then,

agents 1 and 2 will keep trading among themselves and the mechanism does not terminate.

2.3.1 Pareto efficiency, weak-core and generality of the Trading mecha-

nisms

Our first result is that every member of the family of mechanisms described in Algorithm 1

produces a Pareto efficient allocation.

Theorem 1. Whenever the selection rule leads to termination, the allocation given by

Algorithm 1 is Pareto efficient.

Proof. Consider the allocation given by the mechanism. Let T1 be the first step in which

a terminal sink is removed. Every agent leaving in step T1 will end up owning one of

his top-ranked objects. Let T2 > T1 be the first step after T1 in which a terminal sink

is removed. Any agent leaving in step T2 will end up with one of his top-ranked objects

among those remaining. If an agent leaving in step T2 strictly prefers another object to his

current allocation, that object must have left in step T1. Since all agents who left in step

5The reason termination condition needs to be enforced is that we are not requiring the selection rule

to select only improving cycles. Indeed, in both the works by Alcalde-Unzu and Molis and Jaramillo and

Manjunath, non-improving cycles can be selected. In Section 2.5 we limit our attention to rules which only

select improving cycles, in which case termination is guaranteed.


Original endowments:

ω(1) = a, ω(2) = b, ω(3) = c

Preference lists:

1 a,b, c2 a,b, c

3 a, . . .

Priority ordering:

1, 2, 3

(a) Endowments and preferences.

Objects within braces are in the

same indifference class. Indifference

classes are separated by a comma.

1(a)

2(b)

3(c)

(b) TTC-graph be-

fore the first iteration.

There are no terminal

sinks

1(a)

2(b)

3(c)

(c) Graph obtained

by applying the se-

lection rule. 1 and 2

will trade their ob-

jects.

1(b)

2(a)

3(c)

(d) TTC-graph ob-

tained in the second it-

eration. There are no

terminal sinks

Figure 2.1: Example illustrating how Algorithm 1 might fail to terminate. Consider the

selection rule: each agent points to the highest priority agent different from himself that

owns one of his top ranked objects. In the first iteration, 1 and 2 will trade their objects.

The reader can verify that 1 and 2 will also trade in the second round, and the original

instance will be recovered. Then, agents 1 and 2 will keep trading among themselves and

the mechanism does not terminate.


T1 were in a terminal sink, none of them will be willing to trade his allocated object with

anyone who was not in their sink. Hence, no agent leaving in step T2 can do strictly better

without forcing another agent to do worse.

In general, any agent leaving in step Tk will obtain one of his top choices among the

remaining objects at that step. To strictly improve his allocation, he will need to obtain

an object allocated in a previous step. By the definition of step 1b. in Algorithm 1, this

cannot be achieved without someone leaving in a previous step being worse-off.

Next, we show that the family of mechanisms described in Algorithm 1 always finds a

weak-core allocation.

Theorem 2. Whenever the selection rule leads to termination, the allocation given by

Algorithm 1 is in the weak-core. In particular, the allocation is individually rational.

Proof. Let µ be the allocation given by Algorithm 1 and suppose µ is not in the weak-core.

Then, some subset of agents S can redistribute their original endowments among themselves

in such a way that every agent in S prefers his new allocation to that in µ. Without loss of

generality, let S = 1, . . . , k and assume that in the dominating reallocation, agent 1 gets

object ω(2), agent 2 gets ω(3), . . . , and agent k gets ω(1).

Since agent 1 strictly prefers ω(2) over µ(1), it follows that object ω(2) was not in the

problem the first time agent 1 pointed to µ(1) in the TTC-graph. Hence, it was not in the

problem when agent 1 left. This implies that agent 2 either traded ω(2) or left with that

object before agent 1 is able to leave. Following the same argument, object ω(3) must leave

before agent 2 trades or leaves and so on. Finally, object ω(1) must leave before agent k

trades or leaves, implying that agent 1 can only trade or leave the problem only after ω(1)

has left the problem, which is a contradiction.

Remark. One may suppose that the framework of Algorithm 1 is without loss of generality,

meaning that the framework is rich enough to find any Pareto efficient and weak-core

allocation. That such a result is not true is shown in the example of Figure 2.2. In fact,

the allocation shown in the example cannot be obtained by any strategyproof mechanism:

if agent 2 reports the ranking a, b, c, the only Pareto efficient and individually rational


allocation is the one in which agents 1 and 2 swap their endowments, and 2 strictly prefers

this allocation to obtaining object c.


ω(1) = a, ω(2) = b,

ω(3) = c.

Preference Lists:

1 a,b, c

2 a, c, b

3 b, . . .

(a) Endowments and preferences. Objects

within braces are in the same indifference

class. Indifference classes are separated by a

comma. One can verify that, in this setting,

allocation µ = (a, c, b) is efficient and in the

weak-core.

1(a) 2(b) 3(c)

(b) The allocation µ cannot be implemented

as the TTC-graph is missing an edge from 2

to 3. The edge will only appear once agent 1

departs with a, but in that case we need to

find a different departure condition.

Figure 2.2: The trading algorithms may not find some efficient and weak-core allocations.

2.3.2 Computational complexity

We now analyze the computational complexity of our family of mechanisms. We may assume

that individual preferences are stored as a list of sets representing the indifference classes,

so the graph is readily available. We have an ownership vector indexed by objects, which

keeps track of who is the current owner of each object. We also have a vector indexed by

agents to record the final allocation. We assume that both vectors can be accessed in O(1)

time.

Let the TTC-graph be G = (V,E). The strongly connected components of a graph can

be found in O(|V |+|E|) using Tarjan’s algorithm [Tarjan, 1972]. Identifying all the strongly

connected components that are sinks can be done in O(|V |+|E|), as it is a basic reachability

problem. Detecting whether the sinks are terminal sinks can be done in O(|V |+ |E|), as we


just need to check if all the agents in a sink are satisfied. Before removing an object from

the problem, we update the solution and the ownership record. Therefore, each iteration

in the removal and update phase takes O(|V |+ |E|) time, without counting the removal of

objects.

Removing each object from the preference lists can be done in O(|E|), since there is

no need to iterate through all the preference lists. We can delete the objects leaving the

problem from the individual preference lists when they are among the top choices of some

agent. Every time an agent reveals a new indifference class, it can be updated by deleting

the objects that are no longer in the problem. Then, we will have a total of O(|V |2) updates,

plus the O(|E|) in every iteration. Note that if the preferences are strict, all of the above

reduces to O(|V |) as in the original TTC mechanism.

We now bound the number of iterations of the removal and update phase during any

execution of the algorithm. In each step, there is at least one iteration of the removal

and update phase. Additional iterations only occur if a terminal sink is found. Note that

at most n terminal sinks can be found before the algorithm terminates. Therefore, the

algorithm performs at most (n + # improvement phase steps) executions of the removal

and update phase, each one consisting of O(|V |+ |E|) operations. Hence, the complexity of

the algorithm is basically determined by the number of improvement phase steps, and by

how fast we can implement the selection.

2.4 Strategy-proofness

Our goal is to identify a rich, strategyproof, subfamily of trading algorithms. To that end,

imagine an agent i who submits reports Pi and P ′i , when the rest of the agents report

a fixed profile P−i. The corresponding TTC graphs will differ only in the edges leaving

node i initially. To limit i’s influence, it is natural to require that the choices made by

the selection rule at the other nodes of the TTC graph be the same in both problems.

Given the inductive nature of the family of algorithms under consideration, we impose this

requirement until agent i is satisfied. To formalize this requirement—termed Independence

of Unsatisfied Agents—we let F (G) be the subgraph obtained from the TTC-graph G when


the selection rule F , or simply the F -rule, is applied.

Definition 1. (Independence of unsatisfied agents) The selection rule F satisfies

Independence of unsatisfied agents if for any unsatisfied agent i, and any two TTC graphs

G1 and G2 that differ only in the outgoing edges from i, F (G1) and F (G2) can differ only

in the outgoing edge from i.

Consider the example shown in Figure 2.3, and the following selection rule: Fix a priority

ordering of the agents. Each agent points to the highest priority agent (excluding himself)

who owns one of his most-preferred objects 6. It is easy to verify that this selection rule

satisfies the IUA condition. On the given instance, the first trading cycle is between agents

1 and 3, who exchange their endowments; after this step, agent 3 leaves the problem with

object a; the next trading cycle involves agents 1, 4, and 5, after which agent 5 leaves with

object d. Thus, agent 2 does not obtain either one of his best two objects. However, agent

2 can assure himself of object d by simply reporting it as his top choice (in that case, agents

2 and 4 will form a trading cycle in the first step). Thus, the given selection rule does not

induce a strategy-proof mechanism.

To get more insight into why strategy-proofness is violated by the selection rule just

discussed, it is helpful to re-examine the example. Under the original report, agent 4

wished to trade with agents 2 and 3, and the selection rule picked the arc (4, 2), indicating

that agent 4 wanted to trade with agent 2 initially. However, this arc is not chosen in

the subsequent step, after agents 1 and 3 trade their endowments—agent 4’s best objects

are now held by agents 2 and agent 1, and agent 1 has a higher priority than agent 2,

so the selection rule chose the arc (4, 1) instead. To summarize, agent 2 initially had a

prospective trading partner in agent 4, but this changed even though neither agent 2 nor

agent 4 were involved in a trading cycle. Our next definition formalizes this intuitive idea

into a requirement on the selection rule. Each vertex has out-degree exactly one in F (G),

and so each connected component of F (G) consists of a directed cycle and directed paths

ending in a vertex in that cycle.

6This rule may actually not terminate on some instances. It is possible to overcome this by adding some

additional conditions to ensure termination, but this has no bearing on our main point.


Definition 2. (Persistence) Let P = (a1, . . . , ak) be a path in F (G). We say that path

P is F-persistent or justpersistent if P appears in all the successive steps of the algorithm

until agent ak trades his object or leaves the problem.

The importance of the For any TTC graph G and any unsatisfied agent i, consider

the set of agents that are in a path to i in F (G) and let o be agent i’s most-preferred

object among those owned by the agents in that set. Any selection rule satisfying the IUA

condition needs to guarantee i an object that is at least as good as o, for i could report o as

his top-choice at that point, creating a trading cycle under the F -rule in which he obtains

object o 7.

Selection rules satisfying IUA and persistence can nevertheless violate strategy-proofness.

An example of such a rule is included in Appendix A.

2.4.1 An alternative characterization of strategy-proofness

Throughout this section we fix agent i and assume that i is the only agent misreporting

his preference ordering. Let ∆ = 〈N,ω, P 〉 be the problem with the true preferences

P = (P−i, Pi) and ∆′ = 〈N,ω, P ′〉 be the problem with preferences P ′ = (P−i, P ′i ), which

only differs from ∆ in the preferences reported by agent i. Given a (reduced) problem

∆, let G∆ be the TTC-graph of ∆, i.e., a graph such that every agent in ∆ points to his

top-ranked objects among the remaining ones. Denote by G∆m the graph obtained at the

end of step m (or, equivalently, at the beginning of step m + 1) when we start with the

(reduced) problem ∆. Given a graph G∆, let F (G∆) be the subgraph of G obtained when

the F -rule is applied. Finally, we denote by µ the final allocation obtained by the algorithm

when preferences P and endowments ω are used. We start with two claims that will lead

us to the final result.

Claim 1. Let T be the first iteration (if any) in which agent i trades his original endowment,

and let α be the object he obtains after the trade. Then, µ(i) =Pi α.

7The importance of “persistence” is also noted by Jaramillo and Manjunath. Indeed, they explicitly

enforce it in their rule.


Proof. When agent i is assigned a new object, it must be one of i’s top ranked objects

among those remaining in step T . If agent i never trades the object again, the claim is

immediate. If agent i trades it in step T ′ > T , it must be for one of his top ranked objects

among those remaining in step T ′. Since object α is still in the problem and the objects

remaining at step T ′ are a subset of those remaining in step T , it follows that i must be

indifferent between his top-ranked objects in iteration T ′ and α. Then, α =Pi β, where β

is the new object assigned to i. Claim 1 follows by induction.

Claim 2. Let t (resp. t′) be the first step in which agent i is involved in a trading cycle or

is satisfied in ∆ (resp. ∆′). Then, for every step m′, 0 ≤ m′ < m = min(t, t′), the same

terminal sinks and trading cycles occur in both problems.

Furthermore, at the end of each step m′ < m, the two problems ∆ and ∆′ are identical

except possibly for agent i. In particular, the same set of objects and agents remain, and

each agent is endowed with the same object in the two problems, and the graphs G∆m′ and

G∆′m′ are identical, except possibly for the outgoing edges from i.

Proof. The claim is vacuously true in step 0, so the claim is verified for m = 0 and for

m = 1. Suppose m > 1 and suppose the claim is true for steps 0, 1, . . . ,m′ − 1. Consider

step m′. As m′ < m, none of the terminal sinks (if any) found in the first iteration of the

removal and update phase in ∆ contains i. As all the vertices other than i are pointing to

the same vertices in both problems, it follows that every terminal sink in the first iteration

of step m′ in ∆ will also be a terminal sink in ∆′. Since all agents are endowed with the

same objects in both problems (by the inductive hypothesis), the TTC-graphs obtained

after the removal of those sinks will only differ in the outgoing edges from i. As agent i

remains unsatisfied throughout this step, the same argument applies for every iteration of

the removal and update phase during this step. Thus, at the end of the removal and update

phase, we obtain the same reduced problem in both cases, except possibly for the outgoing

edges from i.

Next, we analyze the trading cycles in the improvement phase. Note that agent i remains

unsatisfied in both problems until step m. Then, by the independence of unsatisfied agents

condition, F (G∆m′) and F (G∆′

m′) differ only in the outgoing edge from i. Since i is not in a


trading cycle in this step in either problem, and all the other agents point to the same object,

the same trading cycles occur. (We emphasize that without the independence of unsatisfied

agents condition we cannot guarantee that the same trading cycles will be formed.) The

claim follows.

We now introduce a property—called local invariance—for the family of mechanisms

described in Algorithm 1. A mechanism M satisfies local invariance if and only if the

following property holds for all P and P ′:

Property 1. (Local Invariance) Let P = (P−i, Pi) be the preference lists of the agents,

where Pi = (p1, . . . , pr) and pj represents the set of objects corresponding to agent i’s jth

indifference class. Suppose that agent i obtains object α ∈ pk (α is in agent i’s kth indif-

ference class) when mechanism M is applied to the preference profile P , and suppose and

α >Pi ω(i). Let P ′ = (P−i, P ′i ), where P ′i = (p1, . . . , pk−1, α, pk\α, pk+1, . . . , pr). Then,

when mechanism M is applied to P ′, agent i still obtains α.

This property of “local invariance” is a key intermediate step in the proofs of strategy-

proofness in both [Alcalde-Unzu and Molis, 2011] and [Jaramillo and Manjunath, 2012],

even though their rules are quite different. That this is not a coincidence is shown in

the following theorem, which asserts that this local invariance property is equivalent to

strategy-proofness in our setting.

Theorem 3. A mechanism satisfying the “Independence of Unsatisfied Agents” and the

“Persistence” properties is strategy proof if and only if it satisfies local invariance.

Proof. It is clear that every strategyproof mechanism must satisfy local invariance: for

otherwise, agent i can manipulate the mechanism by reporting Pi when his true preference

ordering is P ′i , where P and P ′ are the profiles at which local invariance is violated.

To show the converse, suppose that Property 1 holds, but the mechanism is not strategy-

proof. Then, there is some agent i, a pair of preferences Pi and P ′i , a pair of objects α and

α′, and a profile of preferences for the other agents P−i such that i is assigned α at (Pi, P−i);

i is assigned α′ at (P ′i , P−i); and α′ >Pi α. By Property 1, we may assume that α′ is the

only object in its indifference class for P ′i . Let m = min(t, t′), where t (resp. t′) is the first


step in ∆ (resp. ∆′) in which i either trades or is satisfied. By Claim 2, up to the beginning

of step m both reduced problems are the same except for the outgoing edges from i. Now

we consider the following cases:

t ≤ t′: Since agent i has not traded in ∆′ before t, it follows that α′ needs to be in the

problem at the beginning of step t. Then, i has to be pointing to α′ in ∆ at step t,

and since α′ >Pi ω(i) and α′ >Pi α, agent i cannot trade or be satisfied in problem

∆.

t > t′: Given that agent i will not be satisfied before trading in ∆′, let C = (q0, . . . , qk, i) be

the F -trading cycle in which i takes part in step t′ in ∆′. By the persistence and the

IUA conditions, it follows that the path q0 → . . . → qk → i also appears in F (G∆t′ )

and will appear in every step until i trades or is satisfied. This implies that q0 will not

trade his current endowment α′ as long as i does not trade his endowment or leave

the problem. Hence, i will be satisfied before α′ leaves the problem and, by Claim 1,

α ≥Pi α′, which is a contradiction.

An immediate consequence of Theorem 3 is that strategyproofness of a mechanism in

our family can be established by verifying the inefficacy of the limited class of misreports

described in Property 1, instead of all possible misreports of preferences that an agent could

engage in.

2.4.2 Sufficient conditions for local invariance

Theorem 3 motivates us to find simple conditions on selection rules that satisfy local in-

variance (Property 1). As usual, let ∆ be the original problem with the original preferences

P = (P−i, Pi), and suppose i is assigned α in this problem, and α is in his kth indiffer-

ence class. Let ∆′ be the problem with preferences P ′ = (P−i, P ′i ), where α is the unique

best object in his kth indifference class. We would like to find sufficient conditions on the

selection rule that ensures the assignment of α to i.

Let t (resp. t′) be the first step in which agent i is satisfied in ∆ (resp. ∆′), and

let T = min(t, t′). By Claim 2, the state of the problems ∆ and ∆′ are identical at the


beginning of step T except for the outgoing edges from i. So, object α is still in the problem,

and i will be pointing to the owner of α in ∆′ (in ∆, however, there may be additional edges

leaving i in the TTC graph, and the selection rule may choose one of the other edges). If

T = t′, agent i is assigned α in ∆′ and we are done. Hence, we may assume T = t < t′.

We abuse notation and call the reduced problems at the beginning of step T as ∆ and ∆′

respectively. At this point, both problems have the same set of agents endowed with the

same objects and all agents but i have the same preference lists (this follows once more by

Claim 2).

Given an agent j and an object β, we denote by om(j) the object owned by agent j at

step m and by am(β) the agent who owns α at step m, all in problem ∆. Let A∆m (resp.

O∆m) be the set of agents (resp. objects) at step m in problem ∆; and let C∆

m(i) be the set

of vertices that have a directed path to i in F (G∆m). We define A∆′

m , O∆′m , and C∆′

m in a

similar way.

Proposition 1. Consider a selection rule F with the following property: at each step m < t′,

each agent j ∈ A∆′m \ C∆′

m (i) is also in A∆m and is endowed with the same object in the two

problems ∆ and ∆′. In addition, the same trading cycles involving agents outside of C∆′m (i)

are solved in both problems. Then F satisfy local invariance (Property 1).

Proof. Fix a selection rule F satisfies the conditions in the statement of the proposition.

We show that α cannot be in a terminal sink without agent i in problem ∆′. This implies

that i becomes a satisfied agent while α is still in the problem, and so i must obtain α in

∆′.

We show that for each m < t′, every terminal sink in step m in ∆′ is also a terminal

sink in step m in ∆; every trading cycle at step m in ∆′ also occurs at step m in ∆; and

all agents outside of C∆′m (i) end the step with the same endowments. Since every terminal

sink that occurs before t′ in ∆′ is also a terminal sink in ∆, none of them can contain α,

as the unique terminal that contains α in ∆ also contains i and i is not in a terminal sink

before time t′ in ∆′. The proof is by induction on the step number. Recall that step T is

the base case.

Start with problems ∆ and ∆′, which only differ in i’s reported preferences at step T .


As there is an edge from vertex i to the owner of α >Pi ω(i) in the TTC-graph in both

problems, i is an unsatisfied vertex. Hence, i cannot be in a terminal sink in either problem,

thus the set of terminal sinks at step T is the same in both ∆ and ∆′. The improvement

phase starts with the same agents endowed with the same objects, and all agents except

for i have the same outgoing edges in the TTC graphs in ∆ and ∆′. We now consider the

improvement phase of step T . By IUA, F (G∆T ) and F (G∆′

T ) only differ in the outgoing edge

from i. Therefore, every trading cycle in ∆′ is a trading cycle in ∆. This establishes the

basis for the induction argument.

Assume that the induction hypothesis holds for each step m′ < m, that is, every terminal

sink in step m′ in ∆′ is also a terminal sink in step m′ in ∆, every trading cycle that occurs

in ∆′ at step m′ also occurs in ∆ in step m′, and all agents outside of C∆′m′ (i) are endowed

with the same objects in both problems at the end of step m′. We want to show that

this holds at step m, for m < t′. First, notice that every agent in a terminal sink is both

satisfied and outside of C∆′m (i), as i is not satisfied in ∆′ until step t′. Second, the persistence

property implies C∆′T (i) ⊆ C∆′

T+1(i) ⊆ . . . ⊆ C∆′m (i), and so agents involved in a terminal

sink were never part of C∆′m′ (i) for any m′ < m. Furthermore, since the terminal sinks at

step m were not sinks at step m − 1, it must be that at least one trade from the previous

iteration is necessary for that terminal sink to be formed. By the induction hypothesis, the

same trading cycles involving agents outside of C∆′m (i) occur in both problems. Therefore,

this terminal sink must occur at step m in problem ∆ as well. This holds in every iteration

of the removal and update phase in step m. As the TTC graphs for the two problems are

identical (except for the edges leaving i) after the removal and update phase, and as no

trading cycle involves i in ∆′, every trading cycle that occurs in ∆′ also occurs in ∆, and

the agents outside of C∆′m (i) are endowed with the same objects in both problems.

We can use Proposition 1 to describe selection rules that induce strategyproof mecha-

nisms.

Theorem 4 (Sufficient Conditions for Local Invariance). A selection rule F satisfies local

invariance (Property 1) if it satisfies any one of the following conditions:

(a) For any agent j ∈ A∆′m \ C∆′

m (i), if F selects the arc (j, k) in problem ∆ and if k is


endowed with the object β, then F selects the arc (j, k′) in problem ∆′ where k′ is the

owner of β in ∆′.

(b) For any agent j ∈ A∆′m \C∆′

m (i), if F selects the arc (j, k) in problem ∆, then F selects

the arc (j, k) in problem ∆′ as well.

(c) For each agent i, let R be the set of i’s most-preferred objects among the remaining

ones. Let o be the object selected by the F -rule for agent i, that is, i points to the

owner of o in F (G). The F -rule has the following property: for any arbitrary instance

such that S ⊆ R represents agent i’s top-choices and o ∈ S, the F -rule will also select

object o for agent i.

Proofs of (a) and (b) First, note that once an agent is part of C∆′m (i) (for m < t′) it will

continue to be so at least until time t′ by the persistence property. Thus, it suffices to show

that, for all m < t′, all agents outside C∆′m (i) are endowed with the same objects in both

problems. as local invariance will then follow from Proposition 1. We do this by induction.

For the base case of m = T , this is trivial (and is the same as in the proof of Proposition 1).

Suppose this holds for all m′ < m, and consider step m. By the induction hypothesis, at

m′ = m − 1 the agents that are not in C∆′m′ (i) always point to the same objects in both

problems and those objects are owned by the same owners, so all the same trades involving

them took place at m′ in both problems. By the end of iteration m′, the agents outside of

C∆′m′ (i) are endowed with the same objects. Since the removal and update phase does not

shift the endowments and C∆′m−1(i) ⊆ C∆′

m (i), the claim follows. ♦

Proof of (c) Using induction on the step number m, we show that O∆m ⊆ O∆′

m , every

object in O∆′m \O∆

m is owned by an agent in C∆′m (i) and all agents that are not in C∆′

m (i) are

endowed with the same objects in both problems at the end of step m. These statements

imply that all sinks and cycles in ∆′ will also appear in ∆ and the result follows from

Proposition 1.

For m = T , we can use the same arguments as in the proof of Proposition 1 to show

that the same terminal sinks are removed in the “removal and update” phase. Then, by

the end of step m, O∆m ⊆ O∆′

m holds. Furthermore, the improvement phase starts with the


same agents endowed with the same objects and all agents having the same outgoing edges

in the TTC-graph except for i. Then, by the IUA property, the same trading cycles occur

for agents outside C∆′m (i), which establishes the induction basis.

Suppose now that at each step m′ < m, O∆m ⊆ O∆′

m , every object in O∆′m \O∆

m is owned by

an agent in C∆′m (i) and all agents outside of C∆′

m (i) are endowed with the same objects in both

problems at the end of step m′. We shall show that this holds at the end of step m for m < t.

First, every agent in a terminal sink must be a satisfied agent that is not in C∆′m (i), as none

of those agents can be in a terminal sink without i. As C∆′T (i) ⊆ C∆′

T+1(i) ⊆ . . . ⊆ C∆′m (i) by

the persistence property, all the agents involved in a terminal sink in step m were not part

of C∆′m−1(i). Furthermore, since the terminal sinks at step m were not sinks at step m−1, at

least one trade from the previous iteration is necessary for that terminal sink to form. By

the induction hypothesis, the agents outside of C∆′m−1(i) are endowed with the same objects

in both problems in step m − 1 and O∆m−1 ⊆ O∆′

m−1, so the same trading cycles involving

only agents outside of C∆′m−1(i) are solved at step m − 1 in both problems. Hence, every

sink in ∆′ must also be in a sink in ∆ problem at step m and this holds for every iteration

of the removal and update phase in step m. Furthermore, sinks that are in ∆ and not in ∆′

can only include agents (and objects) in C∆′m−1(i). Therefore, O∆

m ⊆ O∆′m and every object

in O∆′m \O∆

m is owned by an agent in C∆′m (i).

During the improvement phase, every cycle in F (G∆′m ) involving only agents outside of

C∆m(i) also occurs in F (G∆

m). To see why, note that by the induction hypothesis, the agents

outside of C∆′m−1(i) are endowed with the same objects in both problems in step m − 1.

Since C∆′m−1(i) ⊆ C∆′

m (i), all agents outside of C∆′m (i) are endowed with the same objects in

the beginning of the improvement phase. Moreover, O∆m ⊆ O∆′

m , which by our hypothesis

implies that every agent not in C∆′m (i) pointing to another agent not in C∆′

m (i) in F (G∆′m )

must point to the same agent in F (G∆m). The result now follows by Proposition 1. ♦

2.5 Selection Rules: Old and New

We first describe the two known strategyproof mechanisms and explain why these mecha-

nisms are particular instances of the family of mechanisms considered here. Alcalde-Unzu


and Molis propose the The Top Trading Absorbing Sets (TTAS) mechanism, which works

as follows. Fix a strict priority ordering of the objects. At each step, examine the sinks of

the TTC graph, and remove any terminal sinks in this graph. Each agent in a non-terminal

sink points to (the owner of) one of his most-preferred objects, breaking ties in favor of the

objects that she has been endowed with the fewest ties, and breaking further ties using the

fixed strict priority ordering. In their mechanism, trading only takes place within a sink.

However, the rule behaves equivalently if applied to the whole TTC-graph at once: suppose

a cycle C = (q0, . . . , qk) is found that is not in a sink. Then, all the vertices that form that

cycle must be in a sink together for the first time and could not have traded their objects

earlier. At that time, qi+1 will still own the highest priority object among qi’s top-ranked

objects. Then, cycle C will be solved. This equivalent view is implicit in their proof of

strategyproofness. To verify that this mechanism belongs to our family of mechanisms:

persistence is satisfied by definition (as only those involved in sink end up trading); IUA is

satisfied because the choice of each agent is independent of the choices of the other agents,

so in particular, it is independent of the choices of the unsatisfied agents; strategyproofness

follows by observing that this rule satisfies condition (a) of Theorem 4. One deficiency of

this mechanism, however, is that it performs a number of “wasteful” trades in which none

of the agents in a trading cycle improves. In fact, Aziz and de Keijzer show that the TTAS

mechanism may require exponentially many iterations to terminate.

A second generalization of the TTC mechanism was given by Jaramillo and Manjunath.

Their mechanism, called the Top Cycles Rule with Priority (TCRP), uses the same depar-

ture condition used in Algorithm 1, and works as follows: Fix a strict priority ordering

of the agents; if agent i pointed to j in the previous iteration and j’s endowment did not

change, then i continues to point to j; otherwise each agent points to the agent on a shortest

path (based on the number of edges) to an unsatisfied agent, breaking ties using the initial

fixed priority ordering on the agents. This selection rule needs to be specified in a particular

order, starting from an agent for whom at least one of his most-preferred objects is held

by an unsatisfied agent. Indeed, their mechanism is a member of our family, as we will

establish later. They consider a unique strict priority ordering over all agents and use it

to create their pointing rule. In their rule, agent will point to a single other agent decided


as follows: Finally, anyone who cannot reach an unsatisfied agent points to the agent with

highest priority, other than himself, holding one of his most preferred objects. To verify that

this mechanism is in our family, observe that persistence is explicitly enforced, and IUA is

automatically satisfied because it only matters who satisfied agents point to. Jaramillo and

Manjunath show that this mechanism runs in O(n6) time: although this mechanism solves

non-improving cycles, one can show that at least one agent improves every n iterations.

2.5.1 Improving-cycles-only rules

As mentioned in Section 2.3.2, the computational complexity of any mechanism in the

class we consider depends on the complexity of implementing the selection rule, and on the

number of steps before termination. One way to ensure that the algorithm runs in O(n)

steps is by guaranteeing that the selection rule solves at least one improving cycle every

O(1) steps. A special class of these rules is one that guarantees that all the cycles solved

at each step are improving. Note that this is not accomplished by any of the existing rules.

We now present a new family of rules, the “Common ordering on agents, individual

ordering on objects” rules, with the property that each member induces a strategy-proof

mechanism in which only improving trading cycles are solved. The selection rule F is best

described by a “labeling” procedure: the labeled agents are those for whom the selection

rule has already determined the unique outgoing edge; and the unlabeled agents are those

for whom the selection rule has yet to make this choice. Every step starts with all agents

unlabeled; and the set of labeled agents gradually grows until all remaining agents are

labeled, at which point the selection rule F has made a choice for every remaining agent.

We shall describe a labeling procedure such that every cycle in F (G) is improving regardless

of the TTC graph G. The idea is to label all unsatisfied agents first, and then label the

unlabeled agents (which can only be satisfied agents) in such a way that every satisfied

agent is in a path to an unsatisfied agent in F (G)8. Therefore, we can guarantee that only

improving cycles are formed.

To decide which edges in the TTC-graph G are be selected to form F (G), the rule

8Clearly, each satisifed agent must be able to reach an unsatisfied agent in the TTC graph, as otherwise

this satisfied agent must be part of a terminal sink and could then be removed from the problem.


will use distinct ordering criteria over both agents and objects. In general, we say that

an ordering criterion over a set S is consistent if, for all sets S1, S2 such that S1, S2 ⊆ S,

the relative ordering of the agents in S1 ∩ S2 is identical to their ordering in S1 and S2.

The selection rule works with n + 1 orderings: first, there is a common ordering of all the

agents; then each of the n agents has their, possibly personalized, ordering over the objects.

The orderings are allowed to change during the course of the algorithm, as long as the

consistency property is maintained for the common ordering over agents.

The family of “Common ordering on agents, individual ordering on objects” rules is

formally described as follows:

Step 1:

(1.a) Each unsatisfied agent points to the owner of the highest priority object (according

to his own ordering) among all of his top-ranked objects. Label all unsatisfied

agents.

(1.b) Repeat until all satisfied agents are labeled:

Using the consistent ordering on the agents, select the highest priority agent

among all the unlabeled agents adjacent to a labeled agent; Make him point

to the owner of the highest priority object (according to his own ordering)

among labeled agents. Label him.

For each satisfied vertex v, we keep track of the first unsatisfied vertex reachable from

v in F (G) and we denote it by X(v). For each unsatisfied vertex v, we denote by X(v) the

vertex he points to in F (G). For step k, the rule is as follows:

Step k:

(k.a) Each agent v for which X(v) still holds the same object as in the previous step

points to the same agent as in the previous step. Label all such agents. All other

agents remain unlabeled.

(k.b) Each unsatisfied unlabeled agent points to the owner of the highest priority object

(according to his own ordering) among all of his top-ranked objects. Label all

unsatisfied agents.


(k.c) Repeat until all the remaining unlabeled agents are labeled:

Using a consistent ordering, select the highest priority agent among all the

unlabeled agents adjacent to a labeled vertex; Make him point to the owner of

the highest priority object (according to his own ordering) among all labeled

agents. Label him.

At the end of each step all satisfied agents are in a path to an unsatisfied vertex in F (G).

Hence, every cycle formed is improving, ensuring termination in O(n) steps. In addition,

persistence is satisfied by construction. Thus, only the “Independence of unsatisfied agents”

property needs to be verified. To that end, note that once an agent is labeled the subsequent

choices are independent of whom he points to. As unsatisfied agents start as labeled, all

choices are independent of whom they point to. We prove strategy-proofness mechanism

by showing that it satisfies condition (a) of Theorem 4.

Theorem 5. The “Common ordering on agents, individual ordering on objects” rules sat-

isfy condition (a) of Theorem 4. Thus, each rule in this family induces a strategy-proof

mechanism.

Proof. Let A∆m (resp. O∆

m) be the set of remaining agents (resp. objects) in the reduced

problem obtained from ∆ at the beginning of the improvement phase at step m. In addition,

let L∆m (resp. UL∆

m) be the set that of vertices are labeled (resp. unlabeled) at the beginning

of the improvement phase at step m in ∆. As in Section 2.4.1, let C∆m(i) be the set of vertices

v such that there is a directed path from v to i in F (G∆m). Given an agent j and an object

α, we denote by om(j) the object owned by agent j at step m and by am(α) the agent

who owns α at step m. We show that “Common ordering on agents, individual ordering on

objects” rules induce strategy-proof mechanisms by showing that they satisfy condition (a)

of Theorem 4.

Let t (resp. t′) be the first step in which agent i trades or is satisfied in ∆ (resp. ∆′).

By induction on the step number m, we show that A∆m ⊆ A∆′

m , O∆m ⊆ O∆′

m , all the agents in

A∆′m \C∆′

m−1(i) remain in ∆ and are endowed with the same objects in both problems, and

condition (a) holds, i.e., each agent j ∈ A∆′m \C∆′

m (i) is in A∆m and if j points to agent k in

F (G∆m) and β = o∆

m(k), then j points to a∆′m (β) in F (G∆′

m ).


As in the proof of Theorem 4, we may assume that we start with the reduced problem

obtained at the beginning of step T = min(t, t′) = t and we denote these reduced problems

by ∆ and ∆′ respectively. At this point, both problems have the same set of agents endowed

with the same objects and all agents except i have the same preference lists. Since we start

our analysis at step T , we rename step T as step 1 before continuing with the proof. At

step 1, the same terminal sinks and cycles will be formed except for the one involving i

by the same arguments used in the proof of Theorem 4. This verifies the base case for the

induction.

Before proceeding to the inductive step, let us illustrate what happens in step 2. Con-

sider an arbitrary terminal sink of ∆′2. Clearly, that sink depended on one or more cycles

of the previous step to be solved and those cycles also appeared in ∆1. Furthermore, as the

set of agents and objects are the same in both problems and none of those vertices has a

path to an object in C∆′1 (i) in ∆′, no such path exists in ∆ either. Therefore, every sink in

∆′ also appears in ∆.Let S be a terminal sink in ∆ but not in ∆′ in step 2. Since S is not a

sink in ∆′, for every vertex in S there is a path to an unsatisfied vertex in ∆′. Furthermore,

all such unsatisfied vertices must be in C∆′1 (i), as otherwise that path will be in ∆ as well.

Hence, all the vertices in S will join C∆′2 (i) during this current step.

The above remarks hold for every iteration of the removal and update phase at this

current step, so we conclude that A∆2 ⊆ A∆′

2 , O∆2 ⊆ O∆′

2 and all the agents in A∆′2 \C∆′

1 (i)

remain in ∆ and are endowed with the same objects in both problems. Given that the

cycles solved during the first step are the same in both cases (except for the one involving

i) and A∆2 ⊆ A∆′

2 , we have L∆2 ⊆ L∆′

2 . Furthermore, L∆′2 \L∆

2 ⊆ C∆′1 (i). In addition, as the

cycles solved during the first step are the same in both cases (except for the one involving

i), every vertex in UL∆′2 \UL∆

2 must be a vertex that is no longer in ∆ (that is, a vertex

in A∆′2 \A∆

2 ), and every unlabeled vertex in UL∆2 \UL∆′

2 must be in C∆′1 (i) and it must be

endowed in ∆ with one of the objects owned by an agent in C∆′1 (i). Finally, every agent

that is in UL∆2 ∩ UL∆′

2 cannot be in C∆′1 (i) and thus is endowed with the same object in

both problems.

Keeping the above properties in mind, let C be a cycle found in ∆′ during the im-

provement phase in step 2. We show that must be C is solved in ∆ and well. Note that


C ∩ C∆′2 (i) = ∅ and C∆′

1 (i) ⊆ C∆′2 (i) . Every satisfied labeled in C must also be satisfied

and labeled in ∆ as the set of labeled vertices only differs in those in C∆′1 (i). Hence, labeled

vertices must point to the same object in both problems and that object is owned by the

same agent as no trading involving that agent occurred in the previous step.

Let v ∈ C be an unlabeled vertex. As argued before, all the vertices that are in a sink in

∆ but not in ∆′ join C∆′2 (i), so v could not have been in a sink in ∆. Then, it must still be

in both problems. Furthermore, since L∆2 ⊆ L∆′

2 , it follows that v must also be unlabeled in

∆. Therefore, v ∈ UL∆2 ∩UL∆′

2 . Finally, all unsatisfied agents in C point to the same object

in both problems, and those objects are owned by the same agents. This follows from the

fact that in both problems all agents in UL∆2 ∩ UL∆′

2 and in L∆2 ∩ L∆′

2 will hold the same

object in both problems by the inductive hypothesis. Since every unsatisfied agent points

to the owner of the highest priority object among his top-choices and O∆2 ⊆ O∆′

2 , the same

choices will be made by those agents in both problems.

From the above, it suffices to show that every vertex in UL∆2 ∩ UL∆′

2 who does not

point to someone in C∆′2 (i) will point to the same object in both problems. We shall show

that, as we grow the set of labeled vertices, at all times we have O(L∆2 ) ⊆ O(L∆′

2 ), i.e., the

objects owned by the set of labeled agents in ∆ is a subset of those owned by labeled agents

in ∆′. Clearly, this holds at the beginning of step 2 since L∆2 ⊆ L∆′

2 and only unlabeled

agents may have changed their endowments during the previous step. Furthermore, all the

unsatisfied agents in ∆ hold their initial endowment in both problems.

Consider now the set of agents in UL∆2 ordered by their priority, and suppose we start

the improvement phase. Let u be the highest priority agent among those in UL∆2 . If

u ∈ UL∆2 \UL∆′

2 , he must be in C∆′1 (i) ⊆ C∆′

2 (i), so we do not care who he points to.

However, since he is endowed with an object owned by someone in C∆′1 (i), the property

O(L∆2 ) ⊆ O(L∆′

2 ) is maintained. If u ∈ UL∆2 ∩ UL∆′

2 and he does not point to one of the

vertices in C∆′2 (i) in the problem ∆′, it means that his top-choice among the remaining

ones in ∆′ is held by an agent that is not in that component. Let that agent be a. Clearly,

a must be labeled as well in ∆ as L∆′2 \L∆

2 ⊆ C∆′1 (i) . Furthermore, since a /∈ C∆′

1 (i),

he must hold the same objects in both problem by inductive hypothesis. Then, u will

end up pointing to a in both problems. In addition, since u /∈ C∆′1 (i), he must hold the


same object in both problems. Therefore, O(L∆2 ) ⊆ O(L∆′

2 ) is maintained. We can reason

inductively to show that, each time a vertex u is labeled in ∆, either u ∈ UL∆2 \UL∆′

2 so

the object he owns is already owned by a labeled agent in ∆′, or u ∈ UL∆2 ∩ UL∆′

2 and it

is endowed with the same object in both problems by the inductive hypothesis. Since the

vertices in u ∈ UL∆2 ∩ UL∆′

2 are added in the same relative order in both problems by the

consistency property, we conclude that O(L∆2 ) ⊆ O(L∆′

2 ) is maintained at all times. Hence,

if u ∈ UL∆2 ∩UL∆′

2 points to an agent a /∈ C∆′1 (i) in ∆′, a is endowed with the same object

in both problems by the inductive hypothesis and since the property O(L∆2 ) ⊆ O(L∆′

2 ) is

maintained at all times, a will also own the highest priority object of u in ∆. Therefore, u

points to a in ∆ and the property holds.

We can continue the proof by induction in the step number using the same arguments

used for step 2 to show that the following holds at each step m < t′: A∆m ⊆ A∆′

m , O∆m ⊆ O∆′

m ,

all agents that are not in C∆′m−1(i) are endowed with the same objects in both problems,

L∆m ⊆ L∆′

m and they only differ in the vertices in C∆′m (i). In addition, every vertex in UL∆′

m

that is not in UL∆m must be a vertex that is no longer in ∆, every unlabeled vertex in UL∆

m

that is not in UL∆′m must be in C∆′

m−1(i) and it must be endowed with one of those objects

in ∆. Finally, every agent that is in UL∆m ∩ UL∆′

m must be endowed with the same object

in both problems.

2.5.1.1 Highest Priority Object Rule

We now focus on the Highest Priority Object (HPO) rule, which is a member of the “Com-

mon ordering on agents, individual ordering on agents” family and therefore induces a

strategy-proof mechanism. We will show the mechanism induced by this rule can be imple-

mented in O(n2 log(n) + n2γ), where γ is the maximum size of an equivalence class in any

preference list.

To properly define this rule, we need to specify the orderings that are used. Fix a

common priority ordering over the objects. Each agent uses this as his “individual” ordering

over the objects. Furthermore, the common priority ordering over the agents is the one

induced by this common ordering over the objects (agent i has higher priority than i′ if the

object i owns has a higher priority than the object i′ owns). Note that this ordering may


vary between steps, as the “endowment” changes. However, it can be seen that the HPO

rule is consistent: the inductive arguments in the proof of Theorem 5 show that, at each

step m, all the unlabeled agents in UL∆m ∩UL∆′

m are endowed with the same object in both

problems, thus inducing the same relative order. Hence, the HPO selection rule induces a

strategy-proof mechanism.

Recall from the description of the “Common ordering on agents, individual ordering on

agents” family that, at each step, only the agents for whom persistence needs to be enforced

start out as labeled and all the others start as unlabeled. Keeping that in mind, the rule

can be summarized as follows:

Rule 1 (Highest Priority Object (HPO)). Fix a common priority ordering over the objects.

Every unsatisfied agent points to the owner of the highest priority object among his top-

choices. Label all unsatisfied agents.

Let L be the set of labeled agents, and let AL be the set of all agents adjacent to a labeled

agent. At each step, select the agent in AL who owns the highest priority object among all

those in AL and make that agent point to the owner of the highest priority object among his

top-choices that are owned by an agent in L. Add this agent to L and all of its neighbors

that are not in L to AL.

A 6-agent example showing how the HPO rule works is in Figure 2.4. Figures 4.a and

4.b show the preferences, endowments, and the common ordering of the objects. The TTC

graph obtained in the first step (G0) is in Figure 4.c. There are no terminal sinks, so the

improvement phase starts right away: The unsatisfied agents (agents 4, 5 and 6) point

to the owner of the highest priority object among his most-preferred objects (3, 6 and 2

respectively). All unsatisfied agents are labeled immediately after and the others remain

unlabeled. The only two agents adjacent to a labeled vertex are 2 and 3. As agent 2 owns

the highest priority object, he points to the labeled agent that owns the highest priority

object among his top-choices. In this case 2 points to 4. Following the same reasoning,

agent 3 points to agent 5 and agent 1 to agent 3. We therefore obtain the graph F (G0)

as shown in Figure 4.d. This graph contains a unique trading cycle formed by agents 1, 3

and 5. After this trade is implemented, the TTC graph G1 is obtained. Since there are no


terminal sinks in G1, we move forward to the improvement phase. According to the rule, the

two unsatisfied vertices (4 and 6) and the one corresponding to agent 2 start as labeled (as

agent 2 point to 4, who still holds the same object as in the previous iteration). Agent 5 is

the only unlabeled agent adjacent to a labeled one, so he points to 6. By following the rule,

agent 1 points to 5 and agent 3 points to 1 to complete F (G1). Note that a trading cycle

involving all agents but 3 is formed, so at the end of this step all the agents are satisfied.

We next note that the HPO rule is different from both previously known rules, as shown

in the examples provided in Figure 2.5.

Implementation. Finally, we discuss the implementation of the mechanism induced by

the HPO rule. Recall that the TCRP mechanism can be implemented in O(n6) and the

Top Trading Absorbing Sets mechanism has been shown to run in exponential time in the

worst case. We show that the mechanism induced by the Highest Priority Object rule can

be implemented in O(n2 log(n) +n2γ), where γ is the maximum size of an equivalence class

in any preference list, which is considerable improvement over the existing mechanisms.

As usual, let G = (V,E) be the TTC-graph at a given iteration. We will maintain

a set AL which will store the all vertices currently pointing to a labeled vertex. Using a

Fibonacci heap to implement AL, we can add vertices in O(1), obtain the next vertex that

must be labeled in O(1) and delete in O(log(|V |)). We can keep track of the sets L and

AL in two arrays, so updating those will take O(|V |) time (throughout all the steps) and

checking whether a vertex is in them can be done in O(1). Hence, deciding whether a vertex

should be added to AL can be done in O(1).

Every time a vertex v is labeled, we need to update AL by checking if each of the vertices

with an incoming edge to v need to be added. By storing the set of incoming edges to a

vertex (which can be done in O(|E|)), we check a total of O(|E|) times per step if a vertex

needs to be added to AL (each takes O(1) time). In addition, we perform O(|V |) insertions

and O(|V |) deletions from AL per step, each of them requiring O(1) and O(log(|V |)) time

respectively. So far, the total number of operations per step are O(|V | log(|V |) + |E|).Selecting the outgoing edge for a vertex v can be done in O(# outgoing edges from v),

as we need to identify the (owner of the) highest priority labeled object. This can be done


for all vertices in O(|E|) and X(v) can be updated in O(1) for each vertex, giving a total

time of O(|V | + |E|). Finding and solving the cycles and updating the endowment lists

can be done in O(|V |). Then, the improvement phase at each step can be computed in

O(|V | log(|V |) + |E|) time.

In Section 2.3.2, we argued that at most O(n + # improving phase steps) iterations of

the removal and update phase take place, each one consisting of O(|V |+ |E|) operations. As

the number of steps is O(n), the total running time can be bounded by O(n2 log(n) +n2γ),

where γ is the maximum size of an equivalence class in any preference list.

2.6 Discussion

The TTC mechanism is the only mechanism satisfying individual rationality, Pareto-efficiency

and strategy-proofness on the strict preference domain. However, when indifferences are

permitted, several distinct mechanisms satisfying these properties exist, but a characteriza-

tion of all such mechanisms is still lacking, and would be interesting. In this work, we take

a step forward toward that goal by showing sufficient conditions on a family of mechanisms

that guarantee these properties, but we do not know if these conditions are necessary. Also,

it would be nice to find necessary and new sufficient conditions for Property 1 to hold.

In Section 2.4.2 we added additional conditions for the rules that we were willing to

consider, namely“Persistence” and “Independence of unsatisfied agents”. It would also be

interesting to explore the necessity of these conditions. As mentioned earlier, without the

“Independence of unsatisfied agents” condition being enforced proving that an abstract

selection rule induces a strategy proof mechanism seems to be difficult.

Finally, we note that while the “Common ordering on agents, individual ordering on

agents” rules and the rule described in the appendix corresponding to this chapter are very

similar (they only differ in the individual orderings used), the former induces strategy-proof

mechanism while the latter does not. It would be interesting to be able to formalize what

fails in the rule described in the appendix as an intermediate step towards finding necessary

conditions for a mechanism to be strategy proof.



ω(1) = a, ω(2) = b,

ω(3) = c, ω(4) = d

ω(5) = e.

Agent order:

1, 2, 3, 4, 5.

(a) Endowments and orders.

Agent 1 has the highest priority

followed by agents 2, 3, 4 and 5.

Preference Lists:

1 c, e, . . .

2 a, d , . . .

3 a, . . .

4 b, c, . . .

5 d, . . .

(b) Preferences.

1(a)

2(b)

3(c)

4(d) 5(e)

(c) G0. There are no terminal sinks in

this graph.

1(a)

2(b)

3(c)

4(d) 5(e)

(d) F (G0). This graph has a

unique cycle 1 → 3 → 1, so

agents 1 and 3 trade their ob-

jects.

1(c)

2(b)

3(a)

4(d) 5(e)

(e) G1. In the second step 3 is a

terminal sink, so agent 3 and ob-

ject a are removed from the prob-

lem. After that, agent 2 points to

agent 4, the owner of object a.

1(c)

2(b)

4(d) 5(e)

(f) F (G1). The trade between

agents 1, 4, 5 takes place and agent

5 leaves the problem with object

d. Hence, agent 2 cannot obtain

any of his first two choices.

Figure 2.3: A selection rule that is not strategy-proof



ω(1) = a, ω(2) = b,

ω(3) = c, ω(4) = d

ω(5) = e, ω(6) = f

Common object order:

a, b, c, d, e, f.


Preference Lists:

1 a, c2 a, b, d3 c, e4 c

5 a, f6 b

(b) Preferences.

1(a)

2(b) 3(c)

4(d) 5(e)6(f)

(c) G0.

1(a)

2(b) 3(c)

4(d) 5(e)6(f)

(d) F (G0).

1(c)

2(b) 3(e)

4(d) 5(a)6(f)

(e) G1.

1(c)

2(b) 3(e)

4(d) 5(a)6(f)

(f) F (G1).

Figure 2.4: Example illustrating the steps of the mechanism induced by the Highest Priority

Object rule. Figures 1.a and 1.b show the original edowments, the preference profile and the

common ordering over the objects. Figure 1.c shows the original TTC graph and Figure 1.d

shows the graph obtained in the improvement phase. Finally, Figure 1.e and 1.f show the

TTC and trading graphs obtained in the second step. Once the improvement phase in the

second step concludes, all agents are satisfied and own their final alocation.



ω = (a, b, c, d)

Common orders:

agents: 1, 2, 3, 4.

objects: a, b, c, d.


The priority order over agents

is used by the TCRP rule,

while the order over objects is

used by the HPO and TTAS

rules

Preference Lists:

1 a, c2 a, b, d3 b

4 b

(b) Using this preference profile,

one can verify that object b is allo-

cated to agent 3 (resp. 4) when us-

ing the mechanism induced by the

HPO rule (resp. TCRP rule)

Preference Lists:

1 a, b, c2 a, b, d3 a

4 a

(c) Using this preference profile,

one can verify that object a is

allocated to agent 3 (resp. 4)

when using the mechanism in-

duced by the HPO rule (resp.

TTAS rule).

Figure 2.5: Examples illustrating that the Highest Priority Object rule is different from the

TCRP and TTAS rules.

CHAPTER 3. COMPLEXITY OF COMPUTING THE RP MATRIX 46

Chapter 3

The Complexity of Computing the

Random Priority Allocation

Matrix

3.1 Introduction

We consider the problem of allocating n objects to n agents, with each agent interested

in consuming at most one unit across all objects. Agents have strict ordinal preferences

over the objects. Perhaps the most common allocation mechanism for this problem is

the priority mechanism (also called the serial dictatorship (SD) mechanism): in such a

mechanism, there is a fixed ordering of the agents and the agents are invited to choose

objects in that order. Thus, the agent who appears first in this ordering will pick his

most-preferred object; the one appearing second will pick his most-preferred object among

the ones that remain, etc. The priority mechanism is Pareto efficient, neutral (invariant

to relabeling of the objects), non-bossy (no agent can alter some other agent’s allocation

without altering his own), strategy-proof, even group strategy-proof, and easy to compute.

Its one major drawback, however, is that it fails anonymity—two agents with identical

preferences and identical claims on the objects are not treated equally by the mechanism

because one of them will appear before the other in the fixed ordering. A standard way


to overcome this in a moneyless market is to randomize the initial ordering of the agents,

yielding the Random Priority (RP) mechanism. In the RP mechanism, an ordering of the

agents is chosen uniformly at random (each of the n! orderings being equally likely), and

the priority mechanism applied to the chosen ordering determines the outcome. Of course,

finding the outcome is an easy task once the ordering is selected.

An alternative way to think about the RP mechanism is in terms of the probabilis-

tic allocation that the agents receive under this mechanism—this can be expressed as a

doubly-stochastic matrix X with xia representing (i) the probability that agent i receives

object a (if the objects are indivisible); or (ii) the fraction of object a allocated to agent

i (if the objects are divisible). The RP mechanism has been extensively analyzed in the

literature for the allocation of both divisible and indivisible objects [Cres and Moulin, 2001;

Satterthwaite and Sonnenschein, 1981], yet the computational complexity of finding the RP

allocation matrix X is not fully understood. Our main result is that determining X exactly

or even approximately is difficult in a sense that can be made precise using the theory of

computational complexity.

In computational complexity theory, a decision problem is a question with a yes or no

answer, depending on the values of some input parameters. As an example, the problem

“Given a preference profile P , does agent i get object a in the priority mechanism with

respect to some ordering σ of the agents?” is a decision problem. Indeed, we refer to this

problem as the SD Feasibility problem. Complexity theory is concerned with understand-

ing the computational resources needed to solve a problem, and to categorize problems into

various complexity classes depending on how “easy” or “difficult” it is to find a solution.

Two complexity classes play an important role in this chapter: the class NP, defined as the

set of decision problems having efficiently verifiable solutions and the class #P, introduced

by [Valiant, 1979] and defined as the counting version of the class NP. A decision problem

is in NP if every “yes” instance can be efficiently verified using a polynomial-size certificate.

As an example, consider the SD Feasibility problem stated earlier. If the problem has

n agents, and if indeed there is an ordering σ of the agents such that the SD mechanism

with respect to σ gives object a to agent i, this can be easily verified in polynomial-time by

running the SD mechanism with the ordering σ. The ordering σ serves as the “certificate”


in the definition of NP. Two additional notions are needed in what follows: A problem is

hard for a given complexity class if it is as difficult as the most difficult problems in the

class. When a problem is both a member of the class and hard for that class, it is said to

be complete for that class.

By definition, the problem of determining each entry xia of the RP allocation matrix

X is equivalent to that of counting the number of orderings under which i obtains a when

the SD mechanism is used. Therefore, this problem is in the class #P. We show that com-

puting the RP allocation is indeed #P-complete, and thus suspected to be very difficult.1

Independently of our work, Aziz et al. [Aziz et al., 2013] proved that computing the RP

allocation is #P-complete using a different reduction.

Even though the problem of finding the RP allocation matrix for a given instance is

not directly a decision problem, it can be solved as a sequence of decision problems of the

following form: “Given an integer K, a preference profile P , an agent i and an object a, are

there at least K orderings under which i is assigned a when the SD mechanism is used?”.

Our #P-completeness result already implies that this decision problem is NP-complete if

K is part of the input. Nevertheless, the decision problem might be easy to solve for

some fixed values of K. Towards that end, we study the computational complexity of the

decision problems associated with the two extreme values of K: deciding whether an agent

has probability exactly one of getting an object (K = n!), and deciding whether he has

probability greater than zero (K = 1). We provide a polynomial-time algorithm to solve

the former and show that the latter is NP-complete. We further show that the problem of

deciding whether an agent has a positive probability of obtaining an object is equivalent

to deciding whether there is a Pareto efficient matching in which a subset of objects must

be matched. Computing matchings with constraints has been a topic of interest to the

research community, as it naturally arises in social choice applications in which some type

of affirmative action is imposed [Abdulkadiroglu, 2005; Ehlers et al., 2011].

In spite of the prominence of the RP mechanism, there is surprisingly little prior work

1Toda’s theorem implies that, for any problem in the polynomial hierarchy, there is a deterministic

polynomial-time Turing reduction to a problem in #P, and therefore one call to a #P oracle suffices to solve

any problem in the polynomial hierarchy in deterministic polynomial time [Toda, 1989].


regarding its computational complexity. As already mentioned, Aziz et al. [Aziz et al., 2013]

independently establish the #P-completeness of computing the RP allocation matrix using a

different reduction. Their work, however, does not address the complexity of approximating

the RP allocation. As a corollary of our NP-completeness result, we establish that the RP

allocation matrix is even hard to approximate. Similarly, to the best of our knowledge,

only the work of Kavitha and Nasre [Kavitha and Nasre, 2009] considers the complexity of

finding constrained optimal matchings, but they use the notion of popularity to define an

optimal matching. Instead, we use the notion of Pareto efficiency, which is more standard

and widely used in the literature.

The rest of this chapter is organized as follows. In Section 3.2, we formally state the

problem and provide some useful definitions and notation. In Section 3.3, we show that

computing the RP allocation is #P-complete. In Section 3.4, we discuss the complexity of

two decision problems associated with the RP mechanism. We provide a polynomial-time

algorithm for deciding whether an agent has probability exactly one of getting an object.

In addition, we show that deciding whether an agent has probability greater than zero of

getting an object is NP-complete and discuss the implications of this result. We end with

some suggestions for further research in Section 3.5.

3.2 Preliminaries

An instance of the house allocation problem is a tuple I = (A,O, P ) consisting of a set of

agents A, a set of objects O and a preference profile P = (P1, . . . , P|A|), where each Pi is a

strict ordering of the set of objects O. The ordering Pi represents agent i’s preferences over

the objects: given two objects a and b, we say that i prefers a to b if a appears before b in

the ordering Pi. We write a >Pi b or simply a >i b if i prefers a to b. Our preference model

assumes that each agent finds every object acceptable, although all the results hold more

generally with some obvious modifications. We also assume that |A| = |O| unless otherwise

noted. We will sometimes refer to an instance by just P , as often the sets of agents and

objects are clear from context and are implicit in P anyway.

A matching is a bijective function from the set of agents to the set of objects. Given a


matching µ, we say that µ(i) = a if agent i receives object a in the matching µ. A matching

µ is Pareto-efficient if there is no other matching ν such that ν(i) ≥i µ(i) for all agents

i ∈ A, with at least one inequality strict. That is, a matching is Pareto efficient if no

agent can be better off without requiring another agent to be worse off. A (Pareto-efficient)

deterministic mechanism is a function that assigns a (Pareto efficient) matching to every

preference profile P . A (Pareto efficient) randomized mechanism is a function that maps

each preference profile P to a distribution over (Pareto efficient) matchings.

Let Σ be the set of all orderings of A. For σ ∈ Σ , let σ(k) be the kth agent according

to order σ and σ−1(i) be the position of agent i in σ. Given a preference profile P and an

ordering σ ∈ Σ, the serial dictatorship (SD) mechanism (also known as priority mechanism)

works as follows: agent σ(1) is assigned his most-preferred object, then agent σ(2) is assigned

his most-preferred object among the remaining ones, and so on. The matching found by

the SD mechanism on a preference profile P and ordering σ is denoted SD(P, σ), or simply

µσ when the preference profile P is clear.

The random priority (RP) mechanism (also called random serial dictatorship) selects an

ordering from Σ uniformly at random and then finds the outcome SD(P, σ), where σ is the

selected ordering. The outcome of the RP mechanism is a bi-stochastic allocation matrix X:

the rows are indexed by agents and the columns by objects, with the entry xia indicating the

probability that agent i will receive object a in the RP mechanism (in the case of indivisible

objects), or the fraction of a that i receives (in the case of divisible objects). To explicitly

indicate the dependence of the RP allocation matrix on the preference profile, we denote as

X(P ) orRP (P ), and its (i, a)th entry byX(P, i, a) orRP (P, i, a). It should be clear from the

definition of the RP mechanism that X(P, i, a) = RP (P, i, a) = |σ ∈ Σ : µσ(i) = a|/n!.

3.3 The complexity of Random Priority

As mentioned in Section 3.2, the definition of RP implies RP (P, i, a) = |σ ∈ Σ : µσ(i) =

a|/n!. Therefore, determining an entry (i, a) of the RP allocation matrix is equivalent to

counting the number of orderings under which i gets a. We refer to this problem as the SD

Count problem, which is formally defined as follows:


SD Count

Input. A strict preference profile P , associated with a set of agents A and a set

of objects O, an agent i and an object a

Output. The number of orderings σ ∈ Σ in which µσ(i) = a.

We now examine the complexity of the SD Count problem. Recall that the SD Fea-

sibility problem (as defined in the introduction) is a problem in NP, and, as SD Count

is the counting version of SD Feasibility, it is a problem in the class #P . We will show

that SD Count is indeed #P-complete. To do so, we introduce the Linear Extension

Count problem. A partially ordered set (or poset) is a set Q equipped with an irreflexive

and transitive relation <Q. A linear extension of a poset Q on n elements is a linear or-

dering ≺ of the elements such that x ≺ y whenever x <Q y. The linear extension count

problem is defined as follows:

Linear Extension Count

Input. A partially ordered set Q.

Output. The number N(Q) of linear extensions of Q.

Brightwell and Winkler [Brightwell and Winkler, 1990] proved that Linear Extension

Count is #P-complete. We now show that Linear Extension Count can be reduced

to SD Count, therefore establishing the hardness of the latter.

Theorem 6. SD Count is #P-complete .

Proof. Clearly, given an ordering σ ∈ Σ, one can verify in polynomial time whether µσ(i) =

a, and so SD Count is in #P . To show that SD Count is #P-complete, we reduce Linear

Extension Count to SD Count. Given a poset Q, consider the following instance of SD

Count with

A = i : i ∈ Q ∪ F, and, O = oi : i ∈ Q ∪ oF .

In words, we have one agent and one object for each element of the poset Q, a special agent

F and a special object oF . Each agent i 6= F ranks oj ahead of oi if and only if j <Q i in

the poset Q; he then ranks the object oi followed by the special object oF ; the remaining


objects appear after oF in any arbitrary order. Thus, the preferences for agent i will look

as follows:

Pi = oj : j <Q i︸︷︷︸in any arbitrary order

, oi, oF , oj : j 6= i, F, j 6<Q i︸︷︷︸in any arbitrary order

.

Finally, the special agent F ranks the special object oF last, and has an arbitrary preference

ordering over the remaining objects.

Let σ be a fixed ordering of agents. Recall that σ(k) is the kth agent according to

order σ and σ−1(i) is the position of agent i in σ. We prove the result by showing that

µσ(F ) = oF if and only if σ−1(F ) = n + 1 and σ(1), . . . , σ(n) is a linear extension of Q.

As a consequence, being able to determine the probability that the special agent F gets the

special object oF under the RP mechanism will imply an ability to compute the number of

linear extensions of the given poset Q. Because the latter problem is #P-complete, so is

the former.

Suppose σ is such that µσ(F ) = oF . As oF is agent F ’s last choice, it follows that (i)

σ−1(F ) = n + 1; and (ii) every other agent received an object that they preferred to oF .

Therefore, for each i 6= F , µσ(i) ∈ oj : j <Q i ∪ oi. We claim that, in fact, µσ(i) = oi

for all i and we prove the result by induction on n. The claim is clearly true for all the

minimal elements of Q: if k is such an element, then the preference ordering corresponding

to agent k in the SD Count instance has ok as the first element and oF as the second; as

each such k appears before the special agent F and does not receive the object oF , it must

be the case that each such k receives object ok. Thus removing all the minimal elements

from Q and their corresponding objects from O does not change the assignment for the rest

of the agents. This also implies that for any pair of elements i, j ∈ Q with i <Q j, agent i

appears before agent j in the ordering σ: for otherwise, j appears before i, and object oi is

still available when it is j’s turn to choose an object, so µσ(j) should be at least as good

as oi according to agent j, contradicting the fact that µσ(j) = oj . This establishes that σ

restricted to the first n positions is a linear extension of Q.

For the converse, suppose that σ−1(F ) = n + 1 and that σ(1), . . . , σ(n) is a linear

extension of Q. Then agent σ(1) must correspond to a minimal element of the poset and

must be assigned object oσ(1) as that is his most-preferred object. Removing this agent and


object from the problem, we get a smaller instance with the same properties, and the result

follows by induction, once we observe that the result is trivially true for n = 1.

3.4 Decision problems associated with Random Priority

Given the preference profile P involving n agents and n objects, an integer 1 ≤ k ≤ n!, an

agent i and an object a, one can ask whether the number of orderings of the agents for which

the SD mechanism gives a to i is at least k. The #P-completeness of computing the RP

allocation matrix proves that this problem is NP-complete: for otherwise, we can determine

the exact value of RP (P, i, a) by doing a binary search over k. This would involve solving

log(n!) = Θ(n log n) instances of this problem, each with a different value of k. Here we

consider the same problem, but for fixed k; specifically, the two natural “extremal” values of

k—that of k = n! and k = 1. We address the following two questions: for a given preference

profile, (i) does agent i have a positive probability of getting object a (SD Feasibility)?;

and (ii) does agent i always get object a (SD Unique Assignment)?

3.4.1 The SD Feasibility problem

We now turn our attention to the SD Feasibility problem, which is formally defined as

follows:

SD Feasibility

Input. A preference profile P , an agent i and an object a.

Output. Is there an ordering σ such that i obtains a in SD(P, σ)?

We show the somewhat surprising result that SD Feasibility is NP-complete by con-

structing a reduction from the problem of finding a minimum-cardinality maximal matching

in a subdivision graph.

A matching in a graph G = (V,E) is a subset M of edges such that no two edges in

M share a vertex. The size of a matching M is the number of edges in M . A maximal

matching is a matching M with the property that M ∪ e is not a matching for any

edge e ∈ E \M . A minimum-cardinality maximal matching, or simply, minimum maximal


matching is a maximal matching of minimum size. The decision version of the minimum

maximal matching problem can be stated as follows:

Minimum Maximal Matching

Input. A graph G = (V,E) and an integer K.

Output. Is there a maximal matching in G of size at most K?

Let G = (V,E) be a given graph. The subdivision graph of G is obtained by splitting

each edge e ∈ E, and by locating a new vertex in the middle. Formally, it is the bipartite

graph S(G) with vertex set V ′ = V ∪ E, and edge set

E′ = e, v | e ∈ E, v ∈ V, and v is incident with e in G.

It is known that Minimum Maximal Matching is NP-complete on subdivision graphs [Hor-

ton and Kilakos, 1993].

Theorem 7. SD Feasibility is NP-complete.

Proof. Given σ, one can verify in polynomial time if i gets a under SD(P, σ) and therefore

the problem is in NP. To show that this problem is NP-complete, we reduce Minimum

Maximal Matching on subdivision graphs to SD Feasibility. LetG′ = (V ′ := V ∪E,E′)–a subdivision graph of G = (V,E)– and K –an integer– be an instance of Minimum

Maximal Matching. Suppose V = v1, . . . , vn, E = e1, . . . , em. Each ei ∈ E connects

two different vertices vpi and vqi with pi < qi. Note that the subdivision graph has edges

(ei, vpi) and (ei, vqi) for each ei ∈ E. Without loss of generality, we may assume m ≥ n.

We construct an instance of SD Feasibility as follows:

• There are 3m+1 agents—two agents for each ei ∈ E and m+1 special agents. The two

agents corresponding to each ei are labeled e1i and e2

i ; m special agents are denoted

F1, F2, . . . , Fm and the remaining special agent is denoted D.

• There are 3m+1 objects—m corresponding to the elements of E and labeled o1, o2, . . . , om;

n corresponding to the elements of V and labeled v1, v2, . . . , vn; m + 1 special ob-

jects labeled oF1 , . . . , oFm+1 , and finally m − n additional (dummy) objects, denoted


d1, . . . , dm−n, to enforce the constraint |A| = |O|. Thus the set of objects O =

o1, . . . , om ∪ v1, . . . , vn ∪ oF1 , . . . , oFm+1 ∪ d1, . . . , dm−n.

The preferences are defined as follows:

• P (e1i ) = oi, vpi , vqi , oFm+1 , oFm , . . . , oF1 .

• P (e2i ) = oi, vqi , vpi , oFm+1 , oFm , . . . , oF1 .

• P (Fi) = oF1 , oF2 , . . . , oFm+1 .

• P (D) = oFm+1 , oFm , . . . , oF1 .

In any preference list, the objects not shown can be appended to that list arbitrarily. To

give some intuition behind the preference structure: agents F1, . . . , Fn rank all the special

objects before any other object and rank them in ascending index order. The two edge

agents corresponding to ei rank their edge object oi first, followed by their vertex objects,

but ordered differently: the first copy ranks vpi before vqi whereas the second copy does

the opposite; this is then followed by all the special objects, but arranged in decreasing

index order. Finally, the special agent D ranks all the special objects in decreasing index

order first. The ranking of the other objects in the preference lists is not important. Two

points about the preference structure deserve mention: first, note that the F agents rank

the special objects in ascending index order, whereas all other agents rank these objects in

descending index order. Second, the special agents F could be omitted from the problem

if we are allowed to have more objects than agents in the reduction, and as such they are

introduced only to maintain balance between the number of agents and number of objects;

as we shall see in a moment, there is a lot of freedom in how these agents are treated in the

reduction.

We claim that there is a maximal matching M ⊆ E′ of G′ such that |M | ≤ K if and

only if there exists an ordering σ such that agent D obtains oFK+1in SD(P, σ).

Given a maximal matching M of G′ with |M | = ` ≤ K, we construct an ordering σ

of the agents such that D obtains oFK+1in SD(P, σ). Observe that in the graph G′, the

vertex ei is connected to exactly two vertices vpi and vqi , so at most one of these two edges

can be in M .


• If (ei, vpi) ∈ M , rank agent e2i ahead of e1

i ; if (ei, vqi) ∈ M , rank agent e1i ahead of

e2i . If M has ` edges, this step will determine 2` agents, and these agents are ranked

ahead of all the other agents; but we are free to order these 2` agents any way we

like as long as we respect the relative ranking of the pair of agents corresponding to

a fixed edge ei as just mentioned.

• From the remaining (m− `) edges in G′ that are unmatched in M , select a subset S

of exactly m −K edges (note that this is possible as ` ≤ K). For each ei ∈ S, rank

agent e1i before e2

i .

• Rank agent D.

• Complete the ordering by adding the remaining edge agents and the agents Fj , 1 ≤j ≤ m, in an arbitrary order.

We now show that agent D will receive the object oFK+1in the ordering just constructed.

If ei is matched, then one of its copies will be assigned oi and the other copy will be

assigned vpi or vqi , depending on whether ei was matched to vqi or vpi . In any case, if ei is

matched, both e1i and e2

i will receive one of their first two choices.

Suppose ei is unmatched, and suppose ei ∈ S. Then, agent e1i will get oi and so agent

e2i cannot be assigned oi; moreover, by the maximality of M , it must be that both vpi and

vqi are matched in M (otherwise one could add one of the edges involving ei to M), and

so the objects vpi and vqi are already assigned to a higher priority agent in our ordering.

Thus, agent e2i must be assigned a special object oFj for some j. Since |S| = m − K,

oFm+1 , oFm , . . . , oFK+2will be taken by the agents e2

i : ei ∈ S. The next agent in the

ordering is agent D, who according to his preferences will get oFK+1. Thus, given a maximal

matching of size at most K, the constructed ordering σ is such that agent D obtains oFK+1

in SD(P, σ), which establishes the “only if” part of the claim.

Now suppose that there exists an ordering σ such that agent D obtains oFK+1in

SD(P, σ). We argue that there is an ordering in which all agents of type e appear be-

fore any special agent of type F , and such that D still receives oFK+1. To that end, suppose

D is the lth agent in σ. First, note that at most K agents of type F can be before D in

σ. Furthermore, if we consider an ordering obtained from σ by removing all agents of type


F that appear before D and placing them after D, the allocation of all agents of type e

before position l will remain the same by the structure of the preferences. Therefore, we

may assume that σ only contains agents of type eji before position l. In addition, for every

σ′ that differs from σ in the ordering after position l, the allocation of the first l agents

(including D) will not change. Hence, we may assume that all agents of type eji appear

before agents of type Fj in σ.

Effectively, we have established that if there is an ordering in which D receives object

oFK+1, then there is an ordering in which all the “edge” agents appear before any special

agent of type F , and such that D still receives oFK+1. In the rest of the proof, we assume

that the given ordering is of this type.

Let M = (ei, vj) : e1i or e2

i obtain vj under σ. We argue that M must be a maximal

matching of G. First, we show that M is a matching. Because all agents of type e appear

before any agent of type F in σ, exactly one of e1i , e

2i —the one that appears earlier—will

obtain oi for every 1 ≤ i ≤ m. Hence, at most one of e1i , e

2i can be allocated an object

of type v, implying that each ei appears in at most one edge in M . On the other hand, by

the definition of the serial dictatorship mechanism, each object of type v is allocated to at

most one agent, and therefore it can appear in at most one edge in M . Hence, we conclude

that M is indeed a matching. The maximality of M follows by the preference structure: if

vpi is unmatched, then when it was the turn of the second copy of ei to choose an object,

vpi was not chosen; this can happen only if that copy of ei chose vqi ; in particular, ei must

be matched. A similar argument applies when vqi is unmatched. We conclude that M is

a maximal matching of G. It remains to be shown that M has size at most K. Note that

m agents of type e get their associated objects and at least m − K get objects of type

oFj . Then, at most K agents of type e will get an object of type vj , and |M | ≤ K which

completes the proof.

3.4.1.1 Implications of the hardness of the SD Feasibility problem.

Theorem 7 has two strong implications. The first implication is related to the inapprox-

imability of the RP mechanism. During the past decades, it has been shown that it is

possible to design polynomial-time algorithms for approximately counting the number of


solutions of some #P-complete problems. Indeed, #P-complete problems admit only

two possibilities: they either allow polynomial approximability to any required degree, or

they cannot be approximated [Sinclair and Jerrum, 1989]. The former possibility is cap-

tured in the definition of a fully polynomial randomized approximation scheme (FPRAS).

Formally, consider a problem whose counting version f is #P-complete. A randomized

algorithm A is an FPRAS for this problem if, for each instance x and error parameter

ε > 0, Pr[|A(x)− f(x)| ≤ εf(x)] ≥ 3/4, and the running time of A is polynomial in |x| and

1/ε. If the decision version of a counting problem is NP-complete, the counting problem

itself cannot admit an FPRAS unless NP = RP, which is the complexity class consisting of

problems that can be solved in randomized polynomial time[Jerrum, 2003].2 Therefore, we

have the following corollary:

Corollary 1. The RP mechanism cannot admit an FPRAS unless NP = RP.

Although the RP allocation matrix cannot be efficiently approximated, it is possible to

distinguish efficiently (with high probability) the entries of the RP allocation matrix with

high values from those with low values. Given preference profile P , an agent i and an

object a, suppose we sample r orderings independently and uniformly at random and, for

each ordering σj with 1 ≤ j ≤ r, we set Xj = 1 if SD(i, a, σj) = 1 and Xj = 0 otherwise.

Note that Pr[Xj = 1] = RP (i, a). Let X =∑r

j=1Xj , and let RP r(i, a) = X/r be our

estimate for the real value of RP (i, a) when using a sample of size r. One question that

naturally arises is how large does r need to be in order to be able to distinguish, with high

probability, if a certain entry RP (i, a) = 0 or is it bigger than a certain q > 0.

We can now use the Hoeffding’s inequality [Hoeffding, 1963] to obtain the following

bound:

Pr(|RP r(i, a)−RP (i, a)| ≥ δ

)≤ 2 exp

(−2r2δ2

)This means that the probability that the estimate deviates more than δ from the real

value of the RP entry is exponentially small in r and δ.

2Similarly to the problem P = NP, the problem of whether NP = RP is open and it suspected to be false.


A second implication of Theorem 7 is related to the complexity of computing Pareto

efficient matchings under constraints. We first highlight a different way of thinking about

the SD Feasibility problem: Let U(i, a) be the set of objects agent i prefers over a

according to Pi. We say that M is a partial Pareto efficient matching if every agent in M

prefers the object he is matched to over all objects unmatched in M , and no trade involving

a subset S of agents in M can make all agents in S better off. Note that the unmatched

agents might find unmatched objects admissible, and therefore M may not be a Pareto

efficient matching.

We claim that there exists an ordering under which agent i is allocated object a if and

only if there is a partial Pareto efficient matching in which all the objects in U(i, a) are

matched and a is unmatched. To prove the “if” part, suppose such a matching exists and

denote it by M . Then, there exists an ordering σM involving the |M | agents matched in M

under which the RP mechanism will give matching M as an output. We can now extend σM

to an ordering σ over all agents, by defining σ(j) = σM (j) for all 1 ≤ j ≤M , σ(|M |+1) = i

and adding the remaining agents to σ in any arbitrary ordering. It is easy to verify that

agent i obtains object a under σ. To show the converse, let σ be an ordering under which

i is allocated a. Note that the matching obtained by running the RP mechanism using

ordering σ until position σ−1(i) must be a partial Pareto efficient matching in which all the

objects in U(i, a) are matched and a is unmatched.

We conclude by noting that finding a Pareto efficient matching in which all the objects

in U(i, a) are matched and a is unmatched is equivalent to finding a (partial) Pareto efficient

matching in the reduced instance I\a\i = (A\i,O\a, P\a) with the constraint that

all objects in U(i, a) be matched.3

Based on this idea, we define the Constrained Pareto Efficient Matching prob-

lem as follows:

Constrained Pareto Efficient Matching

Input. A preference profile P in which agents may have inadmissible objects, a

subset of objects Q.

3Here, P\a represents the preferences P truncated so that every agent only lists as admissible those

objects that he strictly prefers to a.


Output. Is there a Pareto efficient matching in which all objects in Q are

matched?

An immediate corollary of Theorem 7 is the following:

Corollary 2. Constrained Pareto Efficient Matching is NP-complete.

In a recent paper, Haeringer and Iehle [Haeringer and Iehle, 2014] study stability in

two-sided matchings when the preferences for only one side of the market are known. In

a two-sided matching model, each side of the market has preferences over the other side.

For consistency with the existing literature, we refer to the sides of the market as men and

women respectively. A matching is said to be stable if every matched agent finds his match

acceptable, and if there is no pair of agents who would prefer to be matched to each other

rather than to their current match (if any). In the model analyzed in [Haeringer and Iehle,

2014], only the preferences of the women are known. While we do not know the preferences

of the men, we do know that a man m finds a women w acceptable if and only if w ranks

m somewhere in her preference ordering. Their goal is to say whether a pair of agents can

be matched at a stable matching for some preference profile. Haeringer and Iehle de signed

a dynamic-programming algorithm for this problem with an exponential running time in

the size of the input, but left open the possibility of a polynomial-time algorithm to solve

this decision problem. Here we show that their problem is closely related to the problem

of finding a constrained Pareto efficient matching in a one-sided matching problem, and

so is NP-complete: the one-sided allocation problem with strict (but possibly incomplete)

preferences is obtained by viewing the women as agents and men as objects. As before, let

U(w,m) be the set of men that w strictly prefers to m.

Claim 3. In the above setting, a woman w and a men m can be matched at a stable matching

for some preference profile if and only if there is a Pareto efficient matching for the women

in which all the men in U(w,m) are matched (in the problem where m and w are omitted).

Proof. Suppose there is a Pareto efficient matching M in which all the men in U(w,m) are

matched. In this matching, clearly no unmatched woman can have an acceptable unmatched

man. Suppose each matched man ranks his partner in M first; m ranks w first; and the


preferences of the unmatched men are arbitrary. The resulting matching is stable: every

matched man is married to his top-ranked woman; and none of the other men are acceptable

to any available woman. This verifies the ”if” part of the Claim.

To show the converse, suppose m and w are matched in some stable matching M . As M

is stable, there are no blocking pairs. This means that all the men in U(w,m) are matched,

and that every woman must prefer her own match over any unmatched man. Thus the

unmatched men in M will play no further role. If M is not Pareto efficient (in the problem

in which m and w are omitted), consider the following reallocation of the matched pairs:

there is a node for each woman; and there is an arc (w′, w′′) if and only if w′′ is matched

to the most-preferred remaining partner of w′; any cycles that form are cleared (so that

the women involved in the cycle effect a Pareto improving swap), and the procedure is

recursively applied.4 Throughout this procedure, the set of matched men does not change;

in particular, every man in U(w,m) remains matched; and the final outcome is a Pareto

efficient matching.

3.4.2 The SD Unique Assignment problem

We now study the SD Unique Assignment problem, which is defined as follows:

SD Unique Assignment

Input. A preference profile P , an agent i and an object a.

Output. Is it true that for every ordering σ ∈ Σ, agent i obtains a in SD(P, σ)?

Given an instance I = (A,O, P ) and an object a ∈ O, we define the reduced instance

I\a\i = (A\i,O\a, P\a), where P\a represents the preferences P truncated so

that every agent only lists as admissible those objects that he strictly prefers to a.

We start with the following lemma.

Lemma 1. Given an agent i and an object a, µσ(i) = a for all σ ∈ Σ if and only if:

(1) a is agent i’s top-choice.

4The reallocation mechanism we just described is the well-known Top-Trading Cycles (TTC) mechanism

proposed by Shapley and Scarf [Shapley and Scarf, 1974].


(2) In every ordering σ such that σ(i) = n, µσ(j) >j a for all j 6= i ∈ A (that is, in every

ordering in which i is the last agent, all other agents get an object they like better than

a).

Proof. Suppose µσ(i) = a for all σ ∈ Σ. Condition (1) must be trivially satisfied, as

otherwise i will not choose a whenever σ−1(i) = 1. Furthermore, consider an ordering σ

such that σ−1(i) = n. Since a is not assigned to any of the first n− 1 agents, it follows that

all of them must get objects they prefer to a. The converse is even simpler: if conditions

(1) and (2) are satisfied, object a would be available when it is agent i’s turn to choose,

and so i will be assigned a.

Lemma 1 forms the basis of Algorithm 2, which solves the SD Unique Assignment

problem by verifying both conditions. Condition (1) can be easily checked. To verify

condition (2), note that every ordering σ ∈ Σ induces a Pareto efficient matching and every

efficient matching can be implemented with (at least) one ordering σ. Hence, condition (2)

fails to hold if and only if there is a Pareto efficient matching in the reduced problem I\a\i

of size at most n− 2. In that case, at least one object and one agent of I\a\i must remain

unmatched. The key idea is to first identify those objects that are candidates to remain

unmatched in a Pareto efficient matching for the reduced problem, and solve a matching

problem for each object in turn to find whether there exists a Pareto efficient matching in

which they remain unmatched.

Theorem 8. Algorithm 2 solves the SD Unique Assignment problem in polynomial time.

Proof. Clearly, Algorithm 2 runs in polynomial time. To show the correctness of the algo-

rithm, we may assume that object a is agent i’s most-preferred object, as otherwise i will

not always be assigned a. Consider the reduced instance I ′ = (A′,O′, P ′) = I\a\i . Since

every ordering σ ∈ Σ induces a Pareto efficient matching and every efficient matching can

be implemented with (at least) one ordering σ, condition (2) fails to hold if and only if we

are able to find an efficient matching in the reduced problem I ′ of size at most n− 2. Note

that, in that case, at least one object and one agent of I ′ must remain unmatched. Let o

be an unmatched object in an efficient matching M . Clearly, o must be inadmissible for

at least one agent in I ′ (in particular, it must be inadmissible for all unmatched agents in


Algorithm 2 SD Unique Assignment

Input: An instance I = (A,O, P ), and agent i ∈ A and an object a ∈ OOutput: Is it true that, for every ordering σ ∈ Σ, agent i obtains a in SD(P, σ)?

If object a is not agent i’s top choice, return FALSE.

Consider the reduced instance I ′ = (A′,O′, P ′) = I\a\i .

Let S = o ∈ O′ : o /∈ P ′j for some j ∈ A′.For each o ∈ S:

Let A(o) = j ∈ A′ : o ∈ P ′j (set of agents that find o admissible).

Consider the bipartite graph G(o) = ((A′,O′\o), E), where (k, j) ∈ E if and only

if agent k finds object j admissible in I ′ and likes j better than o.

Find a maximum matching M in G(o), with the constraint that every vertex in

A(o) must be matched.

If such a matching exists, return FALSE.

Return TRUE.

M). Therefore, we will first identify those objects that are candidates to remain unmatched

in a Pareto efficient matching for the reduced problem, and then we will solve a matching

problem for each object in turn to find whether there exists a Pareto efficient matching in

which they remain unmatched.

Let S = o ∈ O′ : o /∈ P ′j for some j ∈ A′, that is, the objects in S are those that are

inadmissible for at least one agent and thus are candidates for being unmatched in some

efficient matching of I ′. For each object o ∈ S, let A(o) = j ∈ A′ : o ∈ P ′j be the set of

agents that find object o admissible. Whenever o is unmatched, all agents in A(o) must be

matched to an object they like better than o. For each o ∈ S, we can either find a Pareto

efficient matching in which o is unmatched or we can show that no such matching exists as

follows: Consider a bipartite graph G(o) = ((A′,O′\o), E), where (k, j) ∈ E if and only if

agent k finds object j admissible in I ′ and likes j better than o. Find a maximum matching

M in G(o), with the constraint that every vertex in A(o) must be matched. If no such


matching exists, then o must be matched in every Pareto efficient matching of I. Otherwise,

note that M contains at most n− 2 edges as |O\o| = n− 2, but it might not be efficient.

This matching, however, can be transformed into an efficient matching by performing a set

of Pareto improvements, as described in [Abraham et al., 2005]. Nevertheless, no Pareto

improvement can involve o as all agents that find o admissible were assigned better objects

than o and the rest do not find o admissible. Hence, we were able to find a Pareto efficient

matching of size at most n− 2 and thus show that i does not always get a.

3.5 Discussion

Due to its simplicity and compelling properties, the RP mechanism is one of the most

popular mechanisms for allocating objects. The hardness results in this chapter imply that

any mechanism that relies on the knowledge of the RP allocation matrix is likely to be

impractical when the number of objects is large. This is the case, for instance, if one uses

the RP mechanism to allocate divisible objects, assuming agents still have unit demand.

We have shown that the RP allocation is not only hard to compute in general, but also

hard to approximate. However, in some cases in which the preference domain is restricted,

the RP allocation can be easy to compute. One example is the work by [Cres and Moulin,

2001], who consider a scheduling problem involving unit-length jobs and deadlines, which

could be different for different jobs. For this special case, the RP allocation can be computed

efficiently. A natural question of interest is to determine precisely the conditions under

which one can compute the RP allocation in polynomial time, or to identify other natural

problems where such a result is possible.

CHAPTER 4. THE SIZE OF THE CORE IN ASSIGNMENT MARKETS 65

Chapter 4

The Size of the Core in

Assignment Markets

4.1 Introduction

We study bilateral matching markets such as marriage markets, labor markets, and housing

markets, that allow participants to form partnerships with each other for mutual benefit.

The two classical models of such matching markets are the non-transferable utility (NTU)

model of Gale and Shapley [Gale and Shapley, 1962], where payments are not allowed

between the agents; and the Shapley-Shubik-Becker transferable utility (TU) model [Shapley

and Shubik, 1971; Becker, 1973], where transfer payments are allowed between pairs of

agents who form a match. For each of these models the natural solution concept is that of

a stable outcome, in which there is no pair of agents who would be happier with each other

than in their current outcome. In fact, for TU matching markets, it is well known that

the notion of a stable outcome coincides with that of a competitive equilibrium. A stable

outcome is guaranteed to exist in any two-sided market, but is typically not unique. The

concept of stability is widely used as a starting point in theoretical and empirical studies in

the context of matching. A nearly unique stable outcome is required in order to facilitate

predictions, comparative statics and so on, but little is known about when this occurs in


the TU setting.1 In this work, we seek to characterize the size of the set of stable matches

as a function of market characteristics in TU matching markets.

The motivation for our work is twofold. First, uniqueness of the stable outcome is

typically assumed in empirical investigations, though there is insufficient theoretical basis

to justify such an assumption. We ask when such an assumption is justified. Second, it

is of interest to know whether basic market primitives, i.e., the number of agents and the

values of possible matches, are sufficient to determine the outcome of the market, or whether

there is significant ambiguity arising from which equilibrium the market is in. Can a labor

market support higher wages for labor without adding jobs or improving productivity, just

by moving to a different equilibrium? In TU matching markets, market primitives like the

value generated by a pair/match, and even transfers occurring in outcomes are difficult to

observe, which has hindered empirical studies of features like core size (NTU markets are

much easier to study empirically; see footnote 1). This further increases the importance

of generating theoretical predictions of core size, which can also potentially guide future

empirical work.

We consider the assignment game model of Shapley and Shubik [Shapley and Shubik,

1971], consisting of “workers” and “firms” each of whom can match with at most one agent

on the other side. To model the different skills of the workers and the different requirements

of the firms, we assume that there are K types of workers and Q types of firms. Matching

worker i with firm j generates a value Φij (this can be divided between i and j in an arbitrary

manner since transfers are allowed), which we model as a sum of two terms: a term u(·, ·)that depends only on the types of i and j, and a term ψi,j that represents the “idiosyncratic”

contributions of worker i to firm j. In our model the u(·, ·) is assumed to be fixed, but the

ψij is the sum of two random variables, the “productivity” of worker i with respect to the

type of firm j and, symmetrically, the “productivity” of firm j with respect to the type

of worker i. These productivities are assumed to be independently drawn from a bounded

1A small core has been found in special cases of the TU setting as in [Gretsky et al., 1992; Gretsky

et al., 1999; Hassidim and Romm, 2014], which we discuss below. In the case of the NTU setting, real

markets have almost always been found to contain a nearly unique stable outcome, e.g. [Roth and Peranson,

1999], and a body of theory explains this, e.g. [Immorlica and Mahdian, 2005; Kojima and Pathak, 2009;

Ashlagi et al., 2013; Holzman and Samet, 2013; Azevedo and Leshno, 2012].


distribution (satisfying certain assumptions) for each (agent,type) pair. In addition to being

normatively attractive, such a generative model for the value of a match has been used in

empirical studies of marriage markets, starting with Choo and Siow [Choo and Siow, 2006;

Chiappori et al., 2011; Galichon and Salani, 2010].

We study the size of the set of stable outcomes for a random market constructed in

this way. Shapley and Shubik [Shapley and Shubik, 1971] showed that the set of stable

outcomes (which is the same as the core) has a lattice structure, and thus has two extreme

stable matchings: the worker optimal stable match, where each worker earns the maximum

possible and each firm the minimum possible in any stable matching; and the firm optimal

stable matching which is the symmetric counterpart. Also, all stable outcomes live on a

maximum weight matching, which is generically unique. Given these structural properties,

our metric for the size of the core is quite natural: we consider the difference between the

maximum and minimum utility of a worker (equivalently, a firm) in the core, averaged

over matched workers (or firms). Our main result is that the size of the set of stable

matchings, as measured by this metric, is small under some reasonable assumptions on

market structure: specifically, the expected core size is O∗(1/√n) in a problem with n

agents, and at most ` types of agents on each side (with ` fixed). We show that this bound

is essentially tight by constructing a sequence of markets such that the core size is Ω(1/√n).

Thus the core shrinks with market size, and this shrinking is faster when there are fewer

types of agents. Additionally, we obtain a tighter upper bound in the special case with

just one type of employer and more employers than workers. Our upper bound in this case

improves sharply as the number of additional employers m increases; we establish a bound

of O∗(1/(n1/`m1−1/`)), where ` is the number of worker types.

Our model has the following property (here, think of u( · , · ) as being formally incor-

porated in the worker productivity): For every (worker type, firm type) pair, there is a

“price” associated with this type-pair, such that for every matched pair of agents of these

types, the utility of each agent is her productivity (with respect to the type on the other

side), “corrected” additively (in opposite directions) by the price. We show that variation

in these type-pair prices is uniformly bounded as O∗(1/√n) across core allocations, in ex-

pectation, implying the bound on core size. A key component of our analysis is to relate


the combinatorial structure of the core to order statistics of certain independent identically

distributed (i.i.d.) random variables (r.v.s). These r.v.s are one-dimensional projections of

point processes in (particular subregions of) the unit hypercube, where the point processes

correspond to the market realization. An analytical challenge that we face is that the rel-

evant projections as well as the relevant order statistics are themselves a random function

of the market realization. We overcome this via appropriate union bounds. Our analysis

throws light on which aspects of market structure affect the core and its size.

Most of the related literature focuses on the NTU model of Gale and Shapley [Gale and

Shapley, 1962]. For that model, a number of papers establish a small core under various

assumptions such as short preference lists [Immorlica and Mahdian, 2005; Kojima and

Pathak, 2009; Kojima et al., 2013], strongly correlated preferences [Holzman and Samet,

2013; Azevedo and Leshno, 2012]. In a recent paper Ashlagi et al. [Ashlagi et al., 2013]

show that in a random NTU matching market with long lists and uncorrelated preferences,

even a slight imbalance results in a significant advantage for the short side of the market

and that there is approximately a unique stable matching. Further, the near uniqueness of

the stable matching is found to be robust to varying correlations in preferences and other

features, suggesting that a small core may be generic in NTU matching markets. There is an

extensive literature on large assignment games that extends the many structural properties

established by Shapley and Shubik for finite assignment games to a setting in which the

agents form a continuum, see for example Gretzky, Ostroy and Zame [Gretsky et al., 1992;

Gretsky et al., 1999]. Those papers also show convergence of large finite markets to the

continuum limit, including that the core shrinks to a point. However, unlike in our model,

they model the productivity of each partnership as a deterministic function of the pair

of types, with the only randomness being in the number of agents of each type. The

work on assignment games that is most closely related to our work is a recent preprint of

Hassidim and Romm [Hassidim and Romm, 2014]: in their model, all workers (firms) are

a priori identical, and the value of matching worker i to firm j is a random draw from a

bounded distribution, independently for every pair (i, j). For such a model, they establish an

approximate “law of one price,” i.e., that workers are paid approximately identical salaries

in any core allocation, and that the long side gets almost none of the surplus in unbalanced


markets. In contrast, we work with multiple types of workers and firms, and the value of

a match depends on the types of each agent, and random variables that depend on the

identity of one of the agents and the type (but not the identity) of the other agent.

The rest of the chapter is organized as follows. We present our model in Section 5.3, our

results in Section 4.3, and an overview of the proof of our main result in Section 4.4. We

conclude with a discussion in Section 4.5. Several proofs are deferred to the appendices.

4.2 Model Formulation

We consider a two-sided, transferable utility matching market with a finite number of agents.

The sides of the market are represented by the labor (L) and the employers (E). Let nL be

the number of agents in L and nE be the number of agents in E ; we let n := |L|+ |E| denote

the size of the market, i.e., the total number of agents in the problem. We assume that the

underlying graph is complete, that is, all pairs of agents can potentially be matched. Each

side of the market is partitioned into a finite number of types and we let K and Q denote

the number of different types of agents in L and E respectively. We define TL := 1, . . . ,Kand TE := 1, . . . , Q to be the set of types in the labor and employer side respectively.

Let T = TL × TE denote the set of pairs of types. If nL = nE we say that the problem is

balanced. Otherwise, we say that the problem is unbalanced. In addition, for a given type

t ∈ TL ∪ TE , we denote by nt the number of agents of type t. Finally, let τ(a) denote the

type of agent a ∈ L∪E ; given a type t and an agent a, we say that a ∈ t if τ(a) = t. In what

follows we typically use i to denote an individual agent in L, and j to denote an individual

agent in E .

The value of the match between i and j is denoted Φ(i, j). An outcome is a pair (M,γ),

where M is a matching between agents in L and E , and γ is a payoff vector such that

γi + γj = Φ(i, j) for every pair of matched agents i ∈ L, j ∈ E , (i, j) ∈ M . That is,

the vector γ indicates how the value of a match is divided among the agents involved in

the match. In this chapter we shall be concerned with outcomes that are in the core, i.e.,

outcomes such that no coalition of players can produce greater value among themselves

than the sum of their utilities. Shapley and Shubik [Shapley and Shubik, 1971] show that


for this matching market model, an outcome (M,γ) is in the core if and only if it is satisfies

stability. The stability condition requires γi + γj ≥ Φ(i, j) for all i ∈ L and j ∈ E , and

further requires the γ vector to be non-negative.2 The set of stable outcome utilities turns

out to be the set of optima of the dual to the maximum weight matching linear program,

implying in particular that the matching M in a stable outcome must be a maximum weight

matching.

4.2.1 Structure of Φ(i, j)

We assume that Φ(i, j) is additively separable as follows.

Assumption (Separability). Φ(i, j) = u(τ(i), τ(j)) + ετ(i)j + η

τ(j)i .

It is natural to think that the value of matching i and j can be broken down into a sum

of two components: a utility u(τ(i), τ(j)) that depends only on the agents’ types, and a

term ψτ(i),τ(j)i,j which is match specific and potentially depends on both the identity of the

agents as well as their types. The separability assumption states that the match-specific

component is further additively separable into two terms that each depend on the identity

of one of the agents and only the type of the other agent. In particular, for any fixed

employer j and two distinct workers i, i′ ∈ L we have ετ(i)j = ε

τ(i′)j whenever τ(i) = τ(i′), as

the term ε only depends on the type of the agents in L. Analogously, the term η depends

on the individual worker i ∈ L but only the type of the firm j ∈ E .

We model the term u(τ(i), τ(j)) as a fixed constant, whereas the ε and η terms are

modelled as random variables, independent across agent type pairs. The continuum limit

of such a model was introduced by Choo and Siow [Choo and Siow, 2006], who used the

model to empirically estimate certain structural features of marriage markets. Such a

model is attractive in allowing for reasonable heterogeneity and idiosyncratic variation via

the random variables, while still remaining structured due to a fixed number of types.

2Note that in any unstable outcome, there must either be an individual agent who would prefer to not

participate in the matching (because of a negative payoff) or a blocking pair of agents who can both do

better by matching with each other (because the value they generate by matching with each other exceeds

their current payoffs).


While these features have been important in facilitating estimation [Choo and Siow, 2006;

Chiappori et al., 2011], they simultaneously also make this a plausible model of real markets.

We further assume that the terms ετ(i)j , η

τ(j)i are independent random draws from the

uniform [0, 1] distribution. While the assumption of i.i.d. U [0, 1] r.v.s appears quite restric-

tive, our results and proofs extend to arbitrary non-atomic bounded distributions supported

on a closed interval, with positive density everywhere in the support.

4.2.2 Preliminaries

We now state some preliminary observations on the structure of the core under the separa-

bility assumption. We start by showing that the payoffs can be expressed more conveniently.

For each i ∈ L and each type q ∈ TE , let ηqi = u(τ(i), q) + ηqi .

In our market model with probability 1 the maximum weight matching is unique, so we

assume a unique maximum weight matching M to simplify the exposition. We denote by

M(t) the set of agents who are matched to an agent of type t under M . In addition, we use

U to denote the set of unmatched agents under matching M .

Proposition 2. Let M be the unique maximum weight matching. Any core solution (M,γ),

corresponds to a vector α ∈ RK×Q such that the payoffs can be expressed as:

• γi = ηqi − αkq, for all i ∈ L such that τ(i) = k and i ∈M(q).

• γj = εkj + αkq, for all j ∈ E such that τ(j) = q and j ∈M(k).

Proposition 2 follows from stability, and formalizes the existence of a single “price” for

every type-pair (k, q) that is common across all matched pairs of agents with those types.

Based on Proposition 2, any core solution can be expressed in terms of the maximum weight

matching M and the vector α.

The following proposition states necessary and sufficient conditions for (M,α) to be a

core outcome. (The maximum over an empty set is defined as −∞.)

Proposition 3. Let M be the unique maximum weight matching. The following conditions

are necessary and sufficient for (M,α) to be a core solution:


(ST) For every pair of types (k, q), (k′, q′) ∈ T :

mini∈k′∩M(q′)

ηq′i −η

qi + min

j∈q∩M(k)εkj−εk

′j ≥ αk′q′−αkq ≥ max

i∈k∩M(q)ηq′i −η

qi + max

j∈q′∩M(k′)εkj−εk

′j .

(IM) For every pair of types (k, q) ∈ T :

minj∈q∩M(k)

εkj ≥ −αkq ≥ maxj∈q∩U

εkj ,

and mini∈k∩M(q)

ηqi ≥ αkq ≥ maxi∈k∩U

ηqi .

The first set of conditions follow from the non-existence of a blocking pair of matched

agents. The second conditions follow from the fact that utilities are non-negative (implying

the left inequalities) and the non-existence of a blocking pair involving an unmatched agent.

See [Chiappori et al., 2011, Proposition 1] for a proof.

We conclude with a definition of the size of the core, denoted by C. We define C as

the difference between the maximum and minimum utility of a worker (or firm) in the

core, averaged over workers matched under M . This can be equivalently stated in terms of

the vector α. For each pair of types (k, q) ∈ T , let αmaxkq and αmin

kq be the maximum and

minimum possible values of αkq among core α vectors.

Definition 3 (Size of the core). Let M be the unique maximum weight matching. For each

pair of types (k, q) ∈ T , let N(k, q) denote the number of matches between agents of type k

and agents of type q. Then, the size of the core is denoted by C and is defined as:

C =

∑k

∑qN(k, q)|αmax

kq − αminkq |∑

k

∑qN(k, q)

.

4.3 Results

We keep the number of agent types fixed and allow the number of agents to grow, focusing

on how the size of the core scales as the market grows.

Given the stochastic nature of the our problem, the size of the core C is itself a random

variable. Therefore, the main focus of our work is to study how the expected value of Cdepends on the characteristics of the market. In finite markets it is generically possible to

marginally modify some payoffs in a core solution without violating stability and, therefore,


the size of the core is strictly positive [Shapley and Shubik, 1971]. However, as the size of

the market increases (the agent types stay the same), the set of core vectors α should shrink

as an increase in the number of stability constraints limits the possible perturbations to the

payoffs, cf. Proposition 3.

We start by considering the simple case of markets with one type on each side, that is

K = Q = 1. Given that there is only one type of agent on each side, the deterministic utility

term u = u(τ(i), τ(j)) will be the same for all possible matches, regardless the identity of

the agents. The value of a match between agents i ∈ L and j ∈ E is Φ(i, j) = u + ηi + εj .

Suppose u > 0.

Remark 1. In the case of a balanced market, i.e., nL = nE , the above market has C ≥ u

with probability 1. In particular, E[C] = Ω(1).

The idea is the following: all agents will be matched in a stable solution and by Proposi-

tion 2, we can describe the size of the core in terms of a single parameter α; by Proposition 3,

the core consists of all α ∈ [−minj εj , u+ mini ηi]. In other words, the value u that is part

of Φ(i, j) for each (i, j) can be split in an arbitrary fashion between employers and workers.

On the other hand, in case of any imbalance, i.e., nL 6= nE , it turns out that u must go

entirely to the short side of the market, and the size of the core is O(1/n) (the distance

between consecutive order statistics of the εj ’s or the ηi’s). Thus, the core is small and

rapidly shrinking in any unbalanced market in the case of K = Q = 1.

We now consider the general case of K types of labor and Q types of employers. The

following condition generalizes the imbalance condition to the case of multiple types. The

idea is to get rid of the cases that, for certain values of deterministic utilities u( · , · ), may

resemble a balanced problem.

Assumption 1. For every pair of subsets of types S ⊆ TL and S ′ ⊆ TE we must have∑t∈S nt 6=

∑t∈S′ nt. In words, this means that there is no subset of types such that the

submarket formed by agents of those types is balanced.

We highlight that in our setting with fixed K and Q and growing n, “most” markets

satisfy Assumption 1.3

3Consider possible vectors N = (nt)t∈TL∪TE :∑t nt = n describing the number of agents of each type.


We make a further regularity assumption, namely that the number of agents of each

type grows linearly in the size of the market.

Assumption 2. There exists C > 0 such that for all types t ∈ TL ∪ TE , we have nt ≥ Cn.

We now present our main theorem.

Theorem 9. Consider K ≥ 1 types of labor, and Q ≥ 1 types of employers. There exists

f(n) = O∗(

1max(K,Q)√n

)such that under Assumption 1 and Assumption 2, for a market with

n agents we have E[C] ≤ f(n). Further, there exists a sequence of markets with K types of

labor and Q types of employers such that E[C] = Ω(

1max(K,Q)√n

).

In words, our main result says that under reasonable conditions, E[C] is vanishing as

n → ∞, at a rate O∗(

1max(K,Q)√n

)and that this bound is tight in worst case. Thus, the

core size shrinks to zero as the market grows larger, at a rate that is faster (in worst case)

if there are fewer types of agents. We give a proof of our main theorem in Section 4.4.1,

along with Appendices B.2 and B.3.

The upper bound in Theorem 9 can be improved if further constraints are imposed on

the number of types and the imbalance. As an illuminating example, we show that in the

setting in which K ≥ 2, Q = 1 and nE > nL, the size of the core can be bounded above by

a function that depends on both the size of the market and on the size of the imbalance in

the market.

Theorem 10. Consider the setting in which K ≥ 2, Q = 1, nE > nL and let m = nE −nL.

Under Assumption 2, we have E[C] ≤ O∗(

1

n1Km

K−1K

).

For m = O∗(1), the bound in Theorem 10 matches that in Theorem 9. However, the

bound here becomes tighter as the imbalance m grows. In fact, for m = Θ(n), the core size

is bounded as O∗(1/n). It is noteworthy that the scaling behavior here does not depend

on the number of worker types. We also mention here that, using symmetry, an analogous

result can be stated with Q types of employers, only one type of worker, and more workers

than employers.

Then O(1/n) fraction of these vectors violate Assumption 1.


We prove Theorem 10 in Appendix B.4. The idea is to use the unmatched agents and

condition (IM) in Proposition 3 (for the employers) to control absolute variation in one of

the α’s. We separately control the relative variation of the α’s in the core using condition

(ST) in Proposition 3 under Assumption 2. Combining these we obtain the stated bound

on C.

4.4 Overview of the proof of the main result

We now present an overview of our proof of Theorem 9. We first discuss the key steps in

establishing the upper bound (the complete proof can be found in Appendix B.2), and then

sketch the proof of the lower bound in Section 4.4.3 (completed in Appendix B.3).

Throughout this section, we assume that there is a unique maximum weight matching

and we refer to it as M . Given M , recall that N(k, q) is defined as the number of matches

between agents of type k and agents of type q in M .

We start by constructing a graph associated with matching M as follows. Let G(M) be

the bipartite graph whose vertex sets are the types in L and E , and such that there is an

edge between types k ∈ TL, q ∈ TE if and only if there is an agent of type k matched to an

agent of type q in M , i.e., N(k, q) > 0. The following lemma states a key fact regarding the

structure of G(M).

Lemma 2. Let M be the unique maximum weight matching and let G(M) be the associated

type-adjacency graph. Suppose we mark the vertex in G(M) corresponding to type t if and

only if at least one agent of type t is unmatched under M . Then, under Assumption 1, with

probability 1, every connected component in G(M) must contain a marked vertex.

4.4.1 Overview of the upper bound proof

Roughly, the idea of the upper bound proof of Theorem 9 is as follows. We consider some

suitably defined events (which are discussed later), which occur in typical markets. Under

these events, we show that the variation in the type-pair prices is uniformly bounded as

follows,

max(k,q)∈TL×TE ,N(k,q)>0

|αmaxkq − αmin

kq | ≤ f(n),


for some f(n) = O∗(

1n1/max(K,Q)

). To prove this bound, we use the graph G(M) as defined

above. Given a type t ∈ TL ∪ TE , let the distance d(t) be defined as the minimum distance

in G(M) from t to any marked vertex. By Lemma 2, every unmarked vertex t must be at

a finite distance from a marked one. Furthermore, maxt∈TL∪TE d(t) ≤ K +Q regardless the

realization of the graph.

Our argument to control the variation in the α’s is by induction on d(t). To establish our

induction base, we show that the variation in all the relevant α’s associated with marked

types (these types have distance zero) is bounded. In particular, for each marked type t,

we show that maxt′: N(t,t′)>0

(αmaxt,t′ − αmin

t,t′

)≤ O∗

(1

n1/max(K,Q)

). This is done in Lemma 4.

In the inductive step, we assume the bound holds for every α associated with a type

whose distance is d or less, i.e, for every (t, t′) ∈ TL × TE such that min(d(t), d(t′)) ≤ d,

we have αmaxt,t′ − αmin

t,t′ ≤ O∗(

1n1/max(K,Q)

). Then, we use the inductive hypothesis to show

that the result must also hold for all types whose distance is d + 1. By the definition

of distance, for every type t such that d(t) = d + 1, there must exist a type t∗ such

that d(t∗) = d and N(t, t∗) > 0. Therefore, by our inductive hypothesis, we must have

αmaxt,t∗ − αmin

t,t∗ ≤ O∗(

1n1/max(K,Q)

). Using this bound, we further bound the variation in all

α’s associated with type t, by controlling the relative variation of the α’s in the core, i.e.,

by showing that αt,t1 − αt,t2 for types t1, t2 with matches to type t can vary only within a

range bounded by O∗(

1n1/max(K,Q)

). This is formally achieved in Lemma 5.

To conclude, we briefly describe the nature of the events that we argue must hold with

high probability. These events are related to the distance between order statistics of the

projections of points distributed independently in (sub-regions of) a hypercube. Note that,

once we focus on a single type t, the random productivities associated to an agent of type t

can be described by a D(t)-dimensional vector within the [0, 1]D(t)-hypercube, where D(t)

is dimension of the productivity vector of agents of type t (i.e., D(t) = K if t ∈ TE and

D(t) = Q otherwise). Furthermore, the location of these points can be described by a

point process in [0, 1]D(t). Hence, all the conditions in Proposition 3 can be interpreted as

geometric conditions in the unitary hypercube. We use this geometric interpretation and

relate Proposition 3 to the regions, random sets and random variables defined below in

Section 4.4.2.1 to prove our main theorem.


4.4.2 Hypercube definitions and key lemmas

As mentioned in Section 5.1, a key component of our analysis is to relate the combinato-

rial structure of the core to order statistics of certain independent identically distributed

(i.i.d.) random variables. These random variables are one-dimensional projections of point

processes in (particular subregions of) the unit hypercube, where the point processes cor-

respond to the market realization. Next, we formally define the regions, random sets and

random variables that will be useful in our analysis.

4.4.2.1 Hypercube definitions

Consider a type t ∈ TE . For each employer j : τ(j) = t, there is a vector of productivities

εj distributed uniformly in [0, 1]K , independently across employers. In this subsection we

consider these productivities for a given t. We suppress t in the definitions to simplify

notation (so n here corresponds to nt, and so on). Analogous definitions can be made for

t ∈ TL.

Consider n i.i.d. points (εj)nj=1, distributed uniformly in the [0, 1]K-hypercube. Here

εj = (ε1j , ε2j , . . . , ε

Kj ). Let K = 1, 2, . . . ,K denote the set of dimension indices. Define the

region

Rk = x ∈ [0, 1]K : xk ≥ xk′ ∀k′ 6= k, k′ ∈ K (4.1)

For k1, k2 ∈ K, k1 6= k2 and for δ ∈ [0, 1/2], define the region

Rk1,k2(δ) = x ∈ [0, 1]K : xk1 ≥ xk ∀k /∈ k1, k2, k ∈ K, xk1 ≥ δ . (4.2)

Let

Vk = x : x = εkj for j : εj ∈ Rk , (4.3)

and V k = max(Difference between consecutive values in Vk ∪ 0, 1

). (4.4)

Thus, Vk ⊂ [0, 1] is the set of values of the k-th coordinate of the points lying in Rk,and V k ∈ R is the maximum difference between consecutive values in Vk ∪ 0, 1. (As an

example, if Vk = 0.3, 0.4, 0.8, the differences between consecutive values in Vk∪0, 1 are

0.3, 0.1, 0.4, 0.2, resulting in V k = 0.4.) Note that Vk is a random and finite set, and V k is


a random variable. Let

Vk1,k2(δ) = x : x = εk1j − εk2

j for j : εj ∈ Rk1,k2 , (4.5)

and V k1,k2(δ) = max(Difference between consecutive values in Vk1,k2(δ) ∪ −1 + δ, 1

)(4.6)

Thus, Vk1,k2 ⊂ [−1 + δ, 1] is the set of values of the difference between the k1-th and k2-th

coordinate of points lying in Rk1,k2 , and V k1,k2 ∈ R is the maximum difference between

consecutive values in Vk1,k2 ∪ −1 + δ, 1.In addition, for δ ∈ (0, 1/2] and k ∈ K, define

Rk(δ) = x ∈ [0, 1]K : xk′ ≤ δ ∀k′ ∈ K, k′ 6= k . (4.7)

Let

Vk(δ) = x : x = εkj for j : εj ∈ Rk (4.8)

and V k(δ) = max(Difference between consecutive values in Vk(δ) ∪ 0, 1

). (4.9)

We now relate the above definitions to the combinatorial structure of our problem. We

now include the type t explicitly in the names of the associated regions, sets and random

variables, e.g., region Rk(δ) when defined for type t is referred to as Rk(t, δ).The definition of these regions, sets and random variables might seem arbitrary at

first sight. However, it is closely related to the geometric interpretation of the stability

conditions. Intuitively, for a fixed type t ∈ TE with unmatched agents, one can bound

αkt by using condition (IM) in Proposition 3: minj∈t∩M(k) εkj ≥ −αkt ≥ maxj∈t∩U εkj . To

apply this bound, we just care about the projection onto the k-th coordinate of the points

εj with j ∈ M(k) ∪ U . The main analytical challenge we face is that the these relevant

subregions are themselves a random function of the market realization, as both M(k) and U

are themselves random sets. We overcome this by appropriately defining the region Rk(t, δ)so that it only contains points corresponding to agents in M(k) ∪ U . Once we have done

that, it should be easy to see that minj∈t∩M(k) εkj − maxj∈t∩U εkj is upper bounded by the

maximum distance between two consecutive points in Rk(t, δ), when projected onto their

k-th coordinate (the corner cases of all points being in M(k), or in U , turn out to be easy

to handle). This becomes precise once we introduce the set Vk(t) and the random variable


V k(t, δ). Analogously, the regionsRk(t) (for appropriate k) andRk1,k2(t, δ) (for appropriate

k1, k2) allow us to apply the conditions (IM) and (ST) respectively, to bound the variation

of α’s associated with a type t. These relationships are more involved, so the explanation

is delayed to the proofs.

Using the above notation, we now define the two events that will help us prove the

results:

B1(t, δ) =

max(

maxk∈K

V k(t), max(k1,k2)∈K(2)

V k1,k2(t, δ))≤ f1(nt,K)

, (4.10)

for some f1(nt,K) = O∗(1/n1/Kt ) defined in Lemma 6, δ ∈ [0, 1/2] and where K(2) =

(k1, k2) : k1, k2 ∈ K, k1 6= k2. (If K = 1, then K(2) is the empty set ∅ in which case we

follow the convention that max∅[ · ] = −∞.). In addition,

B2(t, δ) =

maxk∈TL

V k(t, δ) ≤ f2(nt)/δK−1

(4.11)

for some f2(nt) = O∗(1/nt) defined in Lemma 7 and δ ∈ (0, 1].

The proof of all lemmas auxiliary to the proof of Theorem 9 assume that these events

(or some subset of them) occur. As shown by the next result (proved in Appendix B.1),

that assumption does not pose a problem as these events simultaneously occur with high

probability.

Lemma 3. There exists C = C(K,Q) < ∞ such that, for any δ = δ(n) ∈ (0, 1/2], the

event⋂t∈TL∪TE (B1(t, δ) ∩ B2(t, δ)) occurs with probability at least 1− C/n.

4.4.2.2 Statements of the key lemmas

For every type t ∈ TL ∪ TE , we define ϑ(t) as ϑ(t) = k ∈ TL : N(k, t) > 0 when t ∈ TEand ϑ(t) = q ∈ TE : N(t, q) > 0 when t ∈ TL. That is, ϑ(t) is the set of neighbors of t in

the graph G(M). Recall that, given a type t ∈ TL ∪ TE we denote by D(t) the dimension

of the productivity vector of agents of type t. That is, D(t) = K if t ∈ TE and D(t) = K if

t ∈ TL.

Lemma 4. Consider the unique maximum weight matching M and a type t ∈ TL ∩TE . Let

F1(t) be the event


F1(t) = t is marked in G(M) and at least one agent in t is matched, (4.12)

that is, t has at least one unmatched and one matched agent. Let the events B1(t, δ) and

B2(t, δ) be as defined by Eqs. (4.10) and (4.11) respectively. Under F1(t)∩B1(t, δ)∩B2(t, δ),

we have

maxt′∈ϑ(t)


t,t′)≤ max

(f1(nt, D(t)) + δ, f2(nt)/δ

D(t)−1),

where f1 and f2 agree with those in the definitions of events B1(t, δ) and B2(t, δ) respectively.

Lemma 5. Consider the unique maximum weight matching M and a type t ∈ TL ∩TE . Let

F2(t) be the event

F2(t) = all agents in t are matched.

Let the event B1(t, δ) be as defined by Eq. (4.10). Under F2(t)∩B1(t, δ), for every t∗ ∈ ϑ(t)

we have maxt′∈ϑ(t)


t,t′

)≤(αmaxt,t∗ − αmin

t,t∗)

+ 2f1(nt, D(t)) + 2δ, where f1 agrees

with the one in the definition of B1(t, δ).

Using the simple lemmas defined above, we provide a sketch of proof that, together with

the explanation in Section 4.4.1, should suffice to roughly convey the idea while avoiding

the technical details. As a reminder, the complete proof of the upper bound in Theorem 9

can be found in Appendix B.2.

Let n∗ = mint∈TL∪TE nt and let δ = 1/(n∗)1/max(K,Q). Under Assumption 2, we have

that n∗ = Θ(n) and therefore δ = Θ(1/n1/max(K,Q)

). Furthermore, now f1(nt, D(t)) + δ

and f2(nt)/δD(t)−1, 2f1(nt, D(t)) + 2δ as defined in the statements of Lemmas 4 and 5 are

all O∗(

1n1/max(K,Q)

). Using this choice of δ together with the inductive argument outlined

in Section 4.4.1, we show that under the event⋂t∈TL∪TE (B1(t, δ) ∩ B2(t, δ)) we must have

max(k,q)∈TL×TE ,N(k,q)>0

(αmaxkq − αmin

kq

)≤ O∗

(1/n1/max(K,Q)

).

4.4.3 Proof of the lower bound

Our lower bound follows from the following proposition, proved in Appendix B.3.

Proposition 4. Consider a sequence of markets (indexed by n) with |TL| = K types of

labor, with n workers of each type, and a single type “1” of employers, with (K − 1)n + 1


employers of this type. (Assumptions 2 and 1 are satisfied.) Set u(k∗, 1) = 0 for some

k∗ ∈ L, and u(k, 1) = 3 for all k ∈ L\k∗. For this market, we have E[C] = Ω∗(1/(n1/K)).

Note that the sequence of markets described can easily be “dressed up” to fill in the

gaps in market sizes4 and to accommodate Q ≤ K types of firms5. If Q > K, we simply

swap the roles of workers and firms in our construction, leading to E[C] = Ω∗(1/(n1/Q)) as

needed. Thus, the lower bound in Theorem 9 follows from Proposition 4.

The rough intuition for our construction in Proposition 4 is as follows: For our choice

of u’s it is not hard to see that all workers of types different from k∗ are always matched in

the core. One employer j∗ is matched to a worker of type k∗. Suppose vector (αk)k∈TL is in

the core. Given that all types k 6= k∗ are a priori symmetric, we would expect that the αk’s

for k 6= k∗ are close to each other (we formalize using Lemma 15 that they are usually no

more than δ ∼ 1/√n apart). Assuming this is the case, we can order employers based on

Xj = maxk 6=k∗ εkj − εk∗j , and j∗ should usually be the employer with smallest Xj , since this

employer has the largest productivity with respect to k∗ relative to the other types. Now,

the Xj ’s are i.i.d., and a short calculation establishes that the distance between the first

and second order statistics of (Xj)j∈E is Θ(1/n1/K). This “large” gap between the first two

order statistics allows for (αk∗ , (αk + θ)k 6=k∗) to remain within the core for a range of values

of θ ∈ R that has expected length Θ(1/n1/2) for K = 2 and Θ(1/n1/K)−Θ(δ) = Θ(1/n1/K)

for K > 2, leading to the stated lower bound on C.We remark that the key quantity here, the gap between the first two order statistics of

(Xj)j∈E , is determined by the tail behavior (both the left and right tails) of the ε’s, along

with the number of types K. See Section 4.5 for further discussion.

4Here n = (2K − 1)n+ 1 for n = 1, 2, . . . but intermediate values of n can be handled by having slightly

fewer workers of type k∗, which leaves our analysis essentially unaffected.

5Let each worker type q 6= 1 have n agents each and u( · , q) = −2. These workers are always unmatched,

leaving the core unaffected.


4.5 Discussion

This chapter quantifies the size of the core in matching markets with transfers, as a function

of market characteristics. We considered a model of an assignment market with a fixed

number of types of workers and firms. We modeled the value of a match between a pair

of agents as a sum of a deterministic term determined by the pair of types, and a random

component which is the sum of two terms, each depending on the identity of one of the

agents and the type of the other. Under reasonable assumptions, we showed that the size

of the core is bounded as O∗(1/n1/`), where each side of the market contains no more than

` types.

Our work answers some questions but raises several others. One question is what hap-

pens if the random productivity terms are drawn from unbounded distributions. For the

market we construct for our lower bound, the core size is determined by the tail behavior

of the random productivities, cf. Section 4.4.3, suggesting that the core could be larger in

worst case if the productivities have an unbounded distribution.

On the other hand, it is of interest to understand the core in typical/average case

markets, as opposed to worst case markets. Our bound of O∗(1/n) for the special case of

only one type of employer and Θ(n) more employers than workers (a corollary of Theorem

10) does not depend on the number of worker types, in contrast to our general bound, which

implies that a relatively larger core can result in worst case from having more types. How

does the core size depend on the number of types in typical/average case markets?

It would be interesting to extend our results to many-to-one markets, where employers

can each have more than one opening. We expect that our results regarding the core (also

our proofs) extend to the case where each employer has capacity bounded by a constant,

and employer utility is additive across matches.

83

Part II

Auction Markets

CHAPTER 5. PROCUREMENT MECH. FOR DIFFERENTIATED PRODUCTS 84

Chapter 5

Procurement Mechanisms for

Differentiated Products

5.1 Introduction

In the last two decades, governments incorporated a new type of procurement mechanism to

acquire goods and services. Instead of each public organization (schools, hospitals, etc.) be-

ing in charge of their own purchases, the central procurement agency selects an assortment

of differentiated products through competitive bidding. Then, whenever a public organi-

zation needs to make a purchase, it buys its most preferred product from the assortment

The rationale behind adopting such a procurement mechanism is to be able to exploit the

purchasing power of a big central buyer (in this case, the central government), while still

providing the heterogeneous organizations with some flexibility to select the product that

best adapts to their needs. These mechanisms, known as framework agreements (FAs), are

used worldwide to acquire both goods and services in a wide range of categories such as

food, office supplies, computers, and medical services. As an example, in 2010 the European

Union awarded e80 billion using FAs, accounting for 17% of the total value of all public

procurement [European Commision, 2012].

In more detail, a FA roughly works as follows. First, the central government specifies a

broad category (e.g., computers), and a succinct description of products and/or services that

are needed within the category (e.g., laptops of certain size and specifications). Suppliers


are allowed to submit bids for any product fitting the description. Then, an auction-type

mechanism is run to select an assortment of differentiated products with posted unit prices.

Once the government decides on the winning bids, the public organizations buy their most

preferred product at the agreed price as needed, without undergoing any additional public

tendering process.

In these agreements, a main challenge for the government is how to account for the

heterogeneous preferences of the organizations. For example, while a public school may

want to buy laptops with attractive graphics features, the department of treasury may need

laptops with high processing power. In addition, some patients might find a prosthesis

of a certain brand to be more comfortable than that of a competing brand, while for

other patients it might work the other way around. Different organizations buying from

the food FA might also have different needs, such as dietary constraints (e.g., hospitals and

environments with kids). In all these cases, the government has a direct interest in providing

variety to its organizations. The main objective of this chapter is to provide insights on

how to achieve (some) variety in a cost efficient way.

In particular, this work is one of the first in the literature to provide a formal economic

analysis of this type of procurement mechanisms. Our contribution is three-fold: we first

introduce a model for the problem faced by the procurement agency, we then characterize

the optimal mechanism for this setting and, finally, we use these results to study the design

of simpler mechanisms that are commonly used in practice. While our main motivation is to

improve our understanding of FAs, these results also shed light on buying mechanisms used

in many real-world settings to construct assortments of differentiated products to satisfy

the demand arising from heterogeneous consumers. Examples include medical formularies

and group purchasing in the healthcare industry (see, for example, [Truong, 2014]). We

describe our main contributions in more detail next.

Our first main contribution is introducing a model capturing the following fundamental

trade-off faced by a procurement agency when buying differentiated products. On one hand,

consumers buying from the assortment usually have heterogeneous preferences. Therefore,

increasing product variety in the assortment may increase consumer satisfaction, as it be-

comes more likely that consumers will find a better product for their needs. On the other


hand, price competition in the auction stage may be depressed if too many products are

included in the assortment. Intuitively, the market share of each product decreases as more

products are included in the assortment, and hence the revenue generated by each product is

expected to decrease; this might provide incentives to suppliers to increase their bids, so as

to compensate for the lost revenue. Our model extends the classic auction and mechanism

design models to study this trade-off between product variety and price competition.

In our model, there is a set of risk-neutral suppliers offering differentiated products,

which are imperfect substitutes of each other. In the tradition of the auctions literature,

we assume that suppliers have private information about their costs. The central procure-

ment agency (designer) uses an auction-type mechanism to determine a menu, that is, an

assortment of differentiated products together with the unit prices. Then, consumers with

private heterogeneous preferences buy their most preferred alternative in the menu, which

induces aggregate demands over products. In the tradition of the assortment literature, we

assume that the aggregate demands as functions of the assortment and prices are common

knowledge, and are an input to the model. Hence, given the demand model, the designer

chooses a mechanism with the objective of maximizing expected consumer surplus; this

objective captures both the value derived from the characteristics of the products being

consumed as well as the importance of low prices.

Our second main contribution is the characterization of the optimal direct-revelation

posted-price mechanism for a broad class of affine demand models. This class includes the

classic horizontal Hotelling demand model and a pure vertical demand model as particular

cases, as well as more general specifications with both horizontal and vertical sources of

product differentiation. Affine demand models are commonly used in competition models

(e.g., [Vives, 2001]) and we think they provide a reasonable balance between tractability

and generality in our setting. Generally, the optimal mechanism may optimally choose to

restrict the entry of some products to the assortment, decreasing expected payments to

suppliers at the expense of reducing variety for consumers. In more detail, the optimal

mechanism typically restricts the entry of close-substitute products to the assortment by

selecting only one or few products from that set; this induces more price competition among

suppliers, without damaging much variety. On the other hand, if a product is perceived by


consumers as “unique” (not easily substitutable by other product), then such a product will

typically be added to the optimal assortment even if the cost is not very competitive, so as

to improve market coverage. The characterization of the optimal mechanism allows us to

formally quantify this optimal trade-off between variety and price competition in terms of

suppliers’ costs, product characteristics, and substitution patterns across products.

Relative to the traditional mechanism design problem, a distinctive feature of our for-

mulation is that the auctioneer cannot directly decide how to allocate demand across the

products. Instead, the auctioneer selects the menu and demands are then determined by

the underlying preferences of the organizations. This difference introduces significant com-

plexities in the analysis of the problem, and makes the analytical characterization of the

optimal mechanisms harder to obtain. In addition, most of the previous work in auction

and mechanism design assumes homogeneous products (with some notable exceptions dis-

cussed in Section 5.2). Our work advances the theory of auctions and mechanism design by

accounting for an endogenous demand system for differentiated products.

Our third main contribution is to improve our understanding of the performance of cer-

tain type of mechanisms used in practice. The optimal mechanisms previously characterized

are rarely implemented in applications due to their complexity, as they would require of-

fering a menu of contracts to each potential supplier. However, they serve as a powerful

tool to study practical mechanisms: optimal mechanisms provide a benchmark on what is

achievable, and their structure provide insights on how to improve current practice. We

are particularly interested in the type of FAs run by our collaborator in this project, the

Chilean government procurement agency (Direccion ChileCompra), which in 2013 bought

US$2 billion worth of goods using FAs.1

An important observation that arises by looking at the data from ChileCompra’s FAs

is that, because product definitions are narrow and auctions for different products are run

independently, there is a single supplier bidding and winning for many products. Hence,

while these suppliers may compete for demand once in the assortment, there is little to

none competition for the market (i.e., at the auction stage). We study whether the current

1This represented a 21% of the value of all public procurement in Chile [Area de Estudios e Inteligencia

de Negocios, Direccion ChileCompra, 2014].


FA performance can be improved by creating thicker markets, making imperfect substitute

products compete to be in the menu. To this end, we provide an extensive theoretical anal-

ysis of the current implementation of ChileCompra’s FAs in a simple model. Then, using

the insights gained from the optimal mechanism, we explore possible changes to ChileCom-

pra’s FA implementation with regards to the set of suppliers to include in the menu. We

show how, in general, using rules that restrict the entry of close substitute products can

significantly improve performance. Intuitively, this rules increase price competition across

suppliers, which translates into a significant decrease in prices, and an increase in expected

consumer surplus. We provide a detailed analysis that illustrates when it is profitable to

restrict the entry as a function of the market primitives. Overall, our results show that

simple modifications to current practice can induce a more aggressive price competition

across suppliers, which translates into a significant increase in performance.

The rest of the chapter is organized as follows. Section 5.2 describes related literature.

In Section 5.3 we formulate the mechanism design problem faced by the designer. In Sec-

tion 5.4, we describe the general solution approach that we use to solve for the optimal

mechanism. In Section 5.5, we characterize the optimal mechanism for affine demand mod-

els. In Section 5.6, we discuss the design of practical mechanisms using ChileCompra as a

case study. We conclude and provide extensions in Section 5.7. All proofs are deferred to

Appendix C.

5.2 Related literature

Our work is related to several streams of literature in economics and operations. As pre-

viously mentioned, our work extends classic work in mechanism design in the tradition of

[Myerson, 1981] by considering an endogenous demands system; this difference adds sig-

nificant challenges when solving for the optimal mechanism. Furthermore, in our problem

the designer maximizes consumer surplus —as opposed to just minimizing payments to

suppliers—, which also depends on the underlying preferences of consumers.

Our work is also related to the oligopoly pricing models that studied the effects of entry

and competition in consumer surplus (e.g., [Tirole, 1988]). The main difference is that,


in our setting, the decision to enter the market is not freely made by firms. Instead, it is

decided by the designer based on the information elicited in the auction. Further, in our

setting there is asymmetric information about firms’ costs.

In that sense, our work is more related to previous papers in procurement and regulation

economics. For example, [Dana and Spier, 1994] studies how to allocate production rights

to firms that have private cost information. An important insight of theirs is that the

optimal market structure may depend on the firms’ bids, which is similar to our result that

the optimal allocation depends on suppliers’ cost declarations. However, their auction only

determines the market structure and lump-sum fees, as opposed to our case in which unit

prices are determined. Similarly, [Anton and Gertler, 2004] and [McGuire and Riordan,

1995] study the optimal mechanism with an endogenous market structure in a Hotelling

model of product differentiation. However, unit prices are not part of the mechanism, and

allocations are determined by the designer and not endogenously by a demand system like

in our case. A general insight of this body of work is that the designer may single-source

more frequently if firms have private cost information, to be able to exert more pressure on

efficient suppliers to reveal their costs; this is similar to some of our insights.

Closer to our work, [Wolinsky, 1997] studies a spatial duopoly model where firms firms

compete in both prices and quality. While the paper considers an endogenous demand, the

analysis is restricted to solutions in which both firms have positive demands. Instead, we

are particularly interested in solutions in which some firms may be left out of the assortment

to induce more competition. In fact, the assortment in the optimal solution in our model

typically does not contain all suppliers.

Another stream of related work that considers endogenous market structures is that of

split-award auctions or dual sourcing in economics and operations [Chaturvedi et al., 2014;

Li and Debo, 2009; Elmaghraby, 2000; Riordan and Sappington, 1989; Anton and Yao,

1989]. These papers do not not assume an underlying set of heterogeneous consumers as

we do; instead, purchases are decided by the auctioneer.

Our work is also related to the operations literature studying assortment planning deci-

sions [Kok et al., 2009]. In these settings, decisions are made by one retailer that carries all

products, and has full information regarding the costs. In our case instead, an assortment


is built using an auction that elicits private cost information from many different suppliers.

Our analysis of ChileCompra’s FAs in Section 5.6 is closely related to the idea of us-

ing a Demsetz auction [Demsetz, 1968] to introduce competition for the market. This

section is also related to papers in group buying that show that committing to an ex-

clusive purchase from a single seller can be convenient for the group, even if the mem-

bers have heterogeneous preferences, because this can reduce buying prices [Dana, 2012;

Chen and Li, 2013]. However, these papers study models of complete information with

suppliers that share the same marginal costs. Our analysis extends theirs to an auction

setting with asymmetric information.

Finally, only two prior papers study framework agreements (FAs), which is one of the

main objectives of our work. [Albano and Sparro, 2008] consider a Hotelling model of

horizontal differentiation, in which firms are located equidistantly and the subset of potential

suppliers with lowest bids are selected in the assortment. In our case, we consider a richer set

of rules in which the assortment can depend on product characteristic or location. Further,

their analysis assumes complete information about firms’ costs. [Gur et al., 2013] consider

a model of FAs that studies the cost uncertainty faced by a supplier over the FA time

horizon when selling a single-item, but does not consider multiple differentiated products

nor heterogeneous consumers.

Overall, to the best of our knowledge, our work is the first to study optimal buying

mechanisms in an asymmetric information setting, with an endogenous market structure,

endogenous demand, and in which prices are determined in the auction.

5.3 Model and Problem Formulation

In this section, we present our model and a formulation of the auctioneer’s problem as a

mechanism design problem.

5.3.1 Model

We introduce a model of procurement mechanisms for differentiated products demand sys-

tems. The agents of the model are (i) an auctioneer (or designer); (ii) suppliers (or agents);


and (iii) consumers. The designer runs an auction-type mechanism to construct a menu

(i.e., an assortment of products with posted prices) based on the suppliers’ offers. Then,

consumers purchase their most preferred product from this menu at the agreed price. We

describe the main elements of the model next.

5.3.1.1 Suppliers

There is an exogenous set N of n potential suppliers indexed by i. Suppliers offer differ-

entiated products that are imperfect substitutes to each other; the characteristics of these

products are common-knowledge. To simplify the exposition, we initially assume that each

supplier offers exactly one product. Hence, unless otherwise stated, firms and products

share the same indexes. In Section 5.7 and Appendix C.4, we discuss the extension to the

multiproduct setting; it is worth highlighting that our main results also hold under this

extension. We assume suppliers are risk-neutral, so they seek to maximize expected profits.

Following the tradition in the auctions’ literature (see, e.g., [Krishna, 2009]), we assume

that suppliers have production costs drawn independently from common-knowledge distri-

butions, whose realizations are the private information of each supplier. Formally, supplier

i has a private cost θi ∈ Θi, associated to producing one unit of its product, where Θi is a

finite set of strictly positive real numbers. We index the elements of Θi, such that θji < θki

whenever j < k, for all θji , θki ∈ Θi. We say that supplier i is of type θi if his cost is θi.

Let fi be a probability mass function over Θi, where fi(θi) represents the probability that

supplier i is of type θi. Let Fi(θji ) =

∑k≤j fi(θ

ki ) be the cumulative probability distribution.

Let Θ = ΠiΘi denote the type space.2 Because suppliers’ types are independent, the joint

probability of θ = (θ1, . . . , θn) is equal to f(θ) = Πni=1fi(θi). We denote the probability

that all suppliers other than i have type θ−i by f−i(θ−i).3

We assume that suppliers have constant marginal costs of production and do not face ca-

pacity constraints. Therefore, the products included in the assortment are always available

and their production costs do not depend on the quantity demanded. These assumptions

are typically reasonable in many settings we have in mind, as usually the quantities that

2We use discrete type distributions for technical convenience as we explain in Section 5.5.1.

3We use boldfaces to denote vectors and matrices throughout the chapter.


suppliers sell through FAs represent only a small fraction of their total production.

5.3.1.2 Consumers

In the tradition of the assortment literature (e.g. [Kok et al., 2009]) and the work in

oligopoly pricing (e.g. [Tirole, 1988]), we assume that aggregate demand functions are

common knowledge and an input to our model. We introduce the following assumption.

Assumption 3 (Demand system). Suppose that, from the set of potential suppliers N , we

fix a subset Q ⊆ N of suppliers to be in the assortment. Let pQ = pii∈Q, be the vector

of their unit prices. Then, for every set Q and vector pQ, we assume that the vector of

demand functions is given by:

d(Q,pQ) = di(Q,pQ)i∈Q, (5.1)

where di(Q,pQ) denotes the expected demand for product i under assortment Q and prices

pQ and is common knowledge. We assume di(Q,pQ) = 0 for i /∈ Q, and∑

i∈Q di(Q,pQ) =

1, for all Q and pQ.

Note that the demand functions d(Q,pQ) depend on the prices and the characteristics of

all the products in the assortment. We assume that total demand for products in the

assortment is normalized to one, which essentially amounts to assuming that there does

not exist an outside option. However, our results extend to the case in which each product

(or a subset of them) is also offered by an outside supplier at a given price. Further, in

Section 5.7 and Appendix C.4.2, we discuss an extension to the case of elastic demand.

The assumption of a known demand system is plausible in the contexts discussed in the

introduction, because a demand system can typically be estimated using available historical

data or consumer surveys ([Ackerberg et al., 2006]). We note that we assume the designer

is able to predict aggregate demands for every fixed set of products and prices; however,

preferences of a specific consumer may be private information.

We will assume that the auctioneer will maximize consumer surplus when solving for

the optimal mechanism. Hence, we will also need a consumer surplus function as an input

to our model. Given a demand function, the study of the ‘integrability problem’ provides

conditions under which the demand function can be derived from the maximization of a


single utility function (see, e.g., [Mas-Colell et al., 1995] and [Anderson et al., 1992]). For

all demand systems that we consider in this chapter, this utility function corresponds to the

consumer surplus function. We formalize this in the following assumption. Let CS(d,p) be

the consumer surplus for demand quantities d and prices p.

Assumption 4 (Consumer Surplus). The expression for consumer surplus must satisfy for

all p:

(d1(N,p), . . . , dn(N,p)) ∈ argmaxx CS(x,p) , (5.2)

s.t.

n∑i=1

xi = 1, xi ≥ 0 ∀i ∈ N .

In addition, we require that for all i ∈ N , there exists a function ki(d) of the quantities

demanded such that:

CS(d,p) =n∑i=1

[ki(d)− pidi] , (5.3)

that is, consumer surplus is quasi-linear.4

The assumption states that (1) the quantities demanded given prices p when all products

are part of the assortment maximize consumer surplus given those prices5; and (2) that

consumer surplus is separably additive in expenditure and the gross surplus associated to

each product i, which is a function of the vector d. We emphasize that Assumption 4 holds

for all demand models that are considered in the chapter.

A natural way of micro-founding an aggregate demand system and an associated con-

sumer surplus function is to start from a discrete choice model that describes individual

consumption decisions. See [Anderson et al., 1992] for a general discussion; [Armstrong and

Vickers, 2014] also provide a more specific discussion for the affine demand models used be-

low. To illustrate, we present a simple example of a Hotelling demand model of horizontal

differentiation with two suppliers and linear ‘transportation costs’.

4The latter assumption is useful to to solve the optimal mechanism design problem.

5Note that the solution of this maximization problem may set some of the demand quantities equal to

zero.


Example 1 (Hotelling model with two suppliers). Consider the unit interval as the product

space, with two potential suppliers located at the extremes of the interval. There is a con-

tinuum of consumers uniformly distributed on the product space. Each consumer demands

one unit of good and incurs transportation costs which are linear in the distance between

the consumer and the supplier. Consumer j located at `j derives the following utilities from

consuming from the set suppliers N = 1, 2:

uj1(p1) = − (δ`j + pi) and uj2(p2) = − (δ(1− `j) + p2) ,

where supplier 1 (resp. 2) is assumed to be located at 0 (resp. 1) and δ is the transportation

cost. As consumers are uniformly distributed on the [0, 1] segment, the aggregate demands

can be derived from individual utilities as follows:

d1(N,p) = max

0,min

1,p2 − p1 + δ

2δ

and d2(N,p) = max

0,min

1,p1 − p2 + δ

2δ

In addition, using the individual utilities we can derive the expression for consumer surplus:

CS(d,p) = −(δ

2

(d2

1 + d22

)+ p1d1 + p2d2

),

where the first terms represent the transportation costs and the latter terms the monetary

costs. Note that in this example ki(x) = − δ2d

2i , which is equivalent to the total transportation

cost incurred by those consumers buying from i.

5.3.1.3 Auctioneer

The role of the auctioneer is to select or design an auction-type mechanism to construct

the menu of products based on the suppliers’ offers. As previously mentioned, the menu

consists of a subset of suppliers and unit prices for their products. Once selected, the rules

of the auction are common-knowledge. The auctioneer is risk-neutral and her objective is

to maximize expected consumer surplus; this objective incorporates both variety considera-

tions and payments to suppliers. Note that achieving variety is a natural objective in many

relevant contexts, such as the case of a government buying food for different populations

(e.g., nut free cookies for schools and regular cookies for ministries), or when buying medical

drugs with different side effects.6

6We note that in other contexts, it may not always be in the auctioneer’s best interest to provide variety.

For example, a government may not be interested in providing too many options regarding certain products


5.3.2 Mechanism Design Problem Formulation

We provide a mechanism design formulation of the auctioneer’s problem. We consider

mechanisms implemented in Bayes Nash equilibria. By invoking the revelation principle,

we restrict attention to direct revelation mechanisms without loss of optimality. Hence,

for given cost declarations, the designer selects a menu which consists of an assortment of

products (or suppliers) and their unit prices. Formally, a direct revelation mechanism can

be specified by (a) the ‘assortment’ functions qi : Θ→ 0, 1 that are equal to 1 if and only

if supplier i is included in the assortment when cost declarations are θ; and (b) the price

functions pi : Θ→ R, where pi(θ) is the unit price for the item offered by supplier i when

cost declarations are θ. Note that this formulation allows for multiple suppliers to be in the

menu. We define q = (q1, ..., qn) and p = (p1, ..., pn). For given cost declarations θ, the menu

is given by (q(θ),p(θ)). We also define the allocation functions xi : Θ→ [0, 1], where xi(θ)

is the quantity allocated to supplier i when cost declarations are θ. Let x = (x1, . . . , xn).

For each realization of θ, given the menu (q(θ),p(θ)), consumer demand is determined

by the underlying demand system. Hence, for given (q,p), the allocation function x is

restricted by the demand constraints in Eq. (5.1). This is in sharp contrast with classic

mechanism design theory, in which the designer specifies a payment (or transfer) function

and an allocation function. In our case, the designer selects an assortment and unit prices

and, given these, allocations are decided by consumers. As discussed below, these con-

straints on the allocations introduce significant additional complexities to the mechanism

design problem.

In the optimal mechanism design problem, the designer maximizes its objective (in

our case, expected consumer surplus) subject to the usual constraints in mechanism design

theory: incentive compatibility (IC), individual rationality (IR), and feasibility of allocations

(Feas). To write these constraints, we define the interim expected utility for supplier i of

such as soft drinks or ink pens. In these cases, our results provide a way of evaluating the cost of incorporating

variety considering consumers’ idiosyncratic preferences, when perhaps the designer prefers to offer one (or

very few) products.


type θi and report θ′i as:

Ui(θ′i|θi) =

∑θ−i∈Θ−i

f−i(θ−i)( (pi(θ

′i,θ−i)− θi

)xi(θ

′i,θ−i)

), (5.4)

where θ−i is the report of supplier i′s competitors. In addition, the problem must also

have constraints to ensure that the allocations are consistent with the underlying demand

system (Demand). Using the above definitions, the auctioneer’s optimal mechanism design

problem can be formulated as follows:

[P0] maxq,p,x

Eθ[CS(x(θ),p(θ))]

s.t. Ui(θi|θi) ≥ Ui(θ′i|θi) ∀i ∈ N, ∀θi, θ′i ∈ Θi (IC)

Ui(θi|θi) ≥ 0 ∀i ∈ N, ∀θi ∈ Θi (IR)∑i∈N

xi(θ) = 1 ∀θ ∈ Θ, xi(θ) ≥ 0 ∀i ∈ N, ∀θ ∈ Θ (Feas)

xi(θ) = di(q(θ),p(θ)) ∀i ∈ N, ∀θ ∈ Θ, (Demand)

where di(·) correspond to the demand system introduced in Assumption 3. Note that we

abused notation to denote by q(θ) the set of suppliers that are in the assortment given

costs θ. In the next section we discuss our approach to solve the optimal mechanism design

problem P0.

5.4 General Solution Approach

Problem P0 is a mixed integer mathematical program. Further, even if one relaxes the inte-

grality of the variables q, the program is typically non-convex, because demand equations

are often non-linear even in simple cases (see Example 1). Our solution approach relies on

relaxing these demand constraints and solving the relaxed problem. The advantage of doing

this is that the relaxed problem admits an analytical solution, which can be obtained by

extending standard mechanism design arguments based on the envelope theorem [Myerson,

1981] adapted for the setting of discrete distributions [Vohra, 2011]. Further, the relaxed

optimal solution has an intuitive interpretation: it is how a central planner would allocate

demands across different suppliers, if she could dictate how consumers should behave (who

should they buy from) to maximize consumer surplus.


Then, we provide conditions that guarantee the existence of unit prices p that are con-

sistent with the optimal solution of the relaxed problem and satisfy the demand constraints.

If such prices p exist, the optimal solution to the relaxed problem can be achieved by the

original problem P0. More specifically, we need to find prices that will satisfy the incentive

compatibility and individual rationality constraints on the supplier side, and provide incen-

tives to consumers so that they behave as the central planner would want them to —the

aggregate demands under these prices will agree with the optimal allocations in the relaxed

problem. In other words, we show the existence of prices that allow us to decentralize the

solution to the relaxed (centralized) problem. We formalize this argument next.

First, we introduce a new set of variables ti : Θ→ R, where ti(θ) = pi(θ)xi(θ) represents

the total transfer (or payment) to supplier i for a given cost declaration θ. Relaxing the

demand constraints from [P0] and noting that interim utilities (Eq. (5.4)) can be written in

terms of total transfers t, we obtain the relaxed problem:

[P1] maxx,t

Eθ

[n∑i=1

[ki(x(θ))− ti(θ)]

]

s.t. Ui(θi|θi) ≥ Ui(θ′i|θi) ∀i ∈ N, ∀θi, θ′i ∈ Θi (IC)

Ui(θi|θi) ≥ 0 ∀i ∈ N, ∀θi ∈ Θi (IR)∑i∈N

xi(θ) = 1 ∀θ ∈ Θ, xi(θ) ≥ 0 ∀i ∈ N, θ ∈ Θ. (Feas)

where∑n

i=1 [ki(x(θ))− ti(θ)] is the expression for consumer surplus given by Eq. (5.3),

where we replaced the second term (price times demand) by transfers.

We highlight that problem [P1] only differs from the classic mechanism design formu-

lation in the objective function; while the traditional objective is to minimize expected

transfers, we aim to maximize expected consumer surplus. Similarly to the setting of con-

tinuous cost distributions, we introduce the following definition of the virtual cost function

for cost distributions with discrete support.

Definition 4 (Virtual costs). For θi ∈ Θi, let ρi(θi) = maxθ′ ∈ Θi : θ′ < θi, that is,

ρi(θi) is the predecessor of θi in Θi.7 Let vi(θi) = θi + Fi(ρi(θi))

fi(θi)(θi − ρi(θi)) be the virtual

cost of supplier i when he has type θi.

7If θi is the lowest in the support, we define ρi(θi) = θi.


We make the standard regularity assumption in mechanism design that we keep through-

out the chapter:

Assumption 5 (Increasing virtual costs). The function vi(θi) is strictly increasing for all

i ∈ N .

Finally, we also define the interim expected allocations and interim expected transfers as

follows:

Xi(θi) ≡∑

θ−i∈Θ−i

f−i(θ−i)xi(θi,θ−i),

Ti(θi) ≡∑

θ−i∈Θ−i

f−i(θ−i)ti(θi,θ−i).

The advantage of solving the relaxed problem [P1] is that we can extend standard

mechanism design arguments to characterize its optimal solution, as we formalize next.

Proposition 5. Suppose that (x, t) satisfy the following conditions:

1. The allocation function satisfies for all θ ∈ Θ,

x(θ) ∈ argmaxn∑i=1

(ki(x(θ))− xi(θ)vi(θi)) (5.5)

s.t.n∑i=1

xi(θ) = 1, xi(θ) ≥ 0 ∀i ∈ N .

2. Interim expected allocations are monotonically decreasing for all i ∈ N , that is,

Xi(θ) ≥ Xi(θ′) for all θ, θ′ ∈ Θi such that θ ≤ θ′.

3. Interim expected transfers satisfy for all i ∈ N and θji ∈ Θi:

Ti(θji ) = θjiXi(θ

ji ) +

|Θi|∑k=j+1

(θki − θk−1i )Xi(θ

ki ) (5.6)

Then, (x, t) is an optimal mechanism for problem P1.

The proof can be found in Appendix C.1. Condition (1) in Proposition 5 states that,

for each θ ∈ Θ, the optimal vector of allocations x(θ) must be a maximizer of the consumer

surplus when prices are set to be the virtual costs, subject to the feasibility constraints (see

Eq. (5.3)). Further, by Eq. (5.2), the optimal solution is of the form xi(θ) = di(N, v(θ)).


Therefore, optimal allocations in [P1] have an intuitive form: they coincide with the demand

functions given by Eq. (5.1) when the unit price of each supplier is exactly his virtual cost.

This follows because, like in classic mechanism design, the equilibrium ex-ante expected

payment that the auctioneer makes to a bidder is equal to the ex-ante expectation of the

virtual cost times the allocation.

It is important to note that, while the optimal demands are completely characterized,

the optimal transfers are not. The only constraint imposed on transfers by the optimal

solution is over interim expected transfers (Condition (3) in Proposition 5). As transfers

are equal to unit price times demand, this implies that the optimal prices in the relaxed

problem are underspecified. This freedom in the definition of optimal prices becomes useful

later on, when we characterize the optimal solution to the original problem.

To illustrate the result, consider Example 1 and suppose both suppliers have the same

cost distribution. Let θ1 and θ2 be the cost realizations of supplier 1 and 2 respectively.

In this case, the relaxed problem P1 yields an optimal allocation characterized by: (1) if

δ > |v(θ1) − v(θ2)|, the demand is split between the two suppliers with x1 = (v(θ2) −v(θ1) + δ)/(2δ) and x2 = (v(θ1) − v(θ2) + δ)/(2δ); and (2) if δ < |v(θ2) − v(θ1)|, all the

demand is awarded to the supplier with the lowest cost realization. Note that the decision

of whether to split or not the demand depends on the cost realizations. In particular, if

the transportation cost is small relative to the differences in virtual costs, then the optimal

solution includes only the supplier with the lowest virtual cost in the assortment. In this

case, it is worth paying the cost of having less variety in the assortment with the upside of

decreasing the expected payments to bidders. By restricting the entry to the assortment

in some scenarios, the auctioneer can reduce these expected payments while still providing

incentives for truthful cost revelation.

Because problem P1 is a relaxation of P0, the optimal objective of the former is an upper

bound on the optimal objective of the latter. The next corollary provides necessary and

sufficient conditions under which P0 indeed attains the optimal objective of P1.


Corollary 3. Let (x, t) be the unique optimal solution to the relaxed problem P1.8 Define

qi(θ) = 1 if and only if xi(θ) > 0, ∀i ∈ N, θ ∈ Θ. (5.7)

Suppose that for all θ ∈ Θ, there exist prices p(θ) such that

xi(θ) = di(q(θ),p(θ)) ∀i ∈ N, ∀θ ∈ Θ (5.8)

where di(p) is given by Eq. (5.1), and∑θ−i∈Θ−i

pi(θi,θ−i)xi(θi,θ−i)f−i(θ−i) = Ti(θi), ∀i ∈ N, ∀θi ∈ Θi , (5.9)

where, for all i ∈ N , Ti(·) is the expected interim transfer function given ti(·). Then, the

optimal objective of P0 is equal to the optimal objective of P1. Moreover, an optimal solution

of P0 is given by (q,p) characterized by Eqs. (5.7), (5.8), and (5.9), and the corresponding

optimal allocation x of P0. Furthermore, the optimal objective of P0 is equal to the optimal

objective of P1 if and only if such solution (q,p) exists.

The corollary suggests the following approach to solving the optimal mechanism design

problem. First, solve the relaxed problem, the solution of which has an appealing structure

—it gives us the solution a central-planner would choose to maximize consumer surplus.

Then, find unit prices that support the optimal relaxed solution. Equivalently, find prices

that allow to decentralize the optimal solution by making the aggregate demands under such

prices agree with the relaxed optimal allocations, while satisfying the individual rationality

and incentive compatibility constraints.9 We use this solution approach in the next section,

where we use it to solve the original problem for different classes of affine demand models.

5.5 Affine Demand Models

The optimal mechanism design problem takes an underlying consumer demand model as an

input. To obtain analytical solutions we will restrict attention to a general class of affine

8Problem P1 admits a unique optimal solution for all demand systems considered in the chapter. If P1

admits more than one solution, our arguments can easily be extended accordingly.

9Here we differ from the topic of decentralizing efficient allocations in competitive equilibria [Mas-Colell

et al., 1995], because we need to find prices that not only yield the desired allocations, but also provide

suppliers’ incentives for truthful revelation through the interim expected transfers.


demand models, that is, models in which for every set Q, demands d as given by Eq. (5.1)

are (piece-wise) affine functions of prices. The advantage of these models is that they admit

a convex and closed-form expression for consumer surplus. More broadly, affine demand

models capture a vast array of substitution patterns including both horizontal and vertical

dimensions of differentiation, while being generally tractable. For these reasons, they have

been extensively used in a variety of game-theoretic models within the operations literature

[Allon and Federgruen, 2007; Cachon and Harker, 2002; Federgruen and Hu, 2014].

It is easy to see that, even under affine demand models, the demand constraints are

piece-wise linear, and problem P0 remains non-convex.10 However, the approach described

above of relaxing these constraints will allow us to solve the problem.11

In the remainder of this section, we discuss the solution to the optimal mechanism

problem when we assume affine demand models. We first explain how to apply the general

solution approach introduced in Section 5.4 to affine demand models. Next, we characterize

the optimal mechanisms for specific linear demand models. We start by analyzing a popular

affine-demand model: the Hotelling model of horizontal differentiation. Then, we provide

the analysis of a general affine demand model that includes the Hotelling model (and a pure

vertical model) as particular cases.

5.5.1 Applying the Solution Approach to Affine Demand Models

We now discuss how to adapt the general solution approach described in Section 5.4 to

affine demand models. Let (x, t) be an optimal solution to the relaxed problem P1. By

10For instance, consider the simple Hotelling model described in Example 1. There, the demand constraints

for agent i ∈ 1, 2 should be expressed as xi(θ) = max

0,min

1,pj(θ)−pi(θ)+δ

2δ

with j ∈ 1, 2, j 6= i,

which yield a non-convex problem.

11An alternative to the class of affine demand models we use in this chapter would be to start with a

parametric discrete choice model, such as the multinomial logit model; these models are typically used

in the assortment literature. Unfortunately, the basic multinomial logit model is not appropriate for our

analysis because of its inability to capture substitution patterns due to the IIA property. An alternative

that overcomes this issue is the multinomial logit model with random coefficients; however, this model is

hard to solve even in the standard assortment problem, let alone in our auction setting. Another option

that is typically more tractable is the nested logit model (see [Li and Rusmevichientong, 2014]). It may be

worth studying in future work whether our framework can be applied to this demand system.


Proposition 5 and the discussion that follows the proposition, we have xi(θ) = di(N,v(θ))

where v(θ) is defined as the vector of virtual costs, i.e., v(θ) = (v1(θ1), . . . , vn(θn)). We

denote Q(θ) as the set of active suppliers (those with strictly positive demands) in the

optimal solution under cost realizations θ. To apply our solution approach, we must find

unit prices that simultaneously satisfy Eqs. (5.8) and (5.9).

Equations (5.8) require that unit prices p induce the optimal allocations x of P1 through

the demand system —as previously discussed, this is like decentralizing the allocations.

By Corollary 3, we need to find unit prices such that di(N, v(θ)) = di(q(θ),p(θ)) for all

i ∈ Q(θ) and θ ∈ Θ. As the demand function is assumed to be affine in prices, these

equations yield linear constraints in prices. Note that the equations are linear because

they require to find prices to generate a given vector of demands x. This imposes |Q(θ)|constraints over the prices p(θ), corresponding to firms with strictly positive demands.12

However, as the allocations must add up to one, one of these constraints is redundant; the

demands for |Q(θ)| − 1 suppliers determines the demand for the remaining active supplier.

Therefore, the equations in (5.8) impose |Q(θ)| − 1 constraints over prices p(θ). The

redundancy of one constraint plays an important role because it induces degrees of freedom

that can be used to satisfy the constraints on expected interim transfers.13

In addition, Eqs. (5.9) require that unit prices p induce the expected interim transfers

Ti in the optimal solution of P1 —that is, the solution is individually rational, incentive

compatible and expected payments to suppliers agree with those in the relaxed optimal

solution. Given an optimal mechanism for P1, (x, t), these equations are also linear in

prices. In particular, once the constraints in Eqs. (5.8) are imposed, the allocations are fixed

and equal to the optimal allocations of P1; therefore, the equations described in (5.9) are

linear in unit prices. Also, observe that if in the optimal solution we have xi(θji ,θ−i) = 0

for all θ−i ∈ Θ−i, then it must be that Ti(θji ) = 0. This follows by conditions (2) and

12 In all demand models considered in the chapter, only prices associated to suppliers with positive demand

appear in the demand equations. This property is natural: if a supplier has zero demand, then its price does

not play a role in the demand equations of competitors.

13The importance of these degrees of freedom is explicitly illustrated when we consider an elastic demand

(Appendix C.4.2).


(3) in Proposition 5. Hence, the previous equations impose∑

i∈N∑

θi∈ΘiI[∃ θ−i : i ∈

Q(θi,θ−i)] ≡ T constraints.

By the observations above, verifying whether OPT (P0) = OPT (P1) is equivalent to

establishing whether the linear system of equations defined by Eqs. (5.8) and Eqs. (5.9)

admits a solution. Let M and m be the coefficient matrix and the corresponding RHS

respectively defined by the linear equations in (5.8) and (5.9), where each column is asso-

ciated with a price pi(θ). We can safely discard the columns corresponding to prices pi(θ)

such that i /∈ Q(θ), as all the coefficients of such columns are zero. The resulting matrix

M will have∑

θ∈Θ |Q(θ)| columns and∑θ |Q(θ)| − |Θ|+ T rows. It is easy to verify that

T ≤ |Θ| and, therefore, the number of columns is larger or equal to the number of rows.

By the Rouche-Frobenius theorem, a system of linear equations Mp = m is consistent

(has a solution) if and only if the rank of its coefficient matrix M is equal to the rank of its

augmented matrix [M |m]. Note that whenever the rows of M are linearly independent the

system is trivially consistent. In the remainder of this section we show that (under additional

conditions) we can guarantee that the associated system of equations is consistent. Hence,

we can characterize the optimal mechanism.14

5.5.2 Optimal Mechanism for Hotelling Demand Model

Having described the general solution approach, we now discuss the structure of the optimal

mechanism when the consumer demand is given by a Hotelling model. This will allow us

to provide intuition on the structure of the optimal mechanism, before discussing the more

general affine demand models in Section 5.5.3. Recall that a simple version of the Hotelling

model was introduced in Example 1.

We now briefly discuss a general Hotelling demand model with an arbitrary number

n of suppliers in the unitary segment. The n potential suppliers are located at 0 ≤ `1 <

`2 < . . . < `n ≤ 1 respectively; the location represents the horizontal characteristic of the

product offered by the supplier relative to the product space. The closer two suppliers

14Assuming discrete types allow us to work with finite dimensional system of equations and to use finite

dimensional linear algebra. In the continuous type setting, we would have to deal with an infinite dimension

space for price variables, and the results would be more technically involved.


are in the product space, the closer substitutes the products they offer are. The locations

of the suppliers are assumed to be common-knowledge. A continuum of consumers, all of

whom must buy one unit of product, are distributed on the product space. To simplify

the exposition, we assume that consumers are uniformly distributed. However, our results

can be easily extended to arbitrary distributions. The utility consumer j obtains from

buying the product offered by i is given by: uji(pi) = − (δ|ì − `j |+ pi), where δ is the

transportation cost and `j is the position of consumer j in the unit line.

Suppose that suppliers have fixed unit prices p = pii∈N . Then, the set of active sup-

pliers with positive demand is given by Q(p) = i ∈ N : pi ≤ mink 6=i pk + δ|`k − ì|,where we abused notation to make the set depend on prices instead of costs. In words,

supplier i will be active if his price is lower than the the total price (unit price plus trans-

portation cost) a consumer at ì will pay if he buys from any other supplier. In this case,

the consumers located in a neighborhood of ì choose to buy from supplier i.

For unit prices p and supplier i ∈ Q(p), let %p(i) (resp. ϑp(i)) denote the supplier

preceding (resp. following) i in Q(p), that is, %p(i) = max j ∈ Q(p) : j < i and

ϑp(i) = min j ∈ Q(p) : j > i. Also, let ι(Q(p)) (resp. η(Q(p))) denote the rightmost

(resp. leftmost) supplier in Q(p). Then, the aggregate demand for product i is given by:

di(p) =

0 if i /∈ Q(p)

ì + 12δ

(pϑp(i) − pi + δ(`ϑp(i) − ì)

)if i = η(Q(p))

12δ

(p%p(i) − pi + δ(ì − `%p(i))

)+ if i ∈ Q(p), i 6= η(Q(p)), ι(Q(p))

12δ

(pϑp(i) − pi + δ(`ϑp(i) − ì)

)12δ

(p%p(i) − pi + δ(ì − `%p(i))

)+ (1− ì) if i = ι(Q(p))

(5.10)

In the Hotelling model, suppliers split the market with their immediate active neighbors.

The equations above can be easily derived by determining the location of the indifferent

consumer between two active neighboring suppliers. Note that the segment between two

consecutive active suppliers i and j (that is, the segment between ì and `j) is divided

proportionally to their prices: i will obtainpj−pi+δ|`j−ì|

2δ and j the rest.

For the Hotelling model, the optimal solution to the relaxed problem P1 is intuitive.

By Proposition 5 and the discussion that follows, the optimal allocations in the relaxed


problem P1 for a cost realization θ are given by the demand characterization (Eq. (5.10))

with prices equal to the vector of virtual costs v(θ). In addition, for a given vector of cost

realizations θ, the optimal assortment is characterized by

Q(θ) = i ∈ N : vi(θi)− vj(θj) ≤ δ|`j − ì| ∀j ∈ N ,

which corresponds to the definition of active suppliers where prices are replaced by virtual

costs.

Similarly to the Hotelling example with two suppliers, the auctioneer may optimally

restrict participation of bidders in the assortment to decrease expected payments. In par-

ticular, if two products are close substitutes (i.e., δ|`j − ì| is relatively small15), then the

optimal assortment will typically contain only the product with the lowest virtual cost. By

doing so, the optimal mechanism is able to allocate more demand to the suppliers with low-

est virtual costs, thus reducing the expected prices. On the other hand, when two products

are not close substitutes (i.e., δ|`j− ì| is relatively big), then the (virtual) cost of one prod-

uct is less likely to affect whether the other product is included or not in the assortment.

Therefore, if products are not close substitutes, it is usually better to have both of them in

the assortment to offer more variety to consumers.

By Corollary 3, if we can find a feasible pair (q,p) for P0 such that the conditions of the

corollary are satisfied, then we have found an optimal solution for the original problem; this

solution will have exactly the same intuitive interpretation as the relaxed solution, because

the assortment, allocations and expected payments agree. Therefore, we now study in which

cases it is possible to achieve the same optimal objective in both the original problem and

the relaxed problem, that is, in which cases OPT (P0) = OPT (P1).

Optimal solution to original problem. Consider the optimal solution of the relaxed

problem as described by Proposition 5. Let q be defined as Corollary 3, that is, qi(θ) = 1

if i ∈ Q(θ) and qi(θ) = 0 otherwise. By comparing the Hotelling demands as described by

Eq. (5.10) with the optimal allocations of P1 as defined in Proposition 5, it should be clear

15Note that whether two products are close substitutes or not depends on both their relative distance in

the product space, |`j− ì|, as well as how much relative weight consumers assign to product characteristics,

summarized by the transportation cost δ.


that the constraints given by Eqs. (5.8) can be summarized as:

pϑθ(i)(θ)− pi(θ) = vϑθ(i)(θϑθ(i))− vi(θi) ∀θ ∈ Θ, i ∈ Q(θ), i 6= ι(θ). (5.11)

These constraints will implement the optimal allocations of P1 using prices p(θ). In words,

the difference in prices between adjacent active suppliers must be equal to the difference

in virtual costs. By Corollary 3, we must also guarantee that the expected transfers agree

with the optimal ones, that is, unit prices should satisfy the constraints given by Eq. (5.9).

Hence, if we can find a feasible pair (q,p) for P0 such that the optimal allocations for P1 can

be supported and the constraints on the expected interim transfers are maintained, then

we have found an optimal solution for the original problem. To show that the system of

linear equations is consistent, we exploit the fact that Eq. (5.11) imposes a very particular

structure on the coefficient matrix of the system.

We start by analyzing the setting in which suppliers have IID costs and are located at

equidistant intervals. Even in this context, the problem is asymmetric whenever we have

three or more suppliers, as the most central agent has an advantage to capture demand.

We have the following result.

Theorem 11 (IID costs). Consider the setting in which for all i ∈ N we have ì = i−1n−1

(agents are located at equidistant intervals), Θi = Θ and fi = f for some support Θ and pdf

f . Then, OPT (P0) = OPT (P1).

The proof of Theorem 11 can be found in the appendix. We show that there is no gap

between the optima of the original and the relaxed problem by showing that the system of

linear equations Mp = m is consistent.16

We now turn our attention to the more general case in which the cost functions are not

IID and locations are arbitrary. Unfortunately, as opposed to our result in Theorem 11, the

optima of both problems might not agree in the general case. This situation is illustrated by

Example 3 presented in Appendix C.2. Therefore, given that in general the optima of the

16Ideally, one would like to show that the rows of the coefficient matrix are linearly independent. However,

this need not be the case. Indeed, the reader can verify that in the simple case of n = 2, Θ = θL, θH and

δ ≥ 1f(θH )

(θH − θL) the rows of the associated matrix of coefficients are linearly dependent.


relaxed problem and the original problem may not agree, we next focus on providing suffi-

cient conditions under which OPT (P0) = OPT (P1). This is summarized by the following

theorem.

Theorem 12. Consider the general setting in which agents have arbitrary locations and

costs distributions. Let c∗ = min1≤i≤n−1(ì+1 − ì). Suppose that the following conditions

are simultaneously satisfied:

1. There is at least one profile θ ∈ Θ such that |vi+1(θi+1)− vi(θi)| ≤ δ(ì+1 − ì)/2 for

all i ∈ N ; and

2. |Θi| ≥ 3 for all i ∈ N , and for every i ∈ N and every θj ∈ Θi, we have vi(θj+1i ) −

vi(θji ) ≤ δc∗

4 .

Then, we have OPT (P0) = OPT (P1).

The complete proof of Theorem 12 can be found in Appendix C.5.17 In the proof,

we show that the rows of the associated coefficient matrix M are linearly independent

and, therefore, there must exist prices that support the optimal allocation and satisfy the

expected interim transfer constraints.

We now briefly discuss the intuition behind the conditions. The second condition essen-

tially requires the difference in the virtual costs between adjacent points in the support to

be bounded by a function of δ. The smaller the δ, the closer the virtual costs should be.

If we think of the discrete distribution as an approximation of an underlying continuous

distribution, then this is equivalent to require the discretization to be thin enough with

respect to δ. Intuitively, when the supports of the cost distributions are coarse, there are

fewer combinations of prices and therefore fewer price vectors. As a result, there are not

enough degrees of freedom to find prices that simultaneously satisfy the demand and the

expected interim transfers constraints.

The first condition is to require the existence of an ‘interior solution’. More precisely, we

impose the existence of a solution in which all n agents are active. We further require that

17In Appendix C.5 we prove a more general theorem. Then, we explain how the general theorem implies

Theorem 12.


there exist other solutions ‘close’ to that one (in which we replace the cost of an agent by

one of his adjacent costs) in which all agents are also active. Guaranteeing the existence of

several cost profiles for which all agents are active translates into a structural relationship

between the expected transfers constraints (Eq. (5.9)) of all the agents. Intuitively, as the

prices become more related with each other, there are more degrees of freedom to find prices

that satisfy both the optimal demand constraints and the expected transfer constraints.

The conditions imposed in Theorem 12 are not too restrictive. In fact, provided that

the discretization of the support of the costs distributions is thin enough (relative to δ),

Condition (2) will be satisfied. In addition, Condition (1) requires the existence of an

interior solution where all n agents active. Note that, whenever the discretization is thin

enough, this condition should be satisfied; otherwise, we can find an agent that is never

active and thus that agent can be removed from the problem. Further, Condition (1) also

requires the existence of other solutions ‘close’ to that one, in which all agents are also active.

Again, provided that the discretization is thin enough, this is achievable; as the difference

between adjacent virtual costs is very small (by Condition (2)), the set of active suppliers

will not vary if we replace the cost of a supplier by one of his adjacent cost. Therefore,

satisfying the conditions in Theorem 12 amounts to guaranteeing that the support of the

cost distributions we are considering is dense enough.

In Appendix C.3, we provide a related characterization and result for a classic model

of pure vertical differentiation. In this model products have different qualities on which

all consumers agree upon; however, consumers have different price sensitives. Here, the

auctioneer faces the trade-off between variety in terms of quality and prices. We show that,

under sufficient conditions that are similar to those in the Hotelling model, the optima

of the relaxed and original problem agree. Therefore, we are able to obtain a similar

characterization for the optimal mechanisms: allocations equal demands when prices are

replaced by virtual costs. Further, these allocations are functions of the ratio between the

difference in virtual costs and the difference in quality of the products in the assortment.


5.5.3 Optimal mechanisms for general Affine Demand models

So far we considered the classic models of demand for products that are horizontally (or

vertically) differentiated. We now study more general affine demand models, that allow

us to combine both vertical and horizontal sources of differentiation. An affine demand

function is one where the relation d(p) = α − Γp holds for all p ∈ p ∈ R : α − Γp ≥ 0.Here, α ≥ 0 represents a quality (or vertical) component; Γij represents the variation in

the demand of product i as a result of a unit change in the price of product j, when all

other prices remain constant. We assume that the products are substitutes, hence, Γij ≤ 0

for i 6= j. Note that the Hotelling model presented in the previous section and the vertical

model studied in the appendix are both particular cases of affine demand models.

For our purposes, it is important to consider the extension of this specification to price

vectors under which some products get zero demand, as introduced by [Shubik and Levitan,

1980] and further analyzed by [Soon et al., 2009]. We formalize this extension in our setting

in which demands must add up to one assuming that a single ‘representative consumer’

maximizes consumer surplus ([Farahat and Perakis, 2010] also use this approach to study

oligopolistic pricing models under affine demand functions).18

We consider a representative consumer with a strictly concave gross utility function

given by u(x) = c′x− 12x′Dx, where D is a positive definite matrix and D−1 is symmetric

positive definite. The vector c′ denotes the transpose of vector c. Here, D = Γ−1 and

c = Γ−1α have been renamed to avoid burdensome notation. The demand function is

defined as the solution of the representative consumer’s maximization problem, whose utility

also corresponds to consumer surplus. That is, for any p ∈ Rn, let d(p) be defined as the

solution of the following maximization problem:

maxx

c′x− 1

2x′Dx− p′x

s.t 1′x = 1

x ≥ 0

(LD(p))

18Alternatively, a general affine demand model can also be micro-founded using consumers’ individual

utilities like in the Hotelling and vertical models [Martin, 2009; Armstrong and Vickers, 2014]. However, we

think the representative consumer approach provides a cleaner analysis.


Clearly, Problem (LD(p)) has a unique solution for every p ∈ Rn, and thus the demand

function d(p) is well defined. To illustrate, we consider the following example:

Example 2. We consider a duopoly where α = (q1, q2) and Γ =( a1 −γ−γ a2

), with all the

parameters positive and with a1 + a2 ≥ 2γ. Under these paramerters, we have D =

1a1a2−γ2

( a2 γγ a1

)and c = 1

a1a2−γ2

( a2q1+γq2a1q2+γq1

). For any given p, the demand function d(p)

is defined as:

d1(p) = max

0, min

(a2 − γ)q1 − (a1 − γ)q2 + a1 − γ − (a1a2 − γ2)(p1 − p2)

a1 + a2 − 2γ, 1

and

d2(p) = max

0, min

(a1 − γ)q2 − (a2 − γ)q1 + a2 − γ − (a1a2 − γ2)(p2 + p1)

a1 + a2 − 2γ, 1

.

These demand functions exhibit natural properties; they are decreasing in a firm’s own price

and increasing in the competitor’s price. Also, depending on the price vector, there could

be one or two firms active. In the appendix, we show some additional properties for the

general case with an arbitrary number of firms. In particular, we show that demands can

be expressed as affine functions of prices of the set of active suppliers only. It is simple to

observe that any increase in price of a product with zero demand will not have an impact

on the demand function either. In addition, we show that demands only depend on price

differences but not on the actual prices. This freedom in setting unit prices is essential

to our proof technique as, similarly to the Hotelling case, we need to find unit prices that

satisfy the same differences induced by the virtual costs and that simultaneously satisfy the

expected interim transfer constraints.

As before, we note that the optimal allocations in the relaxed problem P1 for a cost

realization θ are given by the demand characterization above with prices equal to the

vector of virtual costs v(θ). To illustrate, we discuss the structure of the optimal solution

in Example 2. We focus, w.l.o.g. on supplier 1. For a given θ, he will be in the assortment

(d1(θ) > 0) if an only if (a1a2−γ2)(v1(θ)−v2(θ)) ≤ (a2−γ)q1−(a1−γ)q2+a1−γ. Therefore,

the difference in virtual costs for him to be active needs to be bounded by a quantity that

is increasing in the normalized quality difference (a2 − γ)q1 − (a1 − γ)q2, which is intuitive.

The quantity is also increasing in the price sensitive of the other product (a2) provided that


q1 ≥ a1(p1−p2), in the own-product price sensitivity (a1) provided that (1−q2) ≥ a2(p1−p2),

and the absolute cross price sensitivity provided 2γ(p1 − p2) ≥ 1 + (q1 − q2). The structure

of the relaxed optimal allocation generalizes to the case of more products.

By Corollary 3, the above intuition of the relaxed optimal allocation holds for the

optimal solution of the original problem as well, whenever the optima of the two problems

coincide. Therefore, we next show that (under sufficient mild conditions) we can guarantee

OPT (P0) = OPT (P1), by showing that the rows of the associated matrix of coefficients of

the system of linear equations are linearly independent.

Theorem 13. Consider the general setting in which N ≥ 2 agents have arbitrary costs

distributions. Suppose that the following conditions are simultaneously satisfied:

1. There exists a profile θ ∈ Θ such that Q(θ) = N , and there exists a d∗ ∈ R such that,

for all θ′ ∈ Θ with |θ − θ′|∞ ≤ d∗ we have Q(θ′) = N .

2. |Θi| ≥ 3 for all i ∈ N , and for every i ∈ N and every θj ∈ Θi, we have vi(θj+1i ) −

vi(θji ) ≤ d∗/3.

we have OPT (P0) = OPT (P1).

We highlight that d∗ depends on the primitives of the problem. However, the intuition

agrees with that of the Hotelling and vertical models: we must guarantee the existence of an

‘interior solution’ and impose a ‘thin enough’ cost discretization. To provide more intuition,

consider a duopoly where c = (α, α) and D =( β γγ β

). Note that this is a particular case

of Example 2. In this case, the result will follow for any market satisfying the conditions

with d∗ = β−γ2 .

5.6 Case Study: ChileCompra-Style Framework Agreements

In the previous section, we characterized the optimal directed-revelation posted-price mech-

anism. In practice, however, simpler mechanisms are generally used, as they are easier to ex-

plain to potential suppliers and require simpler management from the procurement agency.

In particular, FAs are usually implemented as first price auctions with some additional rules

to decide which products to include in the assortment. Unfortunately, one can prove that


for all demand systems considered in the chapter, the optimal mechanism in general cannot

be implemented using a first price auction.19 The objective of this section is to evaluate the

performance of the type of FAs run by ChileCompra and provide concrete recommendations

for their improvement. The optimal mechanism is crucial for this purpose: it serves as a

benchmark of what is achievable, and its structure also provides insights on how to modify

the current practice to enhance performance.

The section is organized as follows. We start by describing the competition incentives

that arise in first price auctions when additional rules to determine the assortment are

added. In Section 5.6.2, we describe the FAs run by ChileCompra. In Section 5.6.3, we use

a simple model of horizontal differentiation to derive analytical results on the performance

of ChileCompra-style FAs. Then, we quantify the potential improvements that can be

achieved by introducing simple modifications to the current rules. Finally, in Section 5.6.4

we provide a large set of numerical experiments showing the robustness of the conclusions

drawn from the analytical results in the simple model. Overall, our analysis shows that

ChileCompra FAs induce thin markets and that, by emulating the optimal mechanism to

make close-substitute products compete to be in the assortment, consumer surplus can be

significantly increased.

5.6.1 Competition For the Market and Competition In the Market

As previously mentioned, FAs are usually implemented as a first price auction (FPA) with

some additional rules to decide which products to include in the assortment.20 These rules

are common-knowledge at the time of the auction, and are generally a function of the

suppliers’ bids, the characteristics of the products offered, as well as characteristics of the

demand side. In such mechanisms, there are two different (but possibly complementary)

types of incentives for the suppliers to aggressively compete in prices.

19One can show that, to be able to find prices that simultaneously satisfy Eqs. (5.8) and (5.9), for some

some realizations of cost vectors the prices of some products might need to be lower than their actual costs.

However, in a first price auction no agent will bid lower than his cost.

20By a first price auction we mean that suppliers submit bids, which represent the per-unit price of their

products. If a product is added to the assortment, the bid is taken as the posted price.


First, suppliers compete at the auction stage to become part of the assortment. Whether

a supplier is included or not in the assortment depends on the rules of the auction and the

bids; by placing a lower bid, a supplier (weakly) increases his chances of being part of the

assortment. We refer to the competition at the auction stage as competition for the market.

However, even if a supplier is added to the assortment, he is not guaranteed any fixed

amount of demand. Once in the assortment, a supplier’s final allocation depends on his own

bid, the bids of the other suppliers in the assortment and the underlying demand system.

Therefore, a supplier will be competing against imperfect substitute products for demand

once in the assortment. Naturally, one would expect that by placing a lower bid, a supplier

can (weakly) increase his market share. We refer to the competition for demand once in

the assortment as competition in the market.

In the rest of the section, we study the effect these two types of competition have on

both the final bids (or prices) and the consumer surplus.

5.6.2 ChileCompra’s Framework Agreements

Since their introduction in 2004, FAs have been playing an increasingly important role in

the procurement strategy of the Chilean government. In 2013, ChileCompra spent slightly

more than US$ 2 billion in FAs, which corresponded to 21% of the total public expenditure

in procurement and was twice the amount spent in 2010. Nowadays, more than 95 thousand

products and services including food, office supplies, computers, and medical services can

be acquired through FAs.

To award the FAs in a given category (e.g., food), ChileCompra runs a FPA-type mech-

anism which works as follows. First, ChileCompra announces the types of products needed

within the category (e.g., cereal and pasta). Then, each supplier submits a bid for each

item he intents to offer; an item stands for a completely specified product. For example, a

box of Kellogg’s Corn Flakes containing 15oz. and one containing 17oz. are two different

items. Suppliers can bid for any items they want, as long as the type of these products are

among those required by the government. For example, if “cereal” is among the types of

products required, a bid for any type of cereal is allowed, regardless the brand, size, and so

on.


Bids are then evaluated using a scoring rule; all products whose scores are above a

threshold are offered in the menu at the price specified by the supplier in his bid. In

practice, the scores are essentially dominated by price, so we abstract away from the other

features considered.21 Prices are compared only across identical items. As a result, the

current FA implementation works as if running one first price auction independently for

each item offered by at least one supplier. Furthermore, the price score for an item-supplier

pair is assigned by comparing his price to the minimum price of an identical item. If

there is a unique supplier offering the item, he automatically obtains the maximum score

regardless of the price. As the item definitions are narrow (only identical products are

directly compared), in most cases there is a single supplier bidding for an item.

To illustrate, we consider the FA for food products.22 There, a total of 8091 products

were offered by 116 suppliers. Out of those items, 4549 were offered by a unique supplier

who got the maximum price score for this item. As a result, all items with a single supplier

were added to the menu. Furthermore, even for items with at least two bidders, the data

suggests that the current rules fail to generate competition for the market. In the food FA,

there were over 23, 000 bids and only 5% of these were rejected because bids prices were

too high. Hence, given the current rules, bidders have hardly any incentives to aggressively

compete for the market. We highlight that other FAs, such as office supplies, prosthesis

supplies, cleaning products, and personal care, among others, are similar to the FA for food

products in that they create thin markets.

These observations motivate the following questions: can the performance of the current

FAs be improved if thicker markets are created by making imperfect substitute products

compete to be in the menu? In other words, can competition for the market, in addition to

competition in the market, improve performance?23

21In the scoring rules, each item gets a score in the 0− 100 scale. All items for which the score is at least

75 points is included in the menu. Tipically, around 70 points correspond only to price.

22This FA corresponds to the public auction number 2239 − 20 − LP09, titled “Alimentos Perecibles Y

No Perecibles”, which was valid 2010 through 2014.

23[Engel et al., 2002] also study this question on a stylized model of complete information.


5.6.3 Analytical Evaluation of ChileCompra-Style FAs in Simple Model

Following the auction theory tradition, we assume that firms have private costs and that,

for a given mechanism, they play a pure strategy Bayesian Nash equilibrium (BNE). Hence,

to evaluate the performance of the FA we need to derive such equilibrium bidding strate-

gies. Unfortunately, deriving such strategies analytically under general model primitives is

challenging as demands, and therefore profits, are a function of all bids through the de-

mand system; to compute expected profits a bidder needs to integrate out over all possible

demand realizations given competitors’ bid functions.

Therefore, to be able to derive analytical results we restrict our attention to a simple pure

horizontal differentiation Hotelling model. We consider a problem with two IID potential

sellers located at 0 and 1 respectively in the unit line and with two cost realizations. Let

Θi = θL, θH for i = 1, 2 and let fL and fH denote f(θL) and f(θH), respectively. This

simple model will provide essential insights. Then, we test the robustness of these insights

with numerical experiments. All proofs in this section can be found in the appendix.

5.6.3.1 Analysis of ChileCompra-Style FAs

Supported both by the description of ChileCompra’s mechanism and the analysis of their

data, we propose the following first order approximation to their current FAs: we consider

a procurement mechanism in which there is no competition to be in the menu, but suppliers

must compete for demand inside the menu (i.e., there is no competition for the market but

there is competition in the market). Every supplier whose price does not exceed the reserve

price is added to the menu, and the bids of those suppliers are taken as posted prices. After,

the demand is split among the agents in the menu according to the demand model.24

We provide a theoretical analysis of the equilibrium bid functions and the performance

24Without the existence of a reserve price, these rules provide many incentives for suppliers to collude. As

an example, if both suppliers increase their prices by the same amount in a Hotelling model, they obtain same

allocations but a higher per-unit profit. Although not used currently used by the Chilean government, the

reserve prices in ChileCompra and in the mechanisms introduced in Section 5.6.3.2 will help us mitigate the

collusive behavior. Designing mechanisms that deter collusion is certainly an important practical question,

but it is out of the scope of the current chapter.


of ChileCompra FA’s in the two-by-two Hotelling model just described. We assume the

ChileCompra mechanism imposes a reserve price equal to θH . Note that this mechanism is

equivalent to a pricing game with private costs and a reserve. We analytically calculate the

BNE strategies of this pricing game in Appendix C.6.0.1. Using the equilibrium prices, we

compute the expected consumer surplus (corresponding to the negative of expected supplier

payments –purchasing cost– plus transportation cost) of the ChileCompra mechanism and

compare it to that of the optimal mechanism for different parameter values. To compare

performance in this section, for a given a mechanism M , we define the optimality gap

between the optimal mechanism and mechanism M as (M/OPT −1)∗100, where we abuse

notation and denote by M and OPT the total expected consumer surplus in mechanism

M and the optimal mechanism, respectively. Optimality gaps are shown in column ‘Chile’

in Table 5.1 as a function of both fL and δ. In general, the optimality gaps is between 5%

and 20% for the different combination of parameters.

In this simplified setting, we say that the outcome of the ChileCompra mechanism

is single-award if, whenever agents have different types, the low-cost agent obtains all

the demand when competing in the market. Otherwise, we say that the outcome of the

mechanism is split-award.25 A key difference between ChileCompra mechanism and the

optimal mechanism is that the split-award outcome occurs more frequently in the former

one; this difference helps understanding the optimality gaps. Higher gaps are observed for

the values of δ in which ChileCompra split awards and the optimal mechanism does not.

Intuitively, when δ is close to zero, both mechanisms single-award and the gap is small.

In these cases, because consumers are highly price sensitive, competition in the market

provides sufficient incentives for suppliers to price aggressively. In contrast, for large values

of δ both mechanisms split-award; restricting entry is not profitable as consumers’ value is

mostly derived from variety. Finally, for intermediate values of δ, ChileCompra split awards

and the optimal mechanism does not. Further, as δ increases to the values in which Chile-

25We highlight that the terms single-award and split-award have been used in the literature with a dif-

ferent meaning. Typically, in auction settings, they refer to the outcome of the allocation rule [Anton and

Yao, 1989]. In the context of the ChileCompra mechanism we use the terms to describe the outcome of

competition. However, when referring to the optimal mechanism and the modification of the ChileCompra

mechanism introduced below, we restore to the traditional meaning.


Compra split-awards, prices increase rapidly to the reserve price, and total purchasing costs

are high. In these cases, competition for the market may significantly reduce prices. 26

δfL = 0.1 fL = 0.25 fL = 0.5 fL = 0.75 fL = 0.9

Chile BRE Chile BRE Chile BRE Chile BRE Chile BRE

0.5 0.88 0.88 2.41 2.41 5.59 5.58 4.02 3.94 4.30 4.30

1 0.74 0.27 1.98 0.93 4.34 2.73 7.02 5.62 8.40 7.91

1.5 0.86 0.11 2.36 0.55 5.45 2.23 9.41 5.36 12.18 8.55

2 0.89 0.10 2.55 0.55 6.36 2.38 11.62 6.04 15.70 9.97

2.5 0.71 0.12 2.15 0.64 5.74 2.72 11.12 6.93 15.40 15.40

3 0.58 0.18 1.77 0.74 5.15 2.99 10.55 7.55 14.99 14.99

3.5 0.50 0.28 1.50 0.88 4.57 3.12 10.01 7.93 14.60 14.60

4 0.43 0.40 1.30 1.09 4.00 3.20 9.47 8.05 14.24 14.24

4.5 0.38 0.38 1.14 1.14 3.50 3.27 8.95 8.17 13.86 13.86

5 0.34 0.34 1.02 1.02 3.11 3.11 8.44 8.06 13.49 13.49

5.5 0.30 0.30 0.91 0.91 2.79 2.79 7.94 7.94 13.13 13.13

6 0.27 0.27 0.83 0.83 2.53 2.53 7.46 7.46 12.78 12.78

Table 5.1: Optimality gaps as a function of both the differentiation cost δ and fL. The

parameters are θL = 10, θH = 12. The horizontal lines indicate the point up to which

restricting the entry outperforms ChileCompra’s policy.

5.6.3.2 Analysis of Mechanisms that Introduce Competition For the Market

We now explore how, by introducing simple changes to the rules of ChileCompra’s mech-

anism, we can improve performance. The idea is to design auctions’ rules which generate

competition for the market to emulate the optimal mechanism; by introducing such compe-

tition, we can make the single-award outcome more likely, restricting the entry of inefficient

suppliers, obtaining lower bids.

Following ChileCompra’s original design, we focus on FPA-type of mechanisms. We con-

sider two possible changes in the auctions’ rules: restricting entry ex-ante –before observing

the bids– and restricting ex-post –as a function of the observed bids–.

26Figure 5.1 provides an example in which the expected total cost (equal to minus expected consumer sur-

plus), expected purchasing costs, and number of firms awarded are shown for the optimal and ChileCompra’s

mechanisms as a function of δ.


Ex-Ante Restricted-Entry Mechanism. We start by analyzing what happens if com-

petition for the market is induced by restricting entry before bids are placed. In particular,

suppose that we decide how many agents will be in the menu before observing the bids and

then run a FPA type mechanism to decide the prices. In our simple model, this amounts

to deciding when does choosing a single winner using a FPA outperforms ChileCompra’s

mechanism in which all firms compete in the market (but the highest cost is sometimes

priced out). A detailed analysis is provided in Appendix C.6.0.2, but we now discuss the

main take-away.

We observe that that simple modification to the FA rules can sometimes improve per-

formance over the current mechanism. However, there is still a large set of parameters for

which this is not the case. The main drawback of this type of mechanisms is that they

always choose one supplier (or a fixed number of them) even if they have similar (or the

same) bids. If two suppliers have similar bids, by adding both to the menu we obtain more

variety (decrease transportation cost) at a similar purchasing cost, thus improving consumer

surplus. This lack of flexibility is what damages the performance of mechanisms in which

entry is restricted ex-ante. We discuss the performance of more sophisticated mechanisms

next.

Ex-Post Restricted-Entry Mechanism. The main issue with restricting entry ex-ante

is that such mechanisms do not split-award when suppliers share the same cost, which causes

an increase in the transportation cost. Therefore, we now study a class of mechanisms for

which the decision on whom will be in the menu is contingent on the bids received by the

auctioneer. Note that this emulates more closely the optimal mechanism; in the latter the

assortment decisions are made as a function of the reported costs.

Using the intuition from the optimal mechanism, we propose the following two parameter

restricted-entry (RE) mechanism. There is a reserve price R (which we assume equal to

θH) and a split parameter C. If bids satisfy |b1 − b2| < C, then both suppliers are added

to the menu. If not, only the lowest bid supplier (provided the bid is smaller than R) is

included in the menu. Note that if both suppliers are in the menu they will still compete

in the market as before. Hence, the only difference with ChileCompra’s mechanism is that


we restrict the entry to the menu and the split parameter C quantifies how restrictive

the entry to the market is. Note that whenever C = δ, our mechanism coincides with

ChileCompra’s, because in this simple Hotelling model the two suppliers are active if and

only if the differences in prices is lower than δ.27

For the set of parameters in which ChileCompra single-awards, it can be shown that the

performance of the mechanism cannot be improved by restricting entry.28 Therefore, our

focus is in the settings in which ChileCompra split-awards. For these cases, we find values of

C (smaller than δ) for which the equilibrium bid of the low-type induces single-award. Note

that the equilibrium bid of a high-type is θH and, hence, a natural candidate for low-type

equilibrium bid is θH−C, because it is the highest possible bid that results in single-award.

We have the following result.

Proposition 6. For every set of parameters fL, θH , θL and δ, there exists a (possibly

empty) interval I such that, for all C ∈ I, we have that θH − C is the unique equilibrium

bid for the low type in the RE mechanism with reserve price θH and split parameter C.

In the appendix, we characterize the intervals referred to in the previous proposition as

a function of fL, θH , θL, and δ. Intuitively, if C is too small, bidders have incentives to

undercut each other and a BNE may not exist. On the other hand, if C is too big, an agent

of type θL might prefer to place a bid greater than θH − C to obtain a higher profit per

unit even if that implies splitting the demand with a high-type agent.

For given model primitives, the designer is interested in maximizing consumer surplus. If

restricting entry is a helpful device to achieve this objective, then the auctioneer will choose

the largest C for which a single-award equilibrium exists, because that induces the lowest

bid for the low type. Hence, we define the “best low-type bid” to be θH − C∗, where C∗ is

the highest C for which θH−C is an equilibrium bid for the low-type. The characterization

of the best low-type bids can be found in the appendix, but we briefly discuss the intuition.

Intuitively, the advantage of bidding at θH − C is to capture the whole demand when the

27Whenever C = 0, our mechanism agrees with a FPA. However, in this section we are only going to

consider split parameters C for which a BNE exists; for discrete types a BNE may not exist for small values

of C.

28The proof can be found in the appendix.


other agent has a high cost. As fL becomes close to one, this advantage vanishes; for this

reason, the best low-type bid is increasing in fL. In addition, the best low-type bid is also

increasing in δ; as the transportation cost increases, demands become less sensitive to prices

and, therefore, a supplier can increase his bid without significantly decreasing demand.

Note, however, that restricting entry may cause the performance to be worse than that

of ChileCompra, as single-award increases the transportation cost. To that end, we define

the best restricted-entry mechanism (BRE) as the mechanism that maximizes consumer

surplus. We obtain the following straightforward result.

Proposition 7. For a given set of parameters, the BRE has one of two possible forms:

(1) coincides with the ChileCompra mechanism (C = δ); or (2) uses the value C∗ (< δ)

associated to the best low-type bid.

For a given set of parameters, if BRE improves over ChileCompra it must be by re-

stricting entry; in such case, (2) is optimal. Otherwise, (1) above is optimal. To illustrate,

in Figure 5.1 we plot the outcome of the optimal, ChileCompra and BRE mechanisms as

a function of the transportation cost δ for a given set parameters. As it can be observed,

the BRE mechanism restricts the entry whenever δ ≤ 4.675. By doing so, the assortments

obtained are similar to the ones generated by the optimal mechanism, and the expected

purchasing cost is much closer to the optimal one. However, when δ exceeds 4.675, the sav-

ings obtained in the purchases cannot compensate for the increase in transportation cost

and, therefore, BRE and ChileCompra coincide beyond that point.

More generally, we study when BRE outperforms ChileCompra as a function of the

parameters. We find that, for when δ is relatively small and ChileCompra split-awards,

restricting entry improves over ChileCompra mechanism regardless of the value of other

parameters. In such cases, the decrease in the low-type equilibrium bid results in a consid-

erable decrease in the expected purchasing cost without a major increase in the expected

transportation cost. In addition, as it can be observed in Table 5.1, restricting entry per-

forms better for the middle-values of fL. If fL is too low, the savings are less likely to occur

and therefore the potential impact is smaller. On the other hand, if fL is too high, the

best-low-type-bid tends to increase and the single-award becomes less profitable. This is


0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 611

12

13

Differentiation cost

Exp

ecte

dto

tal

cost

OPT

Chile

BRE

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

11

11.5

12

12.5

13


Exp

ecte

dp

urc

hasi

ng

cost

OPT

Chile

BRE

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6


Optimalsingle split

ChileComprasingle split

BREsingle split

Figure 5.1: (Top) Expected total costs (purchasing plus transportation, equivalent to -

(consumer surplus)) for optimal, ChileCompra and best restricted-entry (BRE) mechanisms

as a function of the differentiation (transportation) cost δ. The parameters are θL =

10, θH = 12, fL = fH = 1/2. (Center) Expected purchasing costs. (Bottom) Single-award

vs. split-award in optimal, ChileCompra’s, and our best mechanism.


BRE < ChileCompra

BRE = ChileCompra (ChileCompra restrics entry)

BRE = ChileCompra (split-award)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

2

3

4

5

fL

Diff

eren

tiati

on

cost

Figure 5.2: For θL = 10, θH = 12, we show when it is profitable to restrict the entry as a

function of the differentiation cost δ and fL. The dashed line represents the cutoff between

single and split award in the optimal mechanism (i.e., δ = 1fH

(θH − θL)).

illustrated by Figure 5.2, where we fix θL = 10, θH = 12, and show when it is profitable to

restrict entry as a function of δ and fL.

We conclude this section with a note on the practical implementability of the restricted

entry mechanisms. The BRE mechanism uses the best split-parameter C that depends on

the problem primitives and therefore it may be hard to estimate in practice. However,

we argue that even implementing the BRE mechanism with a rough estimate of the best

C (but not the exact one) typically improves performance.29 In particular, if restricting

entry is profitable, any smaller C which is relatively close to the best C will induce the

equilibrium bid θH − C. Therefore, if the parameters are in the interior of the gray area

in Figure 5.2 (where restricting entry improves performance), by choosing a conservative C

the auctioneer should be able to increase consumer surplus. Also note that any C larger

than the best C yields the same outcome as the current ChileCompra mechanism, so it will

not damage performance.

29 This is formally shown in the appendix.


5.6.4 Robustness Results: Numerical Experiments

To test the robustness of our intuition, in this section we numerically solve for the equi-

librium strategies for ChileCompra and the BRE mechanism and compare the expected

consumer surplus these mechanisms with that of the optimal. We replicate this simulation

exercise for a range of environments by varying the cost distributions, the number of bidders

and some of the other parameters of the model. The results are summarized next.

More General Cost Distributions. We first consider adding more points to the support

of the cost distributions. To that end, we consider an initial interval and discretize it evenly

into k costs, for k = 2, 3, 5, 7. We consider 4 types of distributions: uniform, left-skewed,

right-skewed, and symmetric-unimodular (normal-like). We highlight that, even though

now we have multiple costs in the support, the auctioneer still must pick a unique split-

parameter that remains fixed throughout the mechanism.

The results of our simulation show that the intuition for the multiple-costs case coincides

with that of the two-by-two simple model and restricting entry improves the performance

of the current mechanism. In general, the optimality gap decreases by at least 40%, and

the differences in the gap becomes smaller as δ increases. Similarly to the two-by-two case,

the relative benefits are greater when the distribution is left-skewed or normal-like where

restricting entry achieves greater reduction in the bids of the low-type. In addition, as the

number of values in the support increases, restricting entry improves performance for a

wider range of values of cost-differentiation, because the auctioneer can use a more refined

splitting rule.

Larger Number of Bidders. We now consider models with more than two agents.

To that end, we consider n agents at equidistant locations with agent i located at ì =

(i − 1)/(n − 1). We test our results for n ∈ 2, 3, 4, 5. The costs are still assumed to be

IID across agents; however, agents are not ex-ante symmetric due to their locations.

Whenever there are more than two agents, the auctioneer can choose whether to restrict

entry as a function of bids or as a joint function of both bids and product characteristics.

We discuss these two options next. We first consider restricting the entry as a function of


both the bids and product characteristics. The mechanism we consider is the restricted-

entry mechanism with the following modification: for split parameter C and bids b1, . . . , bn,

supplier i will be in the menu only if bi − bj < C ∗ |ì − `j | for every j ∈ N with j 6= i.

This rule is intuitive: it induces more price competition for agents that are close-by in the

product space. Next, we consider restricting the entry solely as a function of bids. Similarly

to the case of two agents, for split parameter C and bids b1, . . . , bn, supplier i will be in the

menu only if bi − bj < C for every j ∈ N with j 6= i. As this rule is less sophisticated than

the previous one, a poorer performance is to be expected.

The main findings are as follows. First, the distribution of costs has the same impact

in the performance as in the two-agent case; ChileCompra performs close to optimal for

right-skewed distributions (when low-types rarely occur), but poorly for the other classes

of distributions. Second, the optimality gap increases with the number of agents. The

intuition seems to be the same as in the two-agent case; without competition for the market,

ChileCompra fails to obtain competitive bids for the low-type (relative to the optimum)

and this lack of competition has a higher impact as the number of suppliers increases.

In accordance to what is observed in the two-agent case, restricting the entry improves

performance for the sets of parameters in which ChileCompra split-awards. For the values

of δ in which ChileCompra split-awards, restricting the entry performs better (with respect

to the optimum) than in the two agent case. In general, for fixed number of agents and

cost distributions, the optimality gap decreases by an average of 25% if characteristics are

taken into account, and around 20% if they are not.

More General Demand Models. We now consider a model that includes both horizon-

tal and vertical differentiation. In particular, we focus on the demand model in Example 2.

We vary the qualities of the products, the price sensitivities and cross elasticity. In this

general model, suppliers are generally asymmetric ex-ante, as products can have different

qualities.

Similarly to the simplified two-agent case with horizontal differentiation, introducing

competition for the market is more efficient whenever ChileCompra split-awards and the

optimal mechanism single-awards. Here, qualities play an important role. The highest-


quality agent has an advantage to capture demand, and can exploit such advantage by

placing a higher bid. As a result, bids in the ChileCompra mechanism tend to increase fast

as cross-elasticity (differentiation) or price sensitivities decrease. Introducing competition

for the market, forces the highest-quality supplier to decrease his bid; otherwise, he will not

be added to the assortment. In turn, this has an effect on how the lowest-quality supplier

bids; he has an incentive to further reduce his bid to capture more demand. In general, the

optimality gap decreases by an average of 8% when restricting entry, and we obtain better

results when the difference in qualities is smaller.

5.7 Conclusions and Extensions

In this chapter we study procurement mechanisms for differentiated products demanded

by heterogeneous consumers. First, we characterize the optimal mechanism for important

classes of demand models. Second, we use these results to shed light on the FAs run by the

Chilean government. Our results are useful to improve our understanding of FAs and, more

generally, of buying mechanisms in similar contexts.

Our basic model can be extended in several interesting directions. First, to simplify

the exposition, we assumed that each supplier offers one product. In Appendix C.4.1, we

provide an extension to our model in which we allow for multi-product suppliers. We show

that our solution framework extends to this setting, and we are able to characterize the

optimal mechanism for the multi-product case.

In our basic model we assume an inelastic total demand, which may be reasonable

for some products, like medicines, but perhaps less so for others. In Appendix C.4.2, we

consider a model with an elastic total demand. We show that, in general, our main result

fails to hold and a gap between the optima of the original and the relaxed problem exists.

However, the preliminary computational results show that this gap is typically small and

that the assortments are usually similar in both the relaxed and the original problems.

In addition, it would be interesting to further explore whether the insights are affected

if other demand systems, such as a nested logit model, are assumed. As future work, one

might want to use econometric techniques to estimate important parameters of the model,


such as those related to the underlying preferences of the organizations in the Chilean

procurement setting, with the objective of sharpening the design recommendations. We

leave all these directions for future research.

127

Part III

Bibliography

BIBLIOGRAPHY 128

Bibliography

[Abdulkadiroglu and Sonmez, 2003] Atila Abdulkadiroglu and Tayfun Sonmez. School

choice: A mechanism design approach. American Economic Review, 93(3):729–747, 2003.

[Abdulkadiroglu et al., 2009] Atila Abdulkadiroglu, Parag A. Pathak, and Alvin E. Roth.

Strategy-proofness versus efficiency in matching with indifferences: Redesigning the nyc

high school match. American Economic Review, 99(5):1954–78, 2009.

[Abdulkadiroglu, 2005] Atila Abdulkadiroglu. College admissions with affirmative action.

International Journal of Game Theory, 33:525 – 549, 2005.

[Abraham et al., 2005] David J. Abraham, Katarina Cechlarova, David F. Manlove, and

Kurt Mehlhorn. Pareto optimality in house allocation problems. In Algorithms and

Computation, volume 3341 of Lecture Notes in Computer Science, pages 3–15. 2005.

[Ackerberg et al., 2006] D. Ackerberg, C.L. Benkard, S. Berry, and A. Pakes. Econometric

tools for analyzing market outcomes. Handbook of Econometrics, 2006.

[Albano and Sparro, 2008] Gian Luigi Albano and Marco Sparro. A simple model of frame-

work agreements: Competition and efficiency. Journal of Public Procurement, 8(3):356–

378, 2008.

[Alcalde-Unzu and Molis, 2011] Jorge Alcalde-Unzu and Elena Molis. Exchange of indivis-

ible goods and indifferences: The top trading absorbing sets mechanisms. Games and

Economic Behavior, 73(1):1–16, 2011.

[Allon and Federgruen, 2007] Gad Allon and Awi Federgruen. Competition in service in-

dustries. Operations Research, 55(1):37–55, 2007.

BIBLIOGRAPHY 129

[Anderson et al., 1992] Simon P Anderson, Andre De Palma, and Jacques Francois Thisse.

Discrete choice theory of product differentiation. MIT press, 1992.

[Anton and Gertler, 2004] James J Anton and Paul J Gertler. Regulation, local monopolies

and spatial competition. Journal of Regulatory Economics, 25(2):115–141, 2004.

[Anton and Yao, 1989] James J Anton and Dennis A Yao. Split awards, procurement, and

innovation. The RAND Journal of Economics, pages 538–552, 1989.

[Area de Estudios e Inteligencia de Negocios, Direccion ChileCompra, 2014] Area de Estu-

dios e Inteligencia de Negocios, Direccion ChileCompra. Informe de gestion, diciembre

2013. 2014.

[Armstrong and Vickers, 2014] Mark Armstrong and John Vickers. Which demand systems

can be generated by discrete choice? Technical report, 2014.

[Ashlagi et al., 2013] Itai Ashlagi, Yash Kanoria, and Jacob D. Leshno. Unbalanced random

matching markets. In EC, pages 27–28, 2013.

[Azevedo and Leshno, 2012] E. D. Azevedo and J. D. Leshno. A supply and demand frame-

work for two-sided matching markets. Working paper, 2012.

[Aziz and de Keijzer, 2012] Haris Aziz and Bart de Keijzer. Housing markets with indiffer-

ences: A tale of two mechanisms. In Proceedings of AAAI’12, 2012.

[Aziz et al., 2013] Haris Aziz, Felix Brandt, and Markus Brill. The computational com-

plexity of random serial dictatorship. Economics Letters, 121(3):341 – 345, 2013.

[Becker, 1973] Gary S Becker. A theory of marriage: Part i. The Journal of Political

Economy, pages 813–846, 1973.

[Bird, 1984] Charles G. Bird. Group incentive compatibility in a market with indivisible

goods. Economics Letters, 14(4):309 – 313, 1984.

[Bogomolnaia et al., 2005] Anna Bogomolnaia, Rajat Deb, and Lars Ehlers. Strategy-proof

assignment on the full preference domain. J. Economic Theory, 123(2):161–186, 2005.

BIBLIOGRAPHY 130

[Bresnahan, 1987] Timothy F Bresnahan. Competition and collusion in the american au-

tomobile industry: The 1955 price war. The Journal of Industrial Economics, pages

457–482, 1987.

[Brightwell and Winkler, 1990] Graham Brightwell and Peter Winkler. Counting linear ex-

tensions is#p-complete. DIMACS, Center for Discrete Mathematics and Theoretical

Computer Science, 1990.

[Cachon and Harker, 2002] Gerard P Cachon and Patrick T Harker. Competition and out-

sourcing with scale economies. Management Science, 48(10):1314–1333, 2002.

[Chaturvedi et al., 2014] Aadhaar Chaturvedi, Damian R Beil, and Victor Martınez-de

Albeniz. Split-award auctions for supplier retention. Management Science, 60(7):1719–

1737, 2014.

[Chen and Li, 2013] Yuxin Chen and Xinxin Li. Group buying commitment and sellers

competitive advantages. Journal of Economics & Management Strategy, 22(1):164–183,

2013.

[Chiappori et al., 2011] Pierre A Chiappori, Bernard Salanie, and Yoram Weiss. Partner

choice and the marital college premium. 2011.

[Choo and Siow, 2006] Eugene Choo and Aloysius Siow. Who marries whom and why.

Journal of Political Economy, 114(1):pp. 175–201, 2006.

[Cres and Moulin, 2001] Herve Cres and Herve Moulin. Scheduling with opting out: Im-

proving upon random priority. Operations Research, 49(4):565–577, 2001.

[Dana and Spier, 1994] James Jr. Dana and Kathryn E. Spier. Designing a private industry

: Government auctions with endogenous market structure. Journal of Public Economics,

53(1):127–147, January 1994.

[Dana, 2012] James D Dana. Buyer groups as strategic commitments. Games and Economic

Behavior, 74(2):470–485, 2012.

[Demsetz, 1968] Harold Demsetz. Why regulate utilities? Journal of law and economics,

pages 55–65, 1968.

BIBLIOGRAPHY 131

[Durrett, 2010] Rick Durrett. Probability: theory and examples. Cambridge university press,

2010.

[Ehlers et al., 2011] Lars Ehlers, Isa Hafalir, Bumin Yenmez, and Muhammed Yildirim.

School choice with controlled choice constraints: Hard bounds versus soft bounds.

GSIA Working Papers 2012-E20, Carnegie Mellon University, Tepper School of Business,

November 2011.

[Ehlers, 2002] Lars Ehlers. Coalitional strategy-proof house allocation. Journal of Economic

Theory, 105(2):298–317, 2002.

[Elmaghraby, 2000] W. Elmaghraby. Supply contract competition and sourcing policies.

Manufacturing & Service Operations Management, 2(4):350–371, 2000.

[Engel et al., 2002] Eduardo Engel, Ronald Fischer, and Alexander Galetovic. Competition

in or for the field: Which is better? Working Paper 8869, National Bureau of Economic

Research, April 2002.

[Erdil and Ergin, 2006] Aytek Erdil and Haluk Ergin. Two-sided matching with indiffer-

ences. Unpublished mimeo, Harvard Business School, 2006.

[Erdil and Ergin, 2008] Aytek Erdil and Haluk Ergin. What’s the matter with tie-breaking?

improving efficiency in school choice. The American Economic Review, 98(3):669–689,

2008.

[European Commision, 2012] European Commision. Annual public procurement implemen-

tation review, 2012.

[Farahat and Perakis, 2010] Amr Farahat and Georgia Perakis. A nonnegative extension of

the affine demand function and equilibrium analysis for multiproduct price competition.

Operations Research Letters, 38(4):280 – 286, 2010.

[Federgruen and Hu, 2014] Awi Federgruen and Ming Hu. Sequential multi-product price

competition in supply chain networks. Working paper, 2014.

[Gale and Shapley, 1962] D. Gale and L. S. Shapley. College Admissions and the Stability

of Marriage. Amer. Math. Monthly, 69(1):9–15, 1962.

BIBLIOGRAPHY 132

[Galichon and Salani, 2010] Alfred Galichon and Bernard Salani. Matching with trade-offs:

Revealed preferences over competing characteristics. Working paper, 2010.

[Gretsky et al., 1992] Neil E Gretsky, Joseph M Ostroy, and William R Zame. The

Nonatomic Assignment Model. Economic Theory, 2(1):103–27, January 1992.

[Gretsky et al., 1999] Neil E. Gretsky, Joseph M. Ostroy, and William R. Zame. Per-

fect Competition in the Continuous Assignment Model. Journal of Economic Theory,

88(1):60–118, September 1999.

[Gur et al., 2013] Y. Gur, L. Lu, and G.Y. Weintraub. Framework agreements in procure-

ment: An auction model and design recommendation. Working paper, 2013.

[Haeringer and Iehle, 2014] Guillaume Haeringer and Vincent Iehle. Two-sided matching

with one-sided preferences. Preprint, 2014.

[Hansen, 1988] Robert G Hansen. Auctions with endogenous quantity. The RAND Journal

of Economics, pages 44–58, 1988.

[Hassidim and Romm, 2014] Avinatan Hassidim and Assaf Romm. An approximate ”law

of one price” in random assignment games. CoRR, abs/1404.6103, 2014.

[Hoeffding, 1963] Wassily Hoeffding. Probability inequalities for sums of bounded random

variables. Journal of the American statistical association, 58(301):13–30, 1963.

[Holzman and Samet, 2013] R. Holzman and D. Samet. Matching of like rank and the size

of the core in the marriage problem. Unpublished, 2013.

[Horton and Kilakos, 1993] J. Horton and K. Kilakos. Minimum edge dominating sets.

SIAM Journal on Discrete Mathematics, 6(3):375–387, 1993.

[Immorlica and Mahdian, 2005] Nicole Immorlica and Mohammad Mahdian. Marriage,

honesty, and stability. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium

on Discrete Algorithms, pages 53–62 (electronic). ACM, New York, 2005.

BIBLIOGRAPHY 133

[Jaramillo and Manjunath, 2012] Paula Jaramillo and Vikram Manjunath. The difference

indifference makes in strategy-proof allocation of objects. Journal of Economic Theory,

147(5):1913 – 1946, 2012.

[Jerrum, 2003] Mark Jerrum. Counting, Sampling and Integrating: Algorithms and Com-

plexity (Lectures in Mathematics. ETH Zurich). Birkhauser Basel, 1 edition, April 2003.

[Kavitha and Nasre, 2009] Telikepalli Kavitha and Meghana Nasre. Popular matchings

with variable job capacities. In Proceedings of the 20th International Symposium on

Algorithms and Computation, ISAAC ’09, pages 423–433, 2009.

[Kojima and Pathak, 2009] Fuhito Kojima and Parag A. Pathak. Incentives and stability

in large two-sided matching markets. American Economic Review, 99(3):608–27, 2009.

[Kojima et al., 2013] Fuhito Kojima, Parag A. Pathak, and Alvin E. Roth. Matching with

couples: Stability and incentives in large markets. Quarterly Journal of Economics,

128(4):1585–1632, 2013.

[Kok et al., 2009] A Gurhan Kok, Marshall L Fisher, and Ramnath Vaidyanathan. As-

sortment planning: Review of literature and industry practice. In Retail supply chain

management, pages 99–153. Springer, 2009.

[Krishna, 2009] Vijay Krishna. Auction theory. Academic press, 2009.

[Levin, 1997] Jonathan Levin. An optimal auction for complements. Games and Economic

Behavior, 18(2):176–192, 1997.

[Li and Debo, 2009] C. Li and L. Debo. Second sourcing vs. sole sourcing with capacity

investment and asymmetric information. Manufacturing & Service Operations Manage-

ment, 11(3):448–470, 2009.

[Li and Rusmevichientong, 2014] Guang Li and Paat Rusmevichientong. A greedy algo-

rithm for the two-level nested logit model. Operations Research Letters, 42(5):319–324,

2014.

[Ma, 1994] Jinpeng Ma. Strategy-proofness and the strict core in a market with indivisi-

bilities. International Journal of Game Theory, 23(1):75–83, 1994.

BIBLIOGRAPHY 134

[Martin, 2009] S. Martin. Microfoundations for the linear demand product differentiation

model. Working Paper, 2009.

[Mas-Colell et al., 1995] Andreu Mas-Colell, Michael Dennis Whinston, Jerry R Green,

et al. Microeconomic theory, volume 1. Oxford university press New York, 1995.

[McGuire and Riordan, 1995] Thomas G. McGuire and Michael H. Riordan. Incomplete in-

formation and optimal market structure public purchases from private providers. Journal

of Public Economics, 56(1):125–141, January 1995.

[Myerson, 1981] R. Myerson. Optimal auction design. Mathematics of Operations Research,

6(1):58–73, 1981.

[Papai, 2000] Szilvia Papai. Strategyproof assignment by hierarchical exchange. Economet-

rica, 68(6):1403–1433, November 2000.

[Pathak and Sethuraman, 2014] Parag Pathak and Jay Sethuraman. Handling appeals in

school assignment. Working Paper, 2014.

[Plaxton, 2012] C. Greg Plaxton. A simple family of top trading cycles mechanisms for

housing markets with indifferences. Working Paper, 2012.

[Riordan and Sappington, 1989] Michael H Riordan and David EM Sappington. Second

sourcing. The RAND Journal of Economics, pages 41–58, 1989.

[Roth and Peranson, 1999] A. E. Roth and E. Peranson. The redesign of the matching

market for American physicians: Some engineering aspects of economic design. American

Economic Review, 89:748–780, 1999.

[Roth and Postlewaite, 1977] Alvin E Roth and Andrew Postlewaite. Weak versus strong

domination in a market with indivisible goods. Journal of Mathematical Economics,

4(2):131–137, 1977.

[Roth et al., 2004] Alvin Roth, Tayfun Sonmez, and Utku Unver. Kidney exchange. Quar-

terly Journal of Economics, 19(2):457 – 488, 2004.

BIBLIOGRAPHY 135

[Roth, 1982] Alvin Roth. Incentive compatibility in a market with indivisible goods. Eco-

nomics Letters, 9(2):127 – 132, 1982.

[Satterthwaite and Sonnenschein, 1981] Mark A Satterthwaite and Hugo Sonnenschein.

Strategy-proof allocation mechanisms at differentiable points. The Review of Economic

Studies, 48(4):587–597, 1981.

[Shapley and Scarf, 1974] Lloyd Shapley and Herbert Scarf. On cores and indivisibility.

Journal of Mathematical Economics, 1(1):23 – 37, 1974.

[Shapley and Shubik, 1971] L.S. Shapley and M. Shubik. The assignment game i: The core.

International Journal of Game Theory, 1(1):111–130, 1971.

[Shubik and Levitan, 1980] M. Shubik and R. Levitan. Market Structure and Behavior.

Harvard University Press, Cambridge, MA, 1980.

[Sinclair and Jerrum, 1989] Alistair Sinclair and Mark Jerrum. Approximate counting,

uniform generation and rapidly mixing markov chains. Information and Computation,

82(1):93 – 133, 1989.

[Sonmez and Unver, 2011] Tayfun Sonmez and M. Utku Unver. Chapter 17 - matching,

allocation, and exchange of discrete resources. volume 1 of Handbook of Social Economics,

pages 781 – 852. North-Holland, 2011.

[Soon et al., 2009] W. Soon, G. Zhao W., and J. Zhang. Complementarity demand func-

tions and pricing models for multi-product markets. European Journal of Applied Math-

ematics, 20(5):399–430, 2009.

[Strang, 1988] Gilbert Strang. Linear Algebra and Its Applications. Brooks Cole, 3rd edi-

tion, 1988.

[Tarjan, 1972] Robert Tarjan. Depth-First Search and Linear Graph Algorithms. SIAM

Journal on Computing, 1(2):146–160, 1972.

[Tirole, 1988] Jean Tirole. The Theory of Industrial Organization. MIT press, 1988.

BIBLIOGRAPHY 136

[Toda, 1989] Seinosuke Toda. On the computational power of pp and (+)p. In 30th Annual

Symposium on Foundations of Computer Science, pages 514–519, 1989.

[Truong, 2014] Van-Anh Truong. Optimal selection of medical formularies. Journal of

Revenue & Pricing Management, 13(2):113–132, 2014.

[Valiant, 1979] Leslie G. Valiant. The complexity of computing the permanent. Theoretical

Computer Science, 8(2):189 – 201, 1979.

[Vives, 2001] X. Vives. Oligopoly Pricing: Old Ideas and New Tools. MIT Press, 2001.

[Vohra, 2011] R.V. Vohra. Mechanism Design: A Linear Programming Approach. Econo-

metric Society Monographs. Cambridge University Press, 2011.

[Wang and Krishna, 2006] Yu Wang and Aradhna Krishna. Timeshare exchange mecha-

nisms. Manage. Sci., 52(8):1223–1237, August 2006.

[Wolinsky, 1997] Asher Wolinsky. Regulation of duopoly: managed competition vs regu-

lated monopolies. Journal of Economics & Management Strategy, 6(4):821–847, 1997.

137

Part IV

Appendices

APPENDIX A. HOUSE ALLOCATION WITH INDIFFERENCES 138

Appendix A

House Allocation with

Indifferences: A Generalization

and a Unified View

We now present a selection rule which satisfies all of the required conditions (namely, unique

pointing, termination, persistence and “Independence of unsatisfied agents”), and still fails

to be strategy-proof. This rule, called the Common ordering on agents, individual ordering

on agents rule is defined as follows:

There is a common ordering on agents and also each agent will has his own (individual)

ordering of other agents. All orderings will be fixed throughout the algorithm. At each

step, agents will are divided into two sets: labeled agents and unlabeled agents. Labeled

agents are those for whom their outgoing edge according to the selection rule has already

been decided, while unlabeled agents are those for which who should they point to is yet to

be decided. We will grow the set of unlabeled agents until every agent is labeled, and thus

the rule is defined.

In the first step, the selection rule is as follows:

Step 1:

(1.a) Each unsatisfied agent points to the highest priority agent (according to his own


ordering) among those who own one of his top-ranked objects. Label all unsatis-

fied agents.

(1.b) Repeat until all satisfied agents are labeled:

Select the highest priority agent among all the agents adjacent to a labeled

vertex, and make it point to the highest priority agent (according to his own

ordering) owning one of his top-choices among all those which are labeled.

Label the unlabeled agent.

For each satisfied vertex v, we keep track of the first unsatisfied vertex reachable from

v in F (G) and we denote it by X(v). For each unsatisfied vertex v, we denote by X(v) the

vertex he points to in F (G). For step k, the rule is as follows:

Step k:

(k.a) Each agent v for which X(v) still holds the same object as in the previous step

will continue to point to the same agent as in the previous step. Label all such

agents. All other agents remain unlabeled.

(k.b) Each unsatisfied agent v for which X(v) does not hold the same object as in the

previous step will point to the highest priority agent (according to his own or-

dering) among those who own one of his top-ranked objects. Label all unsatisfied

agents.

(k.c) Repeat until all the remaining unlabeled agents are labeled:

Select the highest priority unlabeled agent among all those adjacent to a

labeled vertex, and make it point to the highest priority agent (according to

his own ordering) owning one of his top-choices among all those which are

labeled. Label the unlabeled agent.

Note that by the end of each step all satisfied agents will be in a path to an unsatisfied

vertex in F (G). Hence, every cycle formed is improving, ensuring termination in O(n)

steps. In addition, persistence is satisfied by construction. Finally, the “Independence

of unsatisfied agents” property is satisfied as all unsatisfied agents start as labeled and

unlabeled agents choose who to point to based only on priorities over the labeled agents,

regardless who labeled agents are pointing to.


In Figure A.1, an example is provided which illustrates why this rule fails to induce a

strategy proof mechanism. Agent 1 obtains c in problem ∆. We define ∆′ from ∆ by only

modifying agent 1’s preferences so that he strictly prefers c over g. Agent 1 gets object

c in ∆ but fails to obtain it in ∆′, implying that whenever i has true preferences which

agree with those in ∆′, he is better off by reporting his preferences as in ∆. Therefore, the

mechanism induced by this rule is not strategy-proof.

Those readers who are already familiar with the results in Section 2.4.1 should note

that the definition of ∆′ from ∆ agrees with the one considered in Property 1. Therefore,

this example shows that the mechanism is not strategy-proof by showing that Property 1

fails to hold. Furthermore, we can see that the sufficient conditions provided in Theorem 4

fail to hold as well, as in the example agent 2 points to someone outside C∆′2 (1) in F (G∆′

2 )

(therefore 2 /∈ C∆′2 (1)) but points to agent 1 in ∆. If the choice was made based solely on

objects, then the object would have been be in C∆′2 (1) and thus there will not be a conflict,

as shown by the rules defined in Section 2.5.



ω(1) = a, ω(2) = b,ω(3) = c, ω(4) = d,ω(5) = e, ω(6) = f ,

ω(7) = g.

Common agent ordering:

1, 2, 3, 4, 5, 6, 7.

(a) Original endowments andcommon agent ordering for Δand Δ′

Preference Lists:

1 g, c2 f, g, d3 b, e, c4 e5 d6 b, f7 a

(b) Preferences for Δ.In Δ′, agent 1 strictlyprefers c over g.

Ind. agent orderings:

1 7, 32 4, 1, 6, 73 5, 24 55 46 27 1

(c) Individual agent order-ings for Δ and Δ′.

Final alloc. for Δ:

μ(1) cμ(2) gμ(3) bμ(4) eμ(5) dμ(6) fμ(7) a

(d) Final allocation forproblem Δ.

7(g)

1(a)

2(b)

6(f)

4(d)

5(e)3(c)

(e) GΔ0

7(g)

1(a)

2(b)

6(f)

4(d)

5(e)3(c)

(f) F (GΔ0 )

7(a)

1(g)

2(b)

6(f)

4(e)

5(d)3(c)

(g) GΔ1 . Terminal sinks 4, 5, 7

are removed.

1(g)

2(b)

6(f)

3(c)

(h) F (GΔ1 )

1(c)

2(g)

6(f)

3(b)

(i) GΔ2 : everyone leaves

with their own objects

7(g)

1(a)

2(b)

6(f)

4(d)

5(e)3(c)

(j) F (GΔ′0 )

7(g)

1(a)

2(b)

6(f)

4(e)

5(d)3(c)

(k) GΔ′1

7(g)

1(a)

2(b)

6(f)

3(c)

(l) F (GΔ′1 ): 6 gets b after which 3

leaves with c, so 1 cannot obtain c.

Figure A.1: This example shows that “Common ordering on agents, individual ordering on

agents” is not strategy-proof. Figs. 2.a-2.d show the initial set up for problems ∆ and ∆′.

Problem ∆′ is defined from ∆ by changing agent 1’s preferences so that he strictly prefers c

over g. Figs. 2.e-2.i show how the final allocation is computed in ∆. Figs. 2.j-2.l show that

1 cannot get c in ∆′, implying that, whenever i has true preferences agreeing with those in

∆′, he will be better off by reporting as in ∆. Hence, strategy-proofness is violated.

APPENDIX B. THE SIZE OF THE CORE IN ASSIGNMENT MARKETS 142

Appendix B

The Size of the Core in

Assignment Markets

B.1 Results on point processes in the unit hypercube

Consider the K dimensional unit hypercube [0, 1]K , and the Poisson process of uniform

rate n in this hypercube, leading to N points (εi)Ni=1. (Note that E[N ] = n.) Here εi =

(ε1i , ε2i , . . . , ε

Ki ). Let K = 1, 2, . . . ,K denote the set of dimension indices.

Let Rk be the region defined by Eq. (4.1), and let Vk and V k be as defined by Eqs. (4.3)

and (4.4) respectively. Similarly, let Rk1,k2(δ) be the region defined by Eq. (4.2), and let

Vk1,k2(δ) and V k1,k2(δ) be as defined by Eqs. (4.5) and (4.6) respectively.

The following lemma, key to our proof of Theorem 9, says that with high probability, all

the (V k)’s and the (V k1,k2)’s are no larger than a (deterministic) function1 of n that scales

as O∗(1/n1/K).

Lemma 6. Let Rk be the region defined by Eq. (4.1), and let Vk and V k be as defined by

Eqs. (4.3) and (4.4) respectively. Similarly, let Rk1,k2(δ) be the region defined by Eq. (4.2),

and let Vk1,k2(δ) and V k1,k2(δ) be as defined by Eqs. (4.5) and (4.6) respectively. Fix K ≥ 1.

Then there exists f(n,K) = O∗(1/n1/K) such that for any δ = δ(n) ∈ [0, 1/2] the following

1In fact, our proof of Lemma 6 identifies a bound of (C logn/n)1/K where C = 6K(K−1), for sufficiently

large n.


holds: Let

B1 =

max

(maxk∈K

V k, max(k1,k2)∈K(2)

V k1,k2(δ)

)≤ f(n,K)

, (B.1)

where K(2) = (k1, k2) : k1, k2 ∈ K, k1 6= k2. (If K = 1, then K(2) is the empty set ∅ in

which case we follow the convention that max∅[ · ] = −∞.) We have

Pr(B1) ≥ 1− 1/n .

Proof. Let m = b1/(C log n/n)1/Kc for some C < ∞ that we will choose later, and let

∆ = 1/m. Note that

∆ ≥ (C log n/n)1/K . (B.2)

In our analysis of V k (resp. V k1,k2), we will divide the interval [0, 1] (resp. [−1 + δ, 1]) into

subintervals of size ∆ each, and show that with large probability, each subinterval contains

at least one value of εi ∈ Rk (resp. εk1i − εk2

i for i : εi ∈ Rk1,k2). We will find that the

density of points in Vk (resp. Vk1,k2) is smallest near 0 (resp. −1 + δ), but even for the

interval [0,∆] (resp. [−1 + δ,−1 + δ+ ∆]), the number of points is Poisson with parameter

Θ(n∆K) = Θ(log n), allowing us to obtain the desired result for appropriately chosen C.

We first present our formal argument leading to a bound on V k, followed by a similar

argument leading to a bound on V k1,k2 . Let

Bk ≡m−1⋂i=0

[i∆, (i+ 1)∆] ∩ Vk 6= ∅ , (B.3)

where ∅ is the empty set. Clearly, Bk ⇒ V k ≤ 2∆. We now show that for any k ∈ K, we

have Pr(Bk)≤ 1/nK+2, for appropriately chosen C. Define

hj(x, θ) =

xj for x ∈ [θ, 1]

0 otherwise .(B.4)

It is easy to see that Vk follows a Poisson process with density nhK−1( · , 0). The number

of points in interval [i∆, (i+ 1)∆] is hence Poisson with parameter

n

∫ (i+1)∆

i∆hK−1(x) dx = ((i+ 1)K − iK)n∆K/K ≥ n∆K/K ≥ C log n/K ,


where we used the lower bound on ∆ in (B.2). It follows that

Pr([i∆, (i+ 1)∆] ∩ Vk = ∅) ≤ exp(−C log n/K) = 1/nC/K ≤ 1/n3 ,

for C ≥ 3K. We deduce by union bound over i = 0, 1, . . . ,m− 1 and De Morgan’s law on

(B.3) that

Pr(Bk)≤ m/n3 ≤ n1/K/n3 ≤ 1/n2 .

Using union bound over k we deduce that

Pr(∪k Bk

)≤ K/n2 (B.5)

We now present a similar argument to control V k1,k2 when K ≥ 2. Let m′ = (1− δ)/∆.

(To simplify notation we assume m′ is an integer. The case when it is not an integer can

be easily handled as well.) Let

Bk1,k2 ≡m−1⋂i=−m′

[i∆, (i+ 1)∆] ∩ Vk1,k2 6= ∅ , (B.6)

where ∅ is the empty set. Clearly, Bk1,k2 ⇒ V k1,k2 ≤ 2∆. We now show that for any k1 6= k2,

we have Pr(Bk1,k2

)≤ K(K − 1)/n2, for appropriately chosen C. It is easy to see that the

two-dimensional projection (x, y) = (εk1i , ε

k2i ) of points in Rk1,k2 follows a two-dimensional

Poisson process with density hK−2(x)I(y ∈ [0, 1]), cf. (B.4). We deduce that values in Vk

follow a one-dimensional Poisson process with density ng for g = hK−2( · , δ) ∗ I(∈ [−1, 0]),

where ∗ is the convolution operator. A short calculation yields

g(x) =

[(x+ 1)K−1 − δK−1

]/(K − 1) for x ∈ [−1 + δ, 0)[

1− δK−1]/(K − 1) for x ∈ [0, δ)

(1− xK−1)/(K − 1) for x ∈ [δ, 1]

0 otherwise.

The number of points in interval [i∆, (i+ 1)∆] is Poisson with parameter

n

∫ (i+1)∆

i∆g(x) dx .

Below we bound the value of this parameter for different cases on i, obtaining a bound of

(K + 3) log n in each case, for large enough C.


For −m′ ≤ i < 0, the smallest parameter occurs for i = −m′, since g(x) is monotone

increasing in [−1 + δ, 0]. Thus, the Poisson parameter is lower bounded by its value for

i = −m′, which is

n[(

(δ + ∆)K − δK)/K − δK−1∆

]/(K − 1)

≥n∆K/(K(K − 1)) ≥ C log n/(K(K − 1)) ≥ 3 log n ,

for C ≥ 3K(K − 1), using (B.2), and (δ + ∆)K ≥ ∆K +K∆δK−1 + δK .

For 0 ≤ i < m−m′, the Poisson parameter is

n[1− δK−1

]∆/(K − 1) ≥ n∆/(2(K − 1)) ≥ n∆K/(K(K − 1)) ≥ 3 log n ,

using δ ≥ 1/2 and K ≥ 2.

For (m−m′) ≤ i < m, the Poisson parameter is

n(∆−∆K((1 + i)K − iK)/K)/(K − 1) .

A short calculation allows us to again bound this below by (K+3) log n (the bound is slack

for K > 2): Note that

∆K((1 + i)K − iK) ≤ ∆K(mK − (m− 1)K) = 1− (1−∆)K

≤ K∆−K(K − 1)∆2/2 +K(K − 1)(K − 2)∆3/6 ,

where we used that (1 + i)K − iK is monotone increasing in i for i ≥ 0. Substituting back,

we obtain that the Poisson parameter is bounded by

n(1− (K − 2)∆/3)∆2/2 ≥ n∆2/4

for (K − 2)∆/3 ≤ 1/2, which occurs for sufficiently large n. Finally, ∆2 ≥ ∆K , hence

n∆2/4 ≥ n∆K/4 ≥ 3 log n for C ≥ 12.

Choosing C = 6K(K − 1), in all cases the Poisson parameter is bounded below by

3 log n. It follows that

Pr([i∆, (i+ 1)∆] ∩ Vk1,k2 = ∅) ≤ exp(−3 log n) = 1/n3 .

We deduce by union bound over i and De Morgan’s law on (B.6) that

Pr(Bk1,k2

)≤ 2m/n3 ≤ n1/K/n3 ≤ 1/n2 , (B.7)


for large enough n. Using union bound over (k1, k2) we deduce that

Pr(∪(k1,k2) Bk1,k2

)≤ K(K − 1)/n2 (B.8)

Combining (B.7) and (B.8) by union bound and using De Morgan’s law, we deduce that

Pr[(∩k Bk

)∩(∩(k1,k2) Bk1,k2

)]≥ 1−K2/n2

for large enough n. This implies that for large enough n, with probability at least 1−K2/n2

we have

max

(maxk

V k,maxk1,k2

V k1,k2

)≤ 2∆ ≤ 3(C log n/n)1/K = O∗(1/n1/K) ,

implying the main result for large enough n (note that K2/n2 < 1/n for large enough n).

For small values of n, we can simply choose f(n, k) large enough to ensure that the bound

holds with sufficient probability.

.

Lemma 7. For k ∈ K, let Rk(δ), Vk(δ) and V k(δ) be as defined by Eqs. (4.7), (4.8) and

(4.9) respectively. Fix K ≥ 1. There exists f(n) = O∗(1/n) such that for any δ ∈ (0, 1], the

following occurs: Let

B2 ≡

maxk∈K

V k(δ) ≤ f(n)/δK−1

. (B.9)

Then

Pr(B2) ≥ 1− 1/n .

Proof. The values in the set Vk ⊂ [0, 1] follow a one-dimensional Poisson process with rate

nδK−1. Choose f(n) = 6 log n/n. If 6 log n/(nδK−1) ≥ 1 there is nothing to prove, since

maxk∈K V k(δ) ≤ 1 by definition. Hence assume 6 log n/(nδK−1) < 1. Divide [0, 1] into

intervals of length ∆ = f(n)/(3δK−1) = 3 log n/(nδK−1) (to simplify notation, we assume

1/∆ ≥ 2 is an integer. The argument can easily be adapted to handle nδK−1/(3 log n) not

an integer). The probability that any particular interval of length ∆ does not contain a

point is no more than exp(−3 log n) = 1/n3. The number of intervals of length ∆ is 1/∆ =


nδK−1/(3 log n) ≤ n for large enough n. By union bound, with probability at least 1−1/n2,

each ∆-interval contains at least one point, implying that V k(δ) ≤ 2∆ = f(n)/δK−1 with

probability at least 1− 1/n2, as required.

In this section so far we considered the rate n Poisson process in [0, 1]K for convenience.

However, the results we proved can easily be transported to the closely related model of n

points distributed i.i.d. uniformly in [0, 1]K .

Lemma 8. Consider n points distributed i.i.d. uniformly in [0, 1]K . Lemmas 6 and 7 hold

for this model as well.

Proof. We use a standard coupling argument along with monotonicity of the considered

random variables with respect to additional points. Let P be a rate n/2 Poisson process

in [0, 1]K . The N points are distributed i.i.d. uniform [0, 1]K conditioned on the value of

N . Let B be the event N ≤ n. Clearly, B occurs with probability at least 1− 1/n2. Let Ube the process consisting of n points distributed i.i.d. in [0, 1]K . Conditioned on B, we can

couple the process P with the process U such that for every point in the Poisson process,

there is an identically located point in U .

We now show how to establish Lemma 7 for process U using such a coupling. Note

that maxk∈K V k(δ) is monotone non-increasing as we add more points. As such, an upper

bound on this quantity continues to hold if more points are added. For instance, consider

maxk∈K V k(δ). Let B′ be the event that

maxk∈K

V k(δ) ≤ f(n/2)/δK−1

under P. The proof of Lemma 7 shows that Pr(B′) ≥ 1 − (2/n)2. By union bound on Band B′, we deduce that Pr(B ∩ B′) ≥ 1 − 5/n2 ≥ 1 − 1/n, for large enough n. We deduce,

using a coupling as described above, that with probability at least 1 − 1/n, for process Uwe have

maxk∈K

V k(δ) ≤ f(n)/δK−1 ,


where f(n) = f(n/2), for large enough n. (For small values of n, we can simply choose

f(n) large enough to ensure that the bound holds with sufficient probability.) Thus we have

shown that Lemma 7 holds for process U .

Lemma 6 can similarly be established for process U using that

max

(maxk∈K

V k, max(k1,k2)∈K(2)

V k1,k2(δ)

)is monotone non-increasing as we add more points.

We now establish another result about n points (εj)nj=1 distributed i.i.d. uniformly in

[0, 1]K . This result is key to the proof of the tightness of Theorem 9 (Proposition 4).

For δ ∈ [0, 1] let

Rk1,k2(δ) = x ∈ [0, 1]K : xk1 ≥ xk2 − δ ; xk1 ≥ xk ∀k /∈ k1, k2, k ∈ K (B.10)

Let nk1,k2(δ) be the number of points in Rk1,k2(δ).

Lemma 9. Let B3 be the event that there for all k1, k2 ∈ K we have nk1,k2 ≥ 1 + n/K. For

δ = δ(n) ≥ 1/n0.49, we have that B3 occurs with high probability.

Proof. A short calculation shows that the volume of Rk1,k2(δ) is

v =1

K − 1

(1− (1− δ)K

K

)(B.11)

≥ 1

K+

δ

K − 1− δ2

2(B.12)

≥ 1 + δ

K(B.13)

for δ ≤ 2/(K(K − 1)). Now, the probability of εj ∈ Rk1,k2(δ) is exactly v. It follows that

nk1,k2 is distributed as Binomial(n, v). Notice E[nk1,k2 ] = nv ≥ n(1 + δ)/K. We obtain

Pr(nk1,k2 < 1 + n/K) ≤ exp− Ω

(nδ2)

= exp− Ω

(n0.02

)= o(1) (B.14)

using a standard Chernoff bound (e.g., see Durrett [Durrett, 2010]). Using union bound

over pairs k1, k2 we deduce that B3 occurs with probability o(1), i.e., event B3 occurs with

high probability.


B.2 Proof of Theorem 9 upper bound

We now present the complete proof of Theorem 9. We start by proving the lemmas stated

in Section 4.4.

Proof of Lemma 2. Suppose not, and let C be a connected component of G(M) where all

vertices are unmarked. Abusing notation, let CL (resp. CE) denote the types in TL (resp.

TE) that are in C. By the definition of the marks, we know that all agents of types in

CL ∪ CE must be matched. Furthermore, by the definition of G(M), an agent whose type

is in CL can only be matched to an agent whose type is in CE and vice versa. Therefore,

we must have that∑

k∈CL nk =∑

q∈CE nq, which contradicts Assumption 1.

Proof of Lemma 3. By invoking Lemma 6, Lemma 7 and Lemma 8, for each t we have that

w.p. at least 1 − 2nt

the event (B1(t, δ) ∩ B2(t, δ)) occurs. As the total number of types

is upper bounded by K + Q, we apply an union bound to conclude that w.p. at least

1− 2(K+Q)n∗ , the event

⋂t∈TL∪TE (B1(t, δ) ∩ B2(t, δ)) occurs.

Before moving on to the key lemmas, we introduce some definitions. Given a type

t ∈ TL ∪ TE we denote by ν(t) or simple ν, the points in t. That is, for each agent j of type

t, we define νj as follows:

νj =

εj if t ∈ TEηj if t ∈ TL

For a fixed t ∈ TL ∪ TE and t′ ∈ ϑ(t), let βtt′ be defined as:

βtt′ =

−αtt′ if t ∈ TEαtt′ − u(t, t′) if t ∈ TL

Using the above notation, we can re-write the conditions in Proposition 3 associated to

a fixed type t ∈ TL ∪ TE as follows:


(ST) For every k, k′ ∈ ϑ(t):

minj∈t∩M(k)

νkj − νk′j ≥ βkt − βk′t ≥ max

j∈t∩M(k′)νkj − νk

′j .

(IM) For every k ∈ ϑ(t):

minj∈t∩M(k)

νkj ≥ βkt ≥ maxj∈q∩U

νkj .

As all the ν variables are in [0, 1], then the above conditions can be interpreted as

geometric conditions in the [0, 1]D(t)-hypercube.

The proof of Lemma 4 is partitioned into two lemmas. Given a core solution (M,α), let

the event D(t, δ) be defined as:

D(t, δ) = βtz ≥ δ ∀z ∈ ϑ(t). (B.15)

Lemma 10 below deals with D(t, δ) whereas Lemma 11 deals with the complement D(t, δ).

Together they imply Lemma 4.

Lemma 10. Consider a core solution (M,α) and a type t. Let the events F1(t), D(t, δ) and

B2(t, δ) be as defined by Eqs. (4.12), (B.15) and (4.11) respectively. Under F1(t)∩D(t, δ)∩B2(t, δ), we have maxt′∈ϑ(t)


t,t′

)≤ f2(nt)/δ

D(t)−1, where f2 is as defined in the

statement of Lemma 7.

Proof. Let D = D(t). Fix k ∈ ϑ(t) and consider the orthotope Rk = Rk(t, δ) as defined by

Eq. (4.7). As D(t, δ) occurs, βtz ≥ 1/δ for all z ∈ ϑ(t) and therefore Rk can only contain

points corresponding to agents in M(k) ∪ U . By using the notation introduced above,

condition (IM) in Proposition 3 implies: αmaxkt − αmin

kt ≤ minj∈t∩M(k) νkj − maxj∈q∩U νkj .

However, minj∈t∩M(k) νkj − maxj∈q∩U νkj ≤ minj∈Rk∩M(k) ν

kj − maxj∈Rk∩U ν

kj ≤ V k(t, δ),

where V k(t, δ) is as defined by Eq. (4.9). Therefore, for each k ∈ ϑ(t) we must have

αmaxkt − αmin

kt ≤ V k(t, δ). Finally, under B2(t, δ) we have maxk∈ϑ(t) Vk(t, δ) ≤ f2(nt)/δ

D−1,

which completes the result.

Lemma 11. Consider a core solution (M,α) and a type t. Let F1(t) be the event defined

in Eq. (4.12). Let the event B1(t, δ) be as defined by Eq. (4.10), and let the event D(t, δ)

denote the complement of the event defined by Eq. (B.15). Under F1(t) ∩D(t, δ) ∩ B1(t, δ),


we have maxt′∈ϑ(t)


t,t′

)≤ f1(nt, D(t))+δ, where f1 is as defined in the statement

of Lemma 6.

Proof. Suppose t ∈ TE . Consider the unit hypercube in RK . For each j ∈ E such that

τ(j) = t, let εj ∈ [0, 1]K denote the vector of realizations of εkj for every k ∈ TL. By

condition (ST) in Proposition 3, we can partition the [0, 1]K hypercube into |ϑ(t)| regions

such that all the points ε corresponding to agents matched to k ∈ ϑ(t) must be contained in

the corresponding region. In particular, for each k ∈ ϑ(t), we define Z(k) ⊆ [0, 1]K to be the

region corresponding to type k, with Z(k) = ∩k′∈ϑ(t), k′ 6=kx ∈ [0, 1]K : xk−xk′ ≥ αk′t−αkt.Note that the region Z(k) can only contain points corresponding to agents matched to k or

unmatched.

Let k∗ = argmaxk∈TLαtk∗ : k ∈ ϑ(t), and let Rk∗ = Rk∗(t) be as defined by Eq. (4.1).

By condition (ST) in Proposition 3, we have that for all k ∈ ϑ(t):

minj∈t∩M(k∗)

εk∗j − εkj ≥ αkt − αk∗t ≥ max

j∈q∩M(k)εk∗j − εkj .

As αkt − αk∗t ≤ 0 for all k ∈ ϑ(t), we must have Rk∗ ⊆ Z(k∗). Let V k∗ = V k∗(t) be as

defined in Eq. (4.4). We claim that αmaxk∗,t −αmin

k∗,t ≤ V k∗ . To see why this holds, consider two

separate cases. First, suppose there is at least one point corresponding to an unmatched

agent inRk∗ . By condition (IM) in Proposition 3, we must have minj∈t∩M(k∗) εk∗j ≥ −αk∗t ≥

maxj∈t∩U εk∗j . Hence, αmax

k∗,t − αmink∗,t ≤ minj∈t∩M(k∗) ε

k∗j − maxj∈t∩U εk

∗j ≤ V k∗ as desired.

For the second case, suppose that all points in Rk∗ correspond to matched agents. As

maxj∈t∩U εk∗j ≥ 0, we must have αmax

k∗,t − αmink∗,t ≤ minj∈t∩M(k∗) ε

k∗j ≤ minj∈Rk∗ ε

k∗j ≤ V k∗ ,

as the difference between 0 and the minj∈Rk∗ εk∗j is upper bounded by V k∗ . Therefore, we

conclude αmaxk∗,t − αmin

k∗,t ≤ V k∗ .

Next, we consider the bound for any arbitrary type k ∈ ϑ(t). By condition (ST) in

Proposition 3, we have that for all k ∈ ϑ(t):

αmaxk∗t + min

j∈t∩M(k∗)εk∗j − εkj ≥ αkt ≥ max

j∈q∩M(k)εk∗j − εkj + αmin

k∗t .

Therefore,

αmaxkt − αmin

kt ≤ αmaxk∗t − αmin

k∗t + minj∈t∩M(k∗)

εk∗j − εkj − max

j∈q∩M(k)εk∗j − εkj .


From our previous bound, we have that αmaxk∗t − αmin

k∗t ≤ V k∗ . We now want an upper

bound on minj∈t∩M(k∗) εk∗j − εkj − maxj∈q∩M(k) ε

k∗j − εkj . Let Rk∗,k = Rk∗,k(t, delta) and

V k∗,k = V k∗,k(t) be as defined by Eqs. (4.2) and (4.6). We shall show that minj∈t∩M(k∗) εk∗j −

εkj −maxj∈q∩M(k) εk∗j − εkj ≤ V k∗,k + δ. Recall that, under D(t, δ), we have δ ≥ αk∗t.

To that end, note that all points in Rk∗,k must correspond to agents matched to k∗ or

matched to k, as the region Rk∗,k cannot contain unmatched without violating condition

(IM). Furthermore, as Rk∗ ⊆ Z(k∗) and Rk∗ ∩ Rk∗,k 6= ∅, at least one point in Rk∗,k

corresponds to an agent matched to k∗. We now consider two separate cases, depending on

whether Rk∗,k contains a at least one point matched to k. First, suppose Rk∗,k contains a

at least one point matched to k. Then, the bound trivially applies as

minj∈t∩M(k∗)


j∈t∩M(k)εk∗j − εkj ≤ min

j∈Rk∗,k∩M(k∗)εk∗j − εkj − max

j∈Rk∗,k∩M(k)εk∗j − εkj ≤ V k∗,k.

Otherwise, Rk∗,k contains only points matched to k∗. In that case,

minj∈t∩M(k∗)


j∈q∩M(k)εk∗j − εkj ≤ min

j∈Rk∗,kεk∗j − εkj − (1 + αk∗t) ≤ V k∗,k + δ,

as desired. Overall, we have shown that:

maxk∈ϑ(t)

(αmaxtk − αmin

tk

)≤ max

(V k∗ , max

k∈ϑ(t)

(V k∗ + V k∗,k + δ

)).

Under B1(t, δ) we have max(V k∗ ,maxk V

k∗,k)≤ f1(nt,K), implying

max

(V k∗ , max

k∈ϑ(t)

(V k∗ + V k∗,k + δ

))≤ 2f1(nt,K) + δ,

as desired.

To conclude, we briefly discuss the changes when t ∈ TL. Consider the unit hypercube in

RQ. For each j ∈ L such that τ(j) = t, let ηj ∈ [0, 1]Q denote the vector of realizations of ηqj

for every q ∈ TE . For each q ∈ ϑ(t), we define Z(q) ⊆ [0, 1]Q to be the region corresponding

to type q. The main difference with the case in which t ∈ TE is that we need to define the

regions Z(q) in terms of the η instead of η. To that end, let βkq = αkq − u(k, q). By the

(ST) condition in Proposition 3, we must have:

mini∈t∩M(q′)

ηq′i − η

qi ≥ αtq′ − αtq ≥ max

i∈t∩M(q)ηq′i − η

qi ,


or equivalently,

mini∈t∩M(q′)

ηq′i − η

qi ≥ βtq′ − βtq ≥ max

i∈t∩M(q)ηq′i − η

qi .

By using β instead of α, the same geometric intuition as before applies. Then, we define

Z(q) = ∩q′∈ϑ(t), q′ 6=qx ∈ [0, 1]Q : xq − xq′ ≥ βqt − βq′t. To select q∗, we just select the one

with smallest βqt. The rest of the proof remains the same.

Proof of Lemma 4. Lemma 4 immediately follows from Lemmas 10 and 11.

Proof of Lemma 5. Consider a core solution (M,α). Let D = D(t). Fix a type t∗ ∈ ϑ(t),

and let k∗ = argmaxk∈ϑ(t)βtk. We start by showing that, under F2(t) ∩ B1(t, δ), we must

have αmaxtk∗ −αmin

tk∗ ≤(αmaxt,t∗ − αmin

t,t∗)

+f1(nt, D(t))+2δ. If k∗ = t∗, the claim follows trivially.

Otherwise, let Rk∗,t∗ = Rk∗,t∗(t, δ) and V k∗,t∗ = V k∗,t∗(t, δ) be as defined by Eqs. (4.2) and

(4.6). We show that minj∈t∩M(k∗) νk∗j − νt

∗j −maxj∈q∩M(t∗) ν

k∗j − νt

∗j ≤ V k∗,t∗ + δ.

To that end, note that all points in Rk∗,t∗ must correspond to agents matched to k∗ or

matched to t∗, as under F2(t) all agents in t are matched. Furthermore, by the definition of

k∗, Rk∗,t∗ must contain a point corresponding to an agent matched to k∗. We now consider

two separate cases, depending on whether Rk∗,t∗ contains at least one point corresponding

to an agent matched to t∗. First, suppose Rk∗,t∗ contains at least one point corresponding

to an agent matched to t∗. Then,

minj∈t∩M(k∗)

νk∗j −νt

∗j − max

j∈t∩M(t∗)νk∗j −νt

∗j ≤ min

j∈Rk∗,t∗∩M(k∗)νk∗j −νt

∗j − max

j∈Rk∗,t∗∩M(k)νk∗j −νt

∗j ≤ V k∗,t∗ .

Otherwise, Rk∗,t∗ contains only points matched to k∗. In that case,

minj∈t∩M(k∗)

νk∗j − νt

∗j − max

j∈t∩M(t∗)νk∗j − νt

∗j ≤ min

j∈Rk∗,t∗νk∗j − νt

∗j − 1 ≤ V k∗,t∗ + δ,

as desired. By condition (ST) in Proposition 3, we must have:

αmaxtk∗ −αmin

tk∗ ≤ αmaxtt∗ −αmin

tt∗ + minj∈t∩M(k∗)

νk∗j −νt

∗j − max

j∈t∩M(t∗)νk∗j −νt

∗j ≤ αmax

tt∗ −αmintt∗ +V k∗,t∗+δ


Next, consider an arbitrary k ∈ ϑ(t) with k 6= t∗, k∗. By condition (ST) in Proposition 3,

we must have:

αmaxkt − αmin


k∗t + minj∈t∩M(k∗)

νk∗j − νkj − max

j∈q∩M(k)νk∗j − νkj .

Let Rk∗,k = Rk∗,k(t, δ) and V k∗,k = V k∗,k(t) be as defined by Eqs. (4.2) and (4.6).

By repeating the same arguments as before, we can show that minj∈t∩M(k∗) νk∗j − νkj −

maxj∈q∩M(k) νk∗j − νkj ≤ V k∗,k + 2δ. Hence,

αmaxkt − αmin


k∗t + V k∗,k + δ ≤ αmaxtt∗ − αmin

tt∗ + V k∗,t∗ + V k∗,k + 2δ.

To conclude, note that

maxk∈ϑ(t)

(αmaxkt − αmin

kt

)≤(αmaxtt∗ − αmin

tt∗)+2

(maxk∈ϑ(t)

V k∗,k)

+2δ ≤(αmaxtt∗ − αmin

tt∗)+2f1(nt, D)+2δ,

where the last inequality follows from the fact that B1(t, δ) occurs by hypothesis.

We can now proceed to the proof of the main theorem.

Proof of Theorem 9. Let n∗ = mint∈TL∪TE nt. Under Assumption 2, we have that n∗ =

Θ(n). Let δ = 1/(n∗)1/max(K,Q). For each t ∈ TL ∪ TE , let the events B1(t, δ) and B2(t, δ)

be as defined by Eqs. (4.10) and (4.11) respectively. We start by showing that, under⋂t∈TL∪TE (B1(t, δ) ∩ B2(t, δ)), we must have C ≤ O∗

(1

max(K,Q)√n

).

To that end, construct the type-adjacency graph G(M) as defined in Section 4.4. For

each vertex v, we denote by d(v) the minimum distance between v and any marked vertex

(that is, d(v) = 0 if v is marked, d(v) = 1 if v is unmarked and has a marked neighbour, and

so on). By Lemma 2, we know that w.p.1, each connected component of G(M) must contain

at least one marked vertex, so d(v) is well-defined for all v. Let Cd = v ∈ C : d(v) = d,that is Cd is the set of vertices that are at distance d from a marked vertex. We now show the

result by induction in d. In particular, we show that, under⋂t∈TL∪TE (B1(t, δ) ∩ B2(t, δ)),

for each t ∈ Cd we have that maxk∈ϑ(t)

(αmaxtk − αmin

tk

)≤ gd(n

∗,max(K,Q)) for some

gd(n∗,max(K,Q)) = O∗( 1

n1/max(K,Q) ).


We start by showing that the claim holds for the base case d = 0. For each t ∈ C0, either

all agents in t are unmatched or at least one agent is matched. In the former case, we can just

ignore type t as it will not contribute to the size of the core. In the latter, we note that w.p.1

the event F1(t) as defined in the statement of Lemma 4 must hold. Therefore, we can ap-

ply Lemma 4 to obtain maxt′∈ϑ(t)


t,t′

)≤ max

(f1(nt, D(t)) + δ, f2(nt)/δ

D(t)−1),

where f1 and f2 are as defined in the statement of the lemma. To conclude the proof of the

base case, let

g0(n∗,max(K,Q)) = max(f1(n∗,max(K,Q)) + δ, f2(n∗)/δmax(K,Q)−1

).

By the definition of f1, f2, and δ, together with Assumption 2, we have g0(n∗,max(K,Q)) =

O∗( 1n1/max(K,Q) ). Therefore, we have shown that, for every t ∈ C0, we have

maxk∈ϑ(t)

(αmaxtk − αmin

tk

)≤ g0(n∗,max(K,Q)).

Now suppose the result holds for all d′ ≤ d, we want to show it holds for d + 1. Fix

t ∈ Cd+1. By definition of Cd+1, we have that all agents in t must be matched and

therefore w.p.1, the event F2(t) as defined in the statement of Lemma 5 occurs. More-

over, there must exist a t∗ such that the vertex corresponding to t∗ is Cd and t∗ ∈ ϑ(t).

By induction, we have that(αmaxtt∗ − αmin

tt∗)≤ gd(n

∗,max(K,Q)) for gd(n∗,max(K,Q)) =

O∗( 1n1/max(K,Q) ). Further, by Lemma 5, we know that under F2(t) ∩ B1(t, δ), we have

maxt′∈ϑ(t)


t,t′

)≤(αmaxt,t∗ − αmin

t,t∗)

+ 2f1(nt, D(t)) + 2δ, where B1(t, δ) as defined

by Eq. (B.1) and f is as defined in the statement of Lemma 6. Therefore, by letting

gd+1(n∗,max(K,Q)) = gd(n∗,max(K,Q)) + 2f1(n∗,max(K,Q)) + 2δ, we have show that

with probability at least 1− d+1n∗ , we have maxk∈ϑ(t)

(αmaxtk − αmin

tk

)≤ gd+1(n∗,max(K,Q))

with gd+1(n∗,max(K,Q)) = O∗( 1n1/max(K,Q) ).

Next, we note that maxv d(v) is upper bounded by K + Q. Hence, for every t ∈ TL ∪TE , we have maxk∈ϑ(t)

(αmaxtk − αmin

tk

)≤ gK+Q(n∗,max(K,Q)) for gK+Q(n∗,max(K,Q)) =

O∗( 1n1/max(K,Q) ) and therefore

maxt∈TL∪TE

maxk∈ϑ(t)

(αmaxtk − αmin

tk

)≤ gK+Q(n∗,max(K,Q)).

To conclude, by Lemma 3 we have that with probability at least 1− 2(K+Q)n∗ , the event


⋂t∈TL∪TE (B1(t, δ) ∩ B2(t, δ)) occurs. In all other cases, we just use the fact that the size of

the core is upper-bounded by a constant C <∞. Hence,

E[C] =

∑(k,q)∈TL×TE N(k, q)

(αmaxkq − αmin

kq

)∑

(k,q)∈TL×TE N(k, q)

≤ (K +Q)gK+Q(n∗,max(K,Q)) + C2(K +Q)

n∗

= O∗(

1max(K,Q)

√n

)implying the main result for large enough n (note that 2(K+Q)

n∗ = Θ∗(1/n)).

B.3 Theorem 9 lower bound: Proof of Proposition 4

Proof of Proposition 4.

Claim 4. For this market, all labor agents of types different from k∗ will be matched in the

core.

Proof. We know that there is some employer j who is either unmatched or matched to a

labor agent i′ of type k∗. Consider any matching where a labor agent i of type k 6= k∗ is

unmatched. Now Φ(i′, j) = εi′ + ηk∗j ≤ 1 + 1 = 2, whereas Φ(i, j) ≥ u(k, 1) = 3, hence

the weight of such a matching can be increased by instead matching j to i. It follows

that in any maximum weight matching, all labor agents with type different from k∗ are

matched. Finally, recall that every core outcome lives on a maximum weight matching, cf.

Proposition 3

Among agents i ∈ k∗, exactly one agent will be matched, specifically agent i∗ =

arg maxi∈k∗ ηi. Let j∗ be the agent matched to i∗ (break ties arbitrarily). Recall that

core solutions always live on a maximum weight matching, and in case of multiple maxi-

mum weight matchings, the set of vectors α such that (M,α) is a core solution is the same

for any maximum weight matching M . This allows us to suppress the matching, and talk

about a vector α being in the core, cf. Proposition 3. The (IM) condition in Proposition 3


for the pair of types (k∗, 1) are

ηi∗ ≥ αk∗ ≥ maxi∈k∗\i∗

ηi , (B.16)

and the slack condition αk∗ ≥ −εk∗j∗ . The (IM) conditions for types (k, 1) for k 6= k∗ are

3 + mini∈k

ηi∗ ≥ αk ≥ − minj∈M(k)

εkj . (B.17)

The stability conditions are

minj∈M(k)

εkj − εk′j ≥ αk′ − αk ≥ max

j∈M(k′)εkj − εk

′j , (B.18)

for all k 6= k′. It is easy to see that Eq. (B.18) with k′ = k∗ implies αk ≤ 2 for all k 6= k∗.

Hence, the upper bound in Eq. (B.17) is slack. Consider the left stability inequality with

k′ = k∗. As Eq.(B.16) implies αk∗ ≥ 0, we must have

αk ≥ − minj∈M(k)

εkj − εk∗j ≥ − minj∈M(k)

εkj

implying that the lower bound in (B.17) is also slack. Thus a vector α is in the core if and

only if conditions (B.16) and (B.18) are satisfied.

For simplicity, we start with the special case K = 2, with the two types of labor being k

and k∗. To obtain intuition, notice that from Eq. (B.16) we have αk∗n→∞−−−→ 1 in probability,

and when we use this together with Eq. (B.18) we obtain αkn→∞−−−→ 2 in probability. (We do

not use these limits in our formal analysis below.) Hence, we focus on Eq. (B.16) together

with

minj 6=j∗

εkj − εk∗j ≥ αk∗ − αk ≥ εkj∗ − εk∗j∗ . (B.19)

where j∗ = arg minj εkj − εk∗j . Now, Xj = εkj − εk∗j are distributed i.i.d. with density

U [0, 1] ∗ U [−1, 0] which is

f(x) =

1− |x| for |x| ≤ 1

0 otherwise.(B.20)

(Note that if we draw n + 1 samples from this distribution, it is not hard to see that

E[(minj 6=j∗ Xj)−Xj∗ ] = Θ(1/√n).) We lower bound the expected core size as follows: Let

Xj = εkj − εk∗j . Let B be the event that exactly one of the Xj ’s is in [−1,−1 + 1/√n], and


no Xj is in [−1 + 1/√n,−1 + 2/

√n]. Under f the probability of being in [−1,−1 + 1/

√n]

is 1/(2n) and the probability of being in [−1 + 1/√n,−1 + 2/

√n] is 3/(2n). It follows that

Pr(B) =

(n+ 1

1, 0, n

)1

2n

(1− 2/n

)n= Ω(1) . (B.21)

Claim 5. Consider the case K = 2. Under event B, for any core vector (αk∗ , αk), for any

value α′k ∈ [αk∗ + 1 − 2/√n, αk∗ + 1 − 1/

√n], we have that vector (αk∗ , α

′k) is in the core.

In particular, C = Ω(1/√n).

Proof. Eq. (B.19) is satisfied since event B holds. Since, αk′ can take any value in an interval

of length 1/√n, it follows that C = Ω(1/

√n) under B.

Combining Claim 5 with Eq. (B.21), we obtain that E[C] = Ω(1/√n) as desired.

We now construct a similar argument for K > 2, with K = TL\k∗ being the other

labor types, all of whose agents are matched. It again turns out that αk∗n→∞−−−→ 1 in

probability, and when we use this together with Eq. (B.18) we obtain αkn→∞−−−→ 2 ∀k ∈ K

in probability (but we do not prove or use these limits).

Considering only the dimensions in K (recall |K| = K − 1 here) of each εj , let B3 be the

event as defined in Lemma 9 with δ = 1/n0.51.

Claim 6. Let k = arg mink∈K αk and let k = arg maxk∈K αk. Under event B3, we claim

that

αk − αk ≤ δ (B.22)

Proof. From Proposition 3, we know that the set of core α’s is a linear polytope, hence

it is immediate to see that the set of θ’s is an interval. Let k = arg mink∈K αk and let

k = arg maxk∈K αk. Under event B3, we claim that αk − αk ≤ δ. We can argue this by

contradiction: Suppose αk − αk > δ. One can see that all j’s such that εKj ∈ Rk,k(δ), cf.

(B.10), will be matched to type k, with the possible exception of j∗. Thus, under B3, the

number of employers matched to type k is bounded below by

nk,k − 1 ≥ ((K − 1)n+ 1)/(K − 1) > n ,

which is a contradiction, implying (B.22).


The above claim bounds the maximum difference between α’s corresponding to any pair

of types in K. Intuitively, note that all types in K have the same u and therefore the same

distribution for the θ variables of the agents in such type. Moreover, all types in K have

the same number of agents. Hence, one would expect the α’s to be equal. While true in the

limit, for each finite n we need to account for the stochastic fluctuations in given realization.

Therefore, we can show that no pair of α’s in K can differ by more than δ. The next claim

follows immediately from Claim 6.

Claim 7. Let k ∈ K be an arbitrary type. Under event B3, we claim that

maxk′∈K

(εk′j − εkj

)≤ δ ∀j ∈M(k) (B.23)

Proof. By Claim 6, we have that under B3, |αk − αk′ | ≤ δ for all k′ ∈ cK. By the stability

condition in Eq. (B.18), we have

δ ≥ αk − αk′ ≥ εk′j − εkj ∀j ∈M(k), ∀k′ ∈ K.

Therefore, for every j ∈M(k) we must have δ ≥ maxk′∈K εk′j − εkj as desired.

Next, we focus on the stability conditions involving type k∗. For each k ∈ K, the stability

condition is:

εk∗j∗ − εkj∗ ≥ αk − αk∗ ≥ max

j∈M(k)εk∗j − εkj , . (B.24)

where j∗ is the employer matched to i∗. For each j ∈ E , let Xj be defined as Xj =

(maxk∈K εkj )− εk∗j . The Xj are distributed i.i.d. with cumulative distribution F (−1 + θ) =

θK/K for θ ∈ [0, 1] (we will not be concerned with the cumulative for positive values). Let

B be the event that exactly one of the Xj ’s is in [−1,−1 + 1/n1/K ] (this will be Xj∗), and

no Xj is in [−1 + 1/n1/K ,−1 + 2/n1/K ]. Under cumulative F , the probability of being in

[−1,−1 + 1/n1/K ] is 1/(Kn) and the probability of being in [−1 + 1/n1/K ,−1 + 2/n1/K ] is

2K/(Kn). It follows that

Pr(B) =

(n+ 1

1, 0, n

)1

Kn

(1− 2K/(Kn)

)n= Ω(1) . (B.25)

Clearly, under B, we must have j∗ = arg minj∈E Xj . Keeping this in mind, we state and

prove our last claim.


Claim 8. Suppose B3 ∩ B occurs. Take any core vector (αk∗ , (αk)k∈K). Then

θ ∈ R : (αk∗ , (αk + θ)k∈K) is in the core (B.26)

is an interval of length at least 1/n1/K − 2δ = Ω(1/n1/K). In particular, C ≥ Ω(1/n1/K).

Proof. Define

θ = 1− 2/n1/K + δ − αk + αk∗

θ = 1− 1/n1/K − αk + αk∗

We claim that, under B3 ∩ B, we have that α(θ) = (αk∗ , (αk + θ)k∈K) is in the core for all

θ ∈ [θ, θ]. To establish this, we need to show that conditions (B.16) and (B.18) are satisfied.

Since α belongs to the core, we immediately infer that (B.16) holds, and also (B.18) when

k∗ /∈ k, k′ by definition of α(θ). That leaves us with (B.24). Now, for any k ∈ K and

θ ∈ [θ, θ] we have

αk(θ) = αk + θ ≤ αk + θ ≤ αk + θ = 1− 1/n1/K + αk∗ ≤ εk∗j∗ − εkj∗ + αk∗ ,

where used the definitions of k and θ, and the fact that B occurs (so 1−1/n1/K ≤ εk∗j∗ −εkj∗).This establishes the left inequality in (B.24). Similarly, for any k ∈ K we have

αk(θ) = αk + θ ≥ αk + θ ≥ αk + θ = 1− 2/n1/K + δ + αk∗

≥ εk∗j −maxk′∈K

εk′j + δ + αk∗ ≥ εk∗j − εkj + αk∗ ∀j ∈M(k) ,

where used the definitions of k and θ for the first two inqualities, and the fact that B occurs

(so 1 − 2/n1/K ≥ εk∗j −maxk′∈K εk′j , ∀j ∈ M(k)). Finally, the last inequality follows from

B3 and Claim 7 (which implies −maxk′∈K εk′j + δ ≥ −εkj for j ∈M(k)). This establishes the

right inequality in (B.24). Thus, we have shown that α(θ) is in the core for all θ ∈ [θ, θ].

The length of this interval is 1/n1/K − (αk − αk) − δ ≥ 1/n1/K − 2δ = Ω(1/n1/K), using

(B.22). Therefore, that E[C] = Ω(1/n1/K) under B3 ∩ B.

Using Lemma 9 and Eq. (B.25) we have

Pr(B3 ∩ B) = Ω(1) .

Combining with the claim above we obtain that E[C] = Ω(1/n1/K).


B.4 Proof of Theorem 10

We start by restating Theorem 10 and discussing the structure of the proof.

Theorem (Restatement of Theorem 10). Consider the setting in which K ≥ 2, Q = 1,

nE > nL and let m = nE − nL. In addition, suppose that u(k, 1) ≥ 0 for all k ∈ TL. Then,

under Assumption 2, we have E[C] ≤ O∗(

1

n1Km

K−1K

).

Note that Assumption 1 is automatically satisfied under the hypotheses of the theorem.

The idea of the proof is as follows. First, we show a bound on the expectation of

mink∈TLαmaxk −αmin

k . In particular, we show that E[mink∈TLαmax

k − αmink

]= O∗

(1

n1Km

K−1K

).

To do so, we note that by condition (IM) in Proposition 3, we must have

mink

(αmaxk − αmin

k

)≤ min

k∈TL

(min

j∈M(k)εkj −max

j∈Uεkj

).

Then, we consider two separate cases to prove the result, depending the size of the imbal-

ance. When m ≤ log(n), the result is shown in Lemma 12, which we prove via an upper

bound on mink∈TL(

minj∈M(k) εkj

). On the contrary, when m ≥ log(n), the result is shown

in Lemma 15. The proof of Lemma 15 relies mainly on the geometry of a core solution

which (roughly) allows us to first control the largest of the α’s (all α’s must be negative i in

the core since some employers are unmatched, and we control, roughly, the least negative

α).

Next, we then show that, for every pair of types k, q ∈ TL we must have

E

[min

j∈M(k)(εkj − εqj)− max

j∈M(q)(εkj − εqj)

]= O∗

(1

n

).

By Condition (ST) in Proposition 3, this implies that for fixed k, q ∈ TL, the expected

maximum variation in αk − αq in the core is bounded by O∗(

1n

).

Finally, we use the bounds in the first two steps to argue that, for every type k ∈ TL,

E[αmaxk − αmin

k ]

= O∗(

1

n1Km

K−1K

),

which implies E[C] = O∗(

1

n1Km

K−1K

). This is done in the proof of Theorem 10.

We now show our bound on E[minkαmax

k − αmink

]. To that end, let Zk = minj∈M(k) ε

kj

and Uk = maxj∈U εkj . By Condition (IM) in Proposition 3, E[mink |αmax

k − αmink |

]≤


E[minkZk−Uk], and therefore we will focus on bounding E[minkZk−Uk]. As a reminder,

we have defined m = nE −nL and δn = log(n)

n1Km

K−1K

. Also, in all lemmas we are working under

the assumptions of the theorem, that is, K ≥ 2, Q = 1, nE > nL and Assumption 2.

Lemma 12. Suppose m ≤ 6K log(nE). Then, there exists a constant C3 = C3(K) < ∞such that E

[minkαmax

k − αmink

]≤ 2C3

log(n)

n1Km

K−1K

.

Proof. Let Zk = minj∈M(k) εkj , Uk = maxj∈U εkj and δn = log(n)

n1Km

K−1K

. By Condition (IM) in

Proposition 3, E[minkαmaxk −αmin

k ] ≤ E[minkZk−Uk]. As Uk is a non-negative random

variable, we have E[minkZk − Uk] ≤ E[minkZk]. Therefore,

E[minkαmax

k −αmink ] ≤ E

[minkZk − Uk

]≤ E

[minkZk

]≤ C3δn+Pr

(minkZk ≥ C3δn

),

using Zk ≤ 1.

To finish the proof, it suffices to show that Pr (mink Zk ≥ C3δn) ≤ C3δn. Hence, our

next step is to bound Pr (mink Zk ≥ C3δn). Now mink Zk ≥ C3δn implies that all j such

that εj ∈ [0, C3δn]K are unmatched. But there are only m unmatched employers. It follows

that

Pr

(minkZk ≥ C3δn

)≤ Pr

(at most m points in the hypercube [0, C3δn]K

)

Let X ∼ Bin(nE , (C3δn)K

)be defined as the number of points, out of nE in total,

that fall in the hypercube [0, C3δn]K . By assumption, m ≤ 6K log(n) ⇒ (C3δn)K ≥(C3 log n/m)K/n ≥ 2K log nK/n ≥ 4(log n)2/n defining C3 ≥ 12K and using K ≥ 2.

Further using n ≤ 2nE we obtain E[X] = nE (C3δn)K ≥ (n/2)4(log n)2/n = 2(log n)2. It

follows that

Pr

(minkZk ≥ C3δn

)≤ Pr (X ≤ 6K log(n)) ≤ exp(−Ω((log n)2)) ≤ 1

n≤ C3δn

where the second inequality was obtained by applying the Chernoff bound. Hence, we have

shown that

E[minkαmax

k − αmink ] ≤ E

[minkZk

]≤ C3δn + Pr

(minkZk ≥ C3δn

)≤ 2C3δn,


which completes the proof.

We now establish an upper bound for the case in which m ≥ 6K log(nE). For the

following results up to Lemma 15 we shall assume m ≥ 6K log(nE).

Before we move on, we briefly give some geometric intuition regarding the problem. For

each agent j ∈ E , let εj = (ε1j , . . . , εKj ) denote the profile of values assigned by the K types

of agents in L to agent j. Given our stochastic assumptions, all points εj will be distributed

in the [0, 1]K hypercube. Using Proposition 3, we can partition the [0, 1]K-hypercube into

K+1 disjoint regions: K of them containing the nk points corresponding to agents matched

to type k (1 ≤ k ≤ K), and one region containing all unmatched agents. Furthermore, the

region containing the unmatched agents is an orthotope2 that has the origin as a vertex.

This follows for the (IM) constraints in Proposition 3.

To that end, let O be the set of K-orthotopes contained in [0, 1]K that have the origin

as a vertex. Suppose R is expanded by the same amount θ in each coordinate direction.

Define D(R) as the smallest value of θ such that an additional point εj is contained in

the expanded orthotope. (If one of the side lengths becomes 1 before an additional point

is reached, then define D(R) = 0. This will never occur for R that contains only the

unmatched agents.) As usual, let Zk = minj∈M(k) εkj and Uk = maxj∈U εkj . We want to

show that E [minkZk − Uk] ≤ C5δn, for some constant C5 = C5(K) < ∞. To that end,

note that minkZk − Uk is equal to D(R) for some orthotope R ∈ O. In particular,

minkZk −Uk is equal to D(R) when R is the orthotope that “tightly” contains all the m

points in U .

For R ∈ O, let V (R) be defined as the volume of R. In addition, we define |R| to be the

number of points contained in R. We start by showing that, given that m ≥ 6K log(n), an

orthotope in O of volume less than m4nE

in extremely unlikely to contain m points.

Lemma 13. Suppose m ≥ 6K log(n). For R ∈ O such that V (R) < m4nE

, we have

2An orthotope (also called a hyperrectangle or a box) is the generalization of a rectangle for higher

dimensions


Pr (|R| = m) ≤ 1nK+1 , where V (R) denotes the volume and |R| denotes the number of points

in R.

Proof. Let X denote number of points in an orthotope in O of volume m4nE

. Then, X ∼Bin

(nE , m

4nE

). We have µ = E[X] = m/4. Using a Chernoff bound we have,

Pr(X ≥ m) = Pr(X ≥ 4µ) ≤ (e3/44)m/4 ≤ exp(−m/4)

Now m/4 ≥ 6K log n/4 ≥ (K + 1) log n, using K ≥ 2. Substituting back we obtain Pr(X ≥m) ≤ exp(−(K + 1) log n) = 1/nK+1. But |X| stochastically dominates |R| since V (R) <

m4nE

. The result follows.

Our next step will be to bound Pr(D(R) > C4δ

∣∣ E) , for R ∈ O and some constant

C4 = C4(K) <∞ where E is the event defined as E = |R| = m, V (R) ≥ m4nE.

Lemma 14. There exists some constant C4 = C4(K) < ∞ such that, for all R ∈ O with

V (R) ≥ m4nE

, we have that P(D(R) > C4δn

∣∣ |R| = m)≤ 1

nK+1 , where δn = log(n)

n1Km

K−1K

.

Proof. Conditioned on |R| = m, the remaining nL = nE−m points are distributed uniformly

i.i.d. in the complementary region of volume (1− V (R)).

Let FC4δn denote the region swept when R is expanded by C4δn along each coordinate

axis. Clearly, D(R) > C4δn if and only if region FC4δn contains no points.

Let X denote the number of points in FC4δn , and let p denote the volume of FC4δn .

Then, X ∼ Bin(nL, p/(1 − V (R))) and hence stochastically dominates Bin(nL, p)). Note

that such a volume p is at least the volume obtained when expanding the hypercube of side

` = K

√m

4nEby C4δn along each direction and therefore, p ≥ K`(K−1)C4δn. Hence,

P (D(R) > C4δn) = Pr(X = 0) ≤ (1− p)L ≤ exp −Ω(np)

≤ exp−Ω(n(m

n)(K−1)/KC4δn) = exp−Ω(C4 log n) ≤ 1

nK+1,

for appropriate C4, where we have used Assumption 2.


Lemma 15. Suppose m ≥ 6K log(n). Then, there exists a constant C5 = C5(K) < ∞,

such that E[minkαmax

k − αmink

]≤ C5

log(n)

n1Km

K−1K

.

Proof. Let Zk = minj∈M(k) εkj , Uk = maxj∈U εkj and δn = log(n)

n1Km

K−1K

. By Condition (IM) in

Proposition 3, we know that αmaxk − αmin

k ≤ Zk − Uk. Then,

E

[minkαmax

k − αmink

]≤ E

[minkZk − Uk

].

In addition, minkZk − Uk is equal to D(R) for some orthotope R ∈ O. In particular,

minkZk −Uk is equal to D(R) when R is the orthotope that “tightly” contains all the m

points in U . Define R = R ∈ O : |R| = m. Then,

E

[minkZk − Uk

]≤ E

[maxR∈RD(R)

].

To bound E [maxR∈R D(R)], consider the grid that results from dividing each of the

K coordinate axes in the hypercube into intervals of length 1/n. Let ∆ denote that grid.

Suppose we just consider orthotopes in the grid, that is, the orthotopes whose sides are

multiples of 1n . Let R∆ = R ∈ R : R ∈ ∆. Then,

maxR∈RD(R) ≤ max

R∈R∆

D(R)+1

n,

and,

E

[maxR∈RD(R)

]≤ E

[maxR∈R∆

D(R)]

+1

n.

Hence, we just need a bound for E [maxR∈R∆D(R)]. Let V∗ = m

4n . Note that D(R) ≤ 1

for all R ∈ O and therefore,

E

[maxR∈R∆

D(R)]≤ E

[maxR∈R′∆

D(R)]

+ Pr

(minR∈R∆

V (R) < V∗

)where R′∆ = R ∈ R∆ : V (R) ≥ V∗. Now, by union bound

Pr

(minR∈R∆

V (R) < V∗

)≤

∑R∈∆:V (R)<V∗

Pr(|R| = m) ≤ nK · 1/nK+1 = 1/n .

using |R ∈ ∆ : V (R) < V∗| ≤ |R ∈ ∆| = nK and Lemma 13.

Further,

E

[maxR∈R′∆

D(R)]≤ E

[max

R∈∆:V (R)≥V∗D(R)I(|R| = m)

]


Now,

Pr

[max

R∈∆:V (R)≥V∗D(R)I(|R| = m) > C4δn

]≤

∑R∈∆:V (R)≥V∗

Pr(|R| = m) Pr[D(R) > C4δn||R| = m]

≤∑

R∈∆:V (R)≥V∗1 · 1/nK+1 ≤ nK/nK+1 = 1/n

using a union bound and Lemma 14 to bound the probability of D(R) ≥ C4δn. It follows

that

E

[maxR∈R′∆

D(R)]≤ 1 · Pr

[max

R∈∆:V (R)≥V∗D(R)I(|R| = m) > C4δn

]+ C4δn = 1/n+ C4δn

Substituting the individual bounds back, we obtain

E

[maxR∈RD(R)

]= C4δn + 2/n ≤ C5δn .

defining C5 = C4 + 2 and using 1/n ≤ δn.

Overall,

E

[minkαmax

k − αmink

]≤ E

[minkZk − Uk

]≤ E

[maxR∈RD(R)

]≤ C5δn

as claimed.

We now proceed to show that, for every pair of types k, q ∈ TL we have

E

[min

j∈M(k)(εkj − εqj)− max

j∈M(q)(εkj − εqj)

]≤ C2

log(nE)nE

.

for appropriate C2 = C2(K) < ∞. This result is shown in Lemma 18. Along the way, we

establish a couple of intermediate results.

Let Zk = minj∈M(k) εkj and Uk = maxj∈U εkj . Note that Zk is an upper bound for −αk.

By the definition of Zk, all the points corresponding agents in M(k) must be contained in

the orthotope [1− Zk, 1]× [0, 1]K−1. The following proposition establishes that Zk cannot

be arbitrarily close to 1.


Lemma 16. Given a constant c ∈ R, let the event Ec be defined as Ec = maxk minj∈M(k) εkj ≤

1− c. Then, there exist constants θ = θ(K) > 0 and C6 = C6(K) > 0 such that, for large

enough n, Eθ occurs with probability at least 1− exp (−C6n).

Proof. Let Zk = minj∈M(k) εkj . The proof follows from the previous observation that all the

points corresponding agents in M(k) must be contained in the orthotope of volume (1−Zk).Let C < ∞ be such that

nEnL≤ C. By Assumption 2, such a C must exist. Furthermore,

by Assumption 2, there must exists CK ∈ R such that nk ≥ CKn for all k ∈ TL. Let nE be

the total number of points in the cube [0, 1]K . Let X denote the number of points out of

the nE ones that fall in the rectangle defined by [1 − θ, 1][0, 1]K−1. Then, X ∼ Bin(nE , θ).

Suppose we set θ < CK2C .Then, for large enough n and appropriate C6 > 0 we have

Pr(Zk > 1− θ) ≤ Pr(X ≥ CKnL) ≤ Pr

(X ≥ CKnE

C

)≤ exp (−2C6n) ≤ (1/K) exp (−C6n)

where we have used a Chernoff bound, 2nE ≥ n, and exp(−C6n) ≤ (1/K) for large enough

n. The result follows from a union bound over possible k.

Remark 2. Let θ, Eθ and C6 be as defined in the statement of Lemma 16. Define Gk,q as

Gk,q =

x ∈ [0, 1]K :

(xk ≥ 1− θ

2or xq ≥ 1− θ

2

)and xr <

θ

2for all 1 ≤ r ≤ K, r 6= k, q

.

Under event Eθ, we must have Gk,q ⊆M(k) ∪M(q).

The above remark follows from Lemma 16 and the definition of Gk,q. If j : εj ∈ Gk,qwere matched to a type k′ /∈ k, q, that will contradict maximality of the matching as, by

swapping the matches of j′ : j′ ∈ M(k), εkj′ = Zk and j, the overall weight of the matching

strictly increases. A similar argument rules out j being unmatched.

Lemma 17. Let Gk,q be as in Remark 2, and let θ be as defined in Lemma 16. Define G′k,q

as follows:

G′k,q = Gk,q ∩ x ∈ [0, 1]K , |xk − xq| ≤ 1− θ


Let Vkq = x : x = εkj − εqj , εj ∈ G′k,q, and let

V kq = max(Difference between consecutive values in Vkq ∪ −1 + θ, 1− θ).

Then, there exists a function f(n) = O∗(1/n) such that Pr(Bkq)≤ 1/n where Bkq is the

event that V kq ≤ f(n).

The proof of Lemma 17 is omitted as the required analysis is similar to (and much

simpler than) that leading to Lemma 6. Essentially, V kq consists of values taken by Θ(n)

points distributed uniformly and independently in [−1 + θ, 1− θ], so, with high probability,

no two consecutive values are separated by more than f(n) = O(log n/n).

In the next lemma we bound the difference between every pair of α’s.

Lemma 18. Consider types k, q ∈ TL and let f be as defined in the statement of Lemma 17.

Under event Eθ ∩ Bkq, in every stable solution we must have that (αmaxq − αmin

q ) ≤ 2f(n) +

(αmaxk − αmin

k ).

Proof. We claim that under Eθ, we must have αq−αk varies within a range of no more than

V kq within the core, where V kq is as defined in the statement of Lemma 17. By Remark 2,

under event Eθ we must have G′kq ⊂M(k)∪M(q), where G′kq is as defined in the statement

of Lemma 17. Suppose that G′kq contains at least one vertex matched to type k and one to

type q. Then, by Condition (ST) in Proposition 3 we must have:

(αq − αk)max − (αq − αk)min ≤ minj∈M(k)

εkj − εqj − maxj∈M(q)

εkj − εqj

≤ minj∈M(k)∩G′k,q

εkj − εqj − maxj∈M(q)∩G′k,q

εkj − εqj

≤ V kq

Next, consider the case in which all vertices in G′kq are matched to type k (the analogous

argument follows if they are all matched to type q). Under event Eθ, by Condition (IM)

in Proposition 3 we must have 0 ≤ −αk ≤ 1 − θ and 0 ≤ −αq ≤ 1 − θ. Therefore,

αq − αk ∈ [−1 + θ, 1 − θ]. In addition, by Condition (ST) in Proposition 3 we must have


αq − αk ≤ minj∈M(k)εkj − εqj. However,

(αq − αk)max − (αq − αk)min ≤ minj∈M(k)

εkj − εqj − (−1 + θ)

≤ minj∈M(k)∩G′k,q

εkj − εqj − (−1 + θ)

= minj∈G′k,q

εkj − εqj − (−1 + θ)

≤ V kq

It follows that (αmaxq −αmin

q ) ≤ 2V kq + (αmaxk −αmin

k ). By definition, under Bkq we have

V kq ≤ f(n), which completes the proof.

Finally, we complete the last step of the proof by showing the main theorem.

Proof of Theorem 10. By definition, C =∑K

k=1

N(k)|αmaxk − αmin

k |nL

, where N(k) is defined

to be the number of agents of type k that are matched. For a given instance, let k∗ =

argminkαmaxk − αmin

k . Let B = Eθ ∩ (∩k,qBk,q). Note that using Lemmas 2 and 17 and a

union bound, we obtain that

Pr(B) ≤ Pr(Eθ) +∑

k,q∈K:k 6=qPr(Bkq) = O(1/n) .

By Lemma 18, under B, for every k ∈ TL we have

αmaxk − αmin

k ≤ 2f(n) + αmaxk∗ − αmin

k∗ .

Therefore,

E[C] ≤ E[αmaxk∗ − αmin

k∗]

+ 2f(n) + Pr(B) ·O(1)

≤ O

(log(n)

n1Km

K−1K

)+O∗(1/n) +O(1/n)

= O∗(

1

n1Km

K−1K

)where the first inequality follows from the above together with using the upperbound of O(1)

for the core size; the second inequality is obtained by using the bound on E[αmaxk∗ − αmin

k∗]

from Lemma 12 for m ≤ 6K log(n) and Lemma 15 for m ≥ 6K log(n), as well as the

definition of f(n) and Pr(B) = O(1/n) shown above.

APPENDIX C. PROCUREMENT MECH. FOR DIFFERENTIATED PRODUCTS 170

Appendix C

Procurement Mechanisms for

Differentiated Products

C.1 Proof of Proposition 5

Proof of Proposition 5. This proof uses the standard arguments from mechanism design

theory introduced in Myerson’s seminal paper [Myerson, 1981]. Since the supports of our

cost distributions are discrete, we follow the version of these arguments presented by [Vohra,

2011]. Throughout this proof, we define mi to be the number of costs in the support of

agent i, that is, mi = |Θi|.We start by re-stating the IC and IR constraints in P1 in terms of the expected alloca-

tions and transfers:

maxx,t

Eθ

[n∑i=1

[ki(x(θ))− ti(θ)]

]

s.t. Ti(θi)−Xi(θi)θi ≥ Ti(θ′i)−Xi(θ′i)θi ∀i, ∀θi, θ′i ∈ Θi

Ti(θi)−Xi(θi)θi ≥ 0 ∀i, ∀θi ∈ Θi∑i∈N

xi(θ) = 1 ∀θ ∈ Θ, xi(θ) ≥ 0 ∀i ∈ N, θ ∈ Θ,

Recall that Θi = θ1i , ..., θ

mii . If we add a dummy type per agent θmi+1

i such that

Xi(θmi+1i ) = 0 and Ti(θ

mi+1i ) = 0, then we can fold the IR constraints into the IC con-


straints:

Ti(θji )−Xi(θ

ji )θ

ji ≥ Ti(θki )−Xi(θ

ki )θji ∀j ∈ 1, ...,mi, ∀k ∈ 1, ...,mi+1 .

Applying Theorem 6.2.1 in [Vohra, 2011] for our procurement setting we obtain that an

allocation x is implementable in Bayes Nash equilibrium if and only if Xi(·) is monotonically

decreasing for all i = 1, ..., n. 1 Further, by Theorem 6.2.2 in [Vohra, 2011], all IC constraints

are implied by the following local IC constraints: Ti(θji )−Xi(θ

ji )θ

ji ≥ Ti(θ

j+1i )−Xi(θ

j+1i )θji (BNICdi,θ)

Ti(θji )−Xi(θ

ji )θ

ji ≥ Ti(θ

j−1i )−Xi(θ

j−1i )θji (BNICui,θ)

Therefore, we can re-write the problem as:

maxx,t

Eθ

[n∑i=1

ki(x(θ))

]−

n∑i=1

mi∑j=1

fi(θji )Ti(θ

ji ) (obj)

s.t. Ti(θji )−Xi(θ

ji )θ

ji ≥ Ti(θ

j+1i )−Xi(θ

j+1i )θji ∀i ∈ N, ∀j ∈ 1, ...,mi (BNICdi,j)

Ti(θji )−Xi(θ

ji )θ

ji ≥ Ti(θ

j−1i )−Xi(θ

j−1i )θji ∀i ∈ N, ∀j ∈ 2, ...,mi (BNICui,j)

0 ≤ Xi(θmi) ≤ . . . ≤ Xi(θ

1), ∀i ∈ N (M)

n∑i=1

xi(θ) = 1 ∀θ ∈ Θ, xi(θ) ≥ 0 ∀i ∈ N, θ ∈ Θ.

In addition, using standard arguments, we can show that all downward constraints

(BNICdi,j) bind in the optimal solution.2 Hence,

Ti(θji )−Xi(θ

ji )θ

ji = Ti(θ

j+1i )−Xi(θ

j+1i )θji ∀i ∈ N, ∀j ∈ 1, ...,mi.

Further, it is simple to show that in this case, the upward constraints (BNICui,j) are satisfied.

Applying the previous equation recursively we obtain:

Ti(θji ) = θjiXi(θ

ji ) +

mi∑k=j+1

(θk − θk−1)Xi(θki ) . (C.1)

1Note that the results cited in Vohra are for IID bidders, but the extension to bidders with different

distributions is straightforward.

2A formal proof can be obtained by trivially adapting the Lemma 6.2.4 in Vohra to the procurement

case.


Replacing in the objective:

obj = Eθ

[n∑i=1

ki(x(θ))

]−

n∑i=1

mi∑j=1

fi(θji )Ti(θ

ji )

= Eθ

[n∑i=1

ki(x(θ))

]−

n∑i=1

mi∑j=1

fi(θji )

θjiXi(θji ) +

mi∑k=j+1


ki )

= Eθ

[n∑i=1

ki(x(θ))

]−

n∑i=1

mi∑j=1

fi(θji )(θjXi(θ

ji ))−

n∑i=1

mi∑j=1

mi−1∑k=0

fi(θji )(Ik ≥ j(θk+1

i − θki )Xi(θk+1i )

)

= Eθ

[n∑i=1

ki(x(θ))

]−

n∑i=1

mi∑j=1

fi(θji )(θjXi(θ

ji ))−

n∑i=1

mi∑k=1

Fi(θk−1i )(θki − θk−1

i )Xi(θki )

=∑θ∈Θ

f(θ)

(n∑i=1

ki(x(θ))

)−

n∑i=1

mi∑j=1

fi(θji )

((θj +

Fi(θj−1i )

fi(θji )

(θji − θj−1i )

)Xi(θ

ji )

)

=∑θ∈Θ

f(θ)

(n∑i=1

ki(x(θ))

)−

n∑i=1

∑θi∈Θi

fi(θi)vi(θi)Xi(θi)

=∑θ∈Θ

f(θ)

(n∑i=1

ki(x(θ))− vi(θi)xi(θ)

)

The equations follow by simple algebra. In particular, the fourth equation follows by

changing the order of summations.

Therefore, if we find an allocation such that for all θ ∈ Θ and i ∈ N ,

x(θ) ∈ argmax

n∑i=1

(ki(x(θ))− vi(θi)xi(θ))

s.t.

n∑i=1

xi(θ) = 1, xi(θ) ≥ 0 ∀i ∈ N ;

and such that the interim expected allocations are monotonic for all i ∈ N , that is, Xi(θ) ≥Xi(θ

′) for all θ ≤ θ′ ∈ Θi; and that the interim expected transfers satisfy Eqs. (C.1), for all

i ∈ N and θ ∈ Θi, then we have found an optimal solution.

C.2 Hotelling GAP

Example 3 (OPT (P0) > OPT (P1)). Consider an instance with only two players located

at the extremes of the unit segment. Let δ = 1 be the transportation cost. Let Θ1 = 1, 2.5,Θ2 = 1, 2, 2.3. The probability functions f1, f2, and v1, v2 are described in the following

tables.


Θ1 1 2.5

f1 1/2 1/2

v1 1 4

Θ2 1 2 2.3

f2 1/2 1/3 1/6

v2 1 3.5 3.8

To show that a gap exists between both problems, we show that it is not possible to

find item prices satisfying the conditions in Corollary 3. To that end, note that the set of

possible outcomes is Θ = (1, 1), (1, 2), (1, 2.3), (2.5, 1), (2.5, 2), (2.5, 2.3). Whenever θ1 = 1

or θ2 = 1 (but not both), only the agent with cost 1 will be active in the optimal solution.

Therefore, whenever agent 2 has cost θ2 = 2 he is only active in one profile, that is, in profile

(2.5, 2). By Eq. (5.9), the price p2(2.5, 2) is completely determined. In addition, Eq. (5.8)

now complete determines price p1(2.5, 2). Similarly, when agent 2 has cost θ2 = 2.3 he is

also active only in profile (2.5, 2.3). Using the same arguments as before, Eq. (5.9) pins-

down p2(2.5, 2.3) and hence Eq. (5.8) fixes price p1(2.5, 2.3). However, once the values of

p1(2.5, 2) and p1(2.5, 2.3) are fixed as explained above, the expected transfer constraint for

T1(2.5) fails to hold and a gap between both problems must exist. In the case, the optimal

objective value of the relaxed and orginal problems are 2.0638 and 2.0645 respectively.

It is easy to verify that condition (2) in Theorem 12 is violated in Example 3. In

particular, |Θ1| = 2 and, furthermore, the difference between consecutive virtual costs in

general exceeds δc∗4 = 1

4 . Intuitively, the support of the cost distributions in the example are

coarse and, therefore, the dimensionality of the price vectors is low. As a result, there are

not enough degrees of freedom to find prices that simultaneously satisfy the demand and

the expected interim transfers constraints. The second condition of the theorem guarantees

this is always the case. In particular, by requiring adjacent virtual costs to be “close”,

the optimal allocations do not vary much if we replace the cost of an agent by one of his

adjacent costs. Then, for a pair θji , θj+1i ∈ Θi, there exists at least some profile θ−i for which

we have i ∈ Q(θji ,θ−i) and i ∈ Q(θj+1i ,θ−i). This is crucial, as it guarantees a structural


relationship between the expected transfers constraints (Eq. (Ti(θji ))) of adjacent costs

(e.g., (Ti(θji )) and (Ti(θ

j+1i ))). Further, by imposing conditions (1) and (2), we guarantee

the existence of several cost profiles for which all agents are active, which translates into

a structural relationship between the expected transfers constraints (Eq. (Ti(θji ))) of all

the agents. As the prices become more related with each other, there are more degrees of

freedom to find prices that satisfy both the optimal demand constraints and the expected

transfer constraints.

C.3 Optimal mechanisms for Vertical Demand Model

We now consider a classic model of pure vertical differentiation (see, e.g., [Bresnahan,

1987]). There are n potential suppliers, supplier i offering a product of quality αi. We

assume, w.l.o.g., that α1 < . . . < αn. The qualities of the products are common-knowledge.

There is a continuum of consumers, all wishing to buy one unit of the good (so the market

is covered), uniformly distributed on the consumer-type space Z = [0, 1]. The type of a

consumer indicates her value for quality. In particular, the utility a consumer of type j ∈ Zobtains from consuming the product offered by supplier i at price pi is given by:

uji(pi) = jαi − pi, (C.2)

Given a set of potential suppliers with fixed unit prices p = pii∈N , the set of active

suppliers with strictly positive demand is given by:

Q(p) =

i ∈ N : max

j∈Zmink 6=ij (αi − αk)− (pi − pk) > 0

.

Namely, a supplier i ∈ N will be active only if there exists a j ∈ Z for which uji(pi) > ujk(pk)

for all k ∈ N with k 6= i.

As in the previous section, for unit prices p and agent i ∈ Q(p), let %p(i) (resp. ϑp(i)) de-

note the agent preceding (resp. following) i in Q(p), that is, %p(i) = max j ∈ Q(p) : j <

i and ϑp(i) = min j ∈ Q(p) : j > i. Also, let ι(Q(p)) (resp. η(Q(p))) denote the

rightmost (resp. leftmost) agent in Q(p). Then, the expected demand for product i is given


by:

di(p) =

0 if i /∈ Q(p)

1 if Q(p) = ipϑp(i)−piαϑp(i)−αi if i = η(Q(p))

pϑp(i)−piαϑp(i)−αi −

pi−p%p(i)

αi−α%p(i)if i ∈ Q(p), i 6= η(Q(p)), ι(Q(p))

1− pi−p%p(i)

αi−α%p(i)if i = ι(Q(p))

(C.3)

The linear constraints imposed by Eq. (C.3) that the prices must satisfy so as to have

OPT (P0) = OPT (P1) agree with those of Hotelling demand case. That is, the prices must

satisfy:

pϑθ(i)(θ)− pi(θ) = vϑθ(i)(θϑθ(i))− vi(θi) ∀θ ∈ Θ, i ∈ Q(θ), i 6= ι(θ), (C.4)

together with the constraints Ti(θji ), ∀i ∈ N, ∀θ

ji ∈ Θi. With this in mind, it is simple to

derive a result analogous to that of Theorem 12.

Theorem 14. Consider the general setting in which agents have arbitrary qualities and costs

distributions. Let b∗ = min1≤i≤n−1(αi+1 − αi). Suppose that the following two conditions

are simultaneously satisfied:

1. There exists θ ∈ Θ and c∗ ∈ R such that vi+2(θi+2)−vi+1(θi+1)αi+2−αi+1

> c∗+ vi+1(θi+1)−vi(θi)αi+1−αi for

all 1 ≤ i ≤ n− 2, v2(θ2)−v1(θ1)α2−α1

> c∗, and, 1− c∗ > vn(θn)−vn−1(θn−1)αn−αn−1

;

2. |Θi| ≥ 3 for all i ∈ N , and, for every i ∈ N and θj ∈ Θi, we have vi(θj+1i )− vi(θji ) ≤

c∗b∗4 .

Then, we have OPT (P0) = OPT (P1).

The intuition behind these two requirements is the same as that of Theorem 12. As

usual, let θ = (θ1, . . . , θn). From the definition of vertical demands (Eq. (C.3)), it is easy

to see that, by condition (1), for n ≥ 2 we must have Q(θ) = N . Hence, the first condition

guarantees the existence of an ‘interior solution’. The second condition imposes a ‘thin

enough’ cost discretization.


C.4 Extensions to our model

We now discuss two important extensions to our model. The first one is related to the

assumption that each supplier offers one product. In Section C.4.1, we provide a reasonable

extension to our model under which suppliers can offer multiple products. We show that our

main result extends accordingly, so we are able to characterize (under additional conditions)

the optimal mechanisms for the multiproduct case.

The second extension is related to the constraint that demand is inelastic. In particular,

we study what happens if we allow the total demand to be elastic in prices instead of

requiring it to be constant. We show that, in general, our main result fails to hold and a

gap between the optima of the original and the relaxed problem exists. However, preliminary

computational results show that the market structures (i.e., which suppliers are in the menu)

are usually similar in both the relax and the original problems.

C.4.1 Extension to multiple products per agents

We now show how to extend our model to the case where suppliers can offer more than

one product. If each agent is assumed to have a different random variable to represent the

cost for each product, then problem involves solving a multidimensional mechanism design

problem. This problem is recognized to be hard. Therefore, our approach is to assume that

suppliers’ costs can be parametrized by a single type, which can be interpreted as if the

auctioneer knows the agents’ cost structures but not their underlying cost parameter. This

approach is commonly used in the literature to overcome the multidimensional mechanism

design problem [Levin, 1997].

For i ∈ N , let Pi denote the set of products offered by supplier i. We assume that

agent i has cost cip(θi) for product p ∈ Pi, where θi is agent is type. The utility function of

supplier i is given by

ui = ti −∑p∈Pi

cip(θi)xip,

where xip is the amount of product p allocated to i, ti is the payment i receives in the

auction, and θi is his type. Similarly, the interim utility for supplier i when he reports cost


θ′i and has true cost θi is given by:

Ui(θ′i|θi) = Ti(θ

′i)−

∑p∈Pi

cip(θi)Xip(θ′i).

For each pair (i, p) with i ∈ N and p ∈ Pi, we define the modified virtual cost as:

vip(θi) = cip(θi) +Fi(ρ(θi))

fi(θi)(cip(θi)− cip(ρ(θi))) .

As usual, we assume virtual costs to be increasing. Furthermore, we require that the

function hi : R|Pi| × R → R defined as hi(xi, θi) =∑

p∈Pi cip(θi)xip satisfies the increasing

differences property. Under these assumptions, the optimal solution to the relaxed problem

is characterized by the following proposition.

Proposition 8. Suppose that (x, t) satisfy the following conditions:

1. The allocation function satisfies for all θ ∈ Θ,

x(θ) ∈ argmax

n∑i=1

∑p∈Pi

kip(x(θ))− vip(θi)xip(θ)

s.t.N∑i=1

∑p∈Pi

xip(θ) = 1, xip(θ) ≥ 0 ∀i ∈ N, p ∈ Pi .

2. Interim expected transfers satisfy for all i ∈ N and θji ∈ Θi:

Ti(θji ) =

∑p∈Pi

cip(θji )Xip(θ

ji ) +

|Θi|∑k=j+1

∑p∈Pi

(cip(θ

ki )− cip(θk−1

i ))Xip(θ

ki )

Then, (x, t) is an optimal mechanism for the relaxed problem.

Ideally, we would like to use the the characterization of the optimal solution to the

relaxed problem to study the original problem. The optimal demands for the relaxed prob-

lem still have an intuitive form, similar to the single-product case. However, the expected

transfers constraints differ. While demands depend on both the individual product and the

cost realization, the expected transfers only depend on the cost realization. Therefore, for

each cost realization, the expected transfers constraints involve terms for potentially many

products. This introduces some additional complexities in the analysis, and the extension

of Theorem 15 to the multiproduct case is not straightforward.


Surprisingly, under sufficient conditions, we are able to show that our main result still

holds. That is, there exists prices under which we have OPT (P0) = OPT (P1). This is

formalized by the following theorem.

Theorem 15. Consider the general setting in which agents have arbitrary costs distributions

and offer any arbitrary number of products. Then, there exists c∗ ∈ N, d∗ ∈ R+ such that,

whenever the following conditions are simultaneously satisfied,

1. There exists a profile θ ∈ Θ such that pi ∈ Q(θ) for all pi ∈ Pi and all i ∈ N .

Furthermore, there exists a d∗ ∈ R such that, for all θ′ ∈ Θ with |θ − θ′|∞ ≤ d∗ we

have Q(θ′) = ∪i∈NPi.

2. |Θi| ≥ c∗ for all i ∈ N , and, for every i ∈ N and θj ∈ Θi, we have maxp∈Pivip(θj+1i )−

vip(θji ) ≤ d∗/3.

we have OPT (P0) = OPT (P1).

The intuition behind the proof of Theorem 15 is similar to the single-product case, but

there are some fundamental differences. For example, the set Q(θ) now denotes the active

products rather than the active suppliers. Note that a single supplier can simultaneously

have many different products in the assortment, which will be reflected in the expected

transfer constraints. In addition, as the cost realization of a supplier is simultaneously valid

for all his products, we need to guarantee that the grid is thin enough for all products

offered by the supplier.

C.4.2 Demand Elasticity

Throughout this work we have assumed that demand is inelastic; regardless the prices,

exactly one unit is consumed across all substitute products. This is a natural constraint

to impose when modeling some specific FAs such as dialysis supply, in which the aggregate

demand is inelastic. In some FAs, however, it is not unreasonable to suppose that the actual

quantity purchased will depend on the prices: for instance, a school seeking to renovate two

computer labs might decide to renovate only one of them if the price of computers is too

high. Therefore, one reasonable extension to our model would be to consider an elastic

demand setting by relaxing the constraint that demands should add up to one.


The problem of auctions with endogenous quantities is not new, it was first introduced

by [Hansen, 1988]. In his paper, even though demand is elastic, the auction has a unique

winner (the lowest-price bidder). Therefore, determining the allocation is easy and only

the quantity is endogenous. This unique-winner assumption is usually common to all the

literature in the area. In our problem instead, both winners (i.e., agents that are in menu)

and quantities should be endogenous, which adds significant difficulties to the analysis.

To illustrate, consider the general affine demand model introduced in Section 5.5.3. By

using the same arguments as in Proposition 5, we know that the optimal solution to the

relaxed problem must satisfy:

x(θ) = argmaxy≥0cy −1

2yTDy − v(θ)y,

where v(θ) is the vector of virtual costs. Unfortunately, given our market primitives, in

the general case this implies a gap between the optimal solutions of the relaxed and the

original problem. As the consumer surplus function is assumed to be strictly concave, it has

a unique optimal solution. Whenever both agents have positive allocations in the optimal

solution to the relaxed problem, the only way to replicate those demands in original problem

is by setting the prices equal to the virtual costs. However, this choice of prices generally

violates the incentive compatibility constraints.

Even though the optimal relaxed solution cannot be mimicked, solving the relaxation

still give us some useful information regarding the original problem. To that end, we

consider the problem of two ex-ante identical agents and two possible types, θL and θH . We

calculated the optimal solution to both problems for different combination of paremeters c,

D, θL and θH and different distributions. In general, we considered own-price elasticities

in the range [−7,−0.3]. We discovered that the optimal solution to the original problem

generally imitates the market structure of the relaxed problem, i.e., the decision on how

many suppliers to include in the assortment agrees in both problems. In addition, the same

constraints bind in optimality: the IR constraint for the high type and the IC constraint for

the low type. Whenever the high-type agents is never in the menu (high elasticity case), the

optimal solution of the relaxed problem can be implemented in the original problem. This

is straightforward; as the demand for the high-type is always zero, the low-type will not

have an incentive to misreport if he is offered his own cost as price. However, in the general


case a gap exists. In such cases, the price of the high-type is set at θH at optimality, and the

prices of the low-type are higher than in the relaxed problem. As a result, when compared

to the relaxation, the demand of low-type agents decreases in the original problem and the

demand of the high-type increases. For all combination of parameters, the gap between the

relaxed and original problem was less than 5%.


C.5 Proof of Main Theorems

In this section we prove our main theorems. In particular, we prove a more general theorem

(Theorem 16), which generalized the statements of Theorem 12, Theorem 14, and Theo-

rem 15. Throughout this section, we use several basic definitions and concepts from linear

algebra. We refer the reader to [Strang, 1988].

C.5.1 The coefficient matrix

LetAij(θ) denote the coefficient of vj(θj) in the equation di(N, v(θ)). In all demand models

considered in the paper, Aij(θ) = 0 for every i ∈ Q(θ) and j /∈ Q(θ). This property is

natural: if a supplier has zero demand, then its price does not play a role in the demand

equations of competitors. Hence, overall, Eqs. (5.8) impose |Q(θ)| − 1 linear constraints

over the |Q(θ)| prices pi(θ) with i ∈ Q(θ). Let ι(Q(θ)) = maxi ∈ N : i ∈ Q(θ). For a

given θ and a given i ∈ Q(θ) with i 6= ι(Q(θ), the constraints imposed by Eqs. (5.8) can be

expressed as:

∑j∈Q(θ)

Aij(θ)pj(θ) =∑

j∈Q(θ)

Aij(θ)vj(θ) (Mi(θ))

We refer to the constraint associated with costs θ and supplier i ∈ Q(θ) (i 6= ι(Q(θ)) as

Mi(θ). Note that any set of prices p(θ) (for all θ ∈ Θ) that satisfy all constraints in the

set Mi(θ) : θ ∈ Θ, i ∈ Q(θ), i 6= ι(Q(θ) implement the optimal allocations given by the

solution of P1.

In addition, by Corollary 3, we need to guarantee that the expected interim transfers

coincide with the optimal ones from P1. We abuse notation and refer to the equality

constraint on the expected transfers corresponding to supplier i and cost θji ∈ Θi by Ti(θji ).

This constraint can be expressed as:

∑θ−i∈Θ−i

f−i(θ−i)xi(θji ,θ−i)pi(θ

ji ,θ−i) = Ti(θ

ji ) ∀i ∈ N, ∀θji ∈ Θi, (Ti(θ

ji ))

Abusing notation, let M and m be the coefficient matrix and the corresponding RHS

respectively defined by linear equations in (Mi(θ)) and (Ti(θji )), where each column is

associated with a price pi(θ). We can safely discard the columns corresponding to prices


pi(θ) such that i /∈ Q(θ), as all the coefficients of such columns are zero. The resulting

matrix M will have∑

θ∈Θ |Q(θ)| columns as we have one price variable per active supplier

and per profile of costs. In addition, for each θ ∈ Θ, there will be |Q(θ)| − 1 rows given

by the constraints in Eqs. (Mi(θ)) and∑

i∈N∑

θi∈ΘiI[∃ θ−i : i ∈ Q(θi,θ−i)] ≤ |Θ| rows

given by the constraints in Eqs. (Ti(θji )). The preceding observations are summarized by

the following remark:

Remark 3 (Dimension of the coefficient matrix). The coefficient matrix M has∑

θ∈Θ |Q(θ)|columns and

∑θ∈Θ |Q(θ)| −Θ +

∑i∈N

∑θi∈Θi

I[∃ θ−i : i ∈ Q(θi,θ−i)] rows. Further, the

number of columns is greater or equal than the number of rows.

By the Rouche-Frobenius theorem, a system of linear equations Mp = m is consistent

(has a solution) if and only if the rank of its coefficient matrix M is equal to the rank of

its augmented matrix [M |m]. To show whether the system of equations has a solution, we

use an equivalent definition of consistency.

Lemma 19 (Consistency of a system of linear equations). Consider the system of linear

equations Mp = m. Let M i,∗ denote the ith row of M . Then, the system is consistent (has

a solution) if and only if for every vector y such that∑

i yiM i,∗ = 0, we have∑

i yimi = 0.

For each row M i(θ), let aiθ denote the associated coefficient. Similarly, we denote by

biθji

the coefficient associated to row Ti(θji ). Let (a, b) be the vector of coefficients we just

described. Then, for a system to be consistent we must have that for every vector (a, b)

such that: ∑θ∈Θ

∑i∈Q(θ)i 6=ι(Q(θ))

aiθM i(θ) +∑i∈N

∑θji∈Θi

biθjiTi(θ

ji ) = 0 (C.5)

the linear combination of the right hand side also equals zero, that is,

∑θ∈Θ

∑i∈Q(θ)i 6=ι(Q(θ))

aiθ

∑j∈Q(θ)

A(θ)ij(θ)vj(θj)

+∑i∈N

∑θji∈Θi

biθji

θjiXi(θji ) +

|Θi|∑k=j+1


ki )

= 0.

(C.6)

To conclude, we note that, whenever the rows of M are linearly independent, the only

vector of coefficients satisfying equation (C.5) is (a, b) = 0 and therefore the system is

trivially consistent.


Through the rest of the section, we consider the general setting of Section 5.5.3. Given

a matrix A, we denote the ith row of A by Ai,∗. Similarly, the jth column is denoted by

A∗,j . For a subset of indices Q ⊂ N , AQ denotes the principal submatrix of A obtained

by selecting only the rows and columns in Q. Similarly, cQ denotes the vector obtained by

selecting only the components in Q and 1Q denotes the vector of ones of dimension |Q|. We

have the following result that characterizes an affine demand function for the set of active

suppliers.

Lemma 20. Given a price vector p and the associated demand d(p), we denote by Q =

Q(p) = i ∈ N : di(p) > 0. Then, demand d(p) can be expressed as:

dQ(pQ) = (DQ)−1

(cQ − pQ +

(1− 1′Q(DQ)−1

(cQ − pQ

)1′Q(DQ)−11Q

)1Q

). (C.7)

Proof. We start by stating the KKT conditions for problem (LD(p)):

c−Dx− p+ λ1 + q = 0 (C.8)

1′x = 1

x ≥ 0

x′q = 0,

where λ is the multiplier associated to the equality constraint and q is the vector of multi-

pliers associated to the non-negativity constraints. Define v = c −Dx − p + λ1. By the

KKT conditions we must have that vi = ci −Di,∗x− pi + λ = 0, for all i ∈ Q. Therefore,

0 = vQ = cQ −DQxQ − pQ + λ1Q.

As D is positive definite and DQ is a principal submatrix of D we have that (DQ)−1 exists

and, furthermore,

xQ = (DQ)−1(cQ − pQ + λ1Q

)In addition, by the feasibility constraint, we must have 1′QxQ = 1 and hence,

1 = 1′QxQ = 1′Q(DQ)−1(cQ − pQ + λ1Q

)which implies

λ =1− 1′Q(DQ)−1

(cQ − pQ

)1′Q(DQ)−11Q

.


Hence,

xQ = (DQ)−1

(cQ − pQ +

(1− 1′Q(DQ)−1

(cQ − pQ

)1′Q(DQ)−11Q

)1Q

),

as desired.

The above demand specification exhibits a natural regularity property: if there is no

demand for a particular product, the price of that product does not affect the demand for

other products. In addition, it is simple to observe that any increase in price of a product

with zero demand will not have an impact on the demand function either.

From Eq. (C.7), it should be clear that whenever two vector of prices pQ and pQ satisfy

(DQ)−1

(pQ −

1′Q(DQ)−1pQ

1′Q(DQ)−11Q1Q

)= (DQ)−1

(pQ −

1′Q(DQ)−1pQ

1′Q(DQ)−11Q1Q

), (C.9)

we must have that dQ(pQ) = dQ(pQ). This observation is useful because it states that

demands only depend on price differences. This freedom in setting unit prices is essential to

our proof technique, as we will find unit prices that satisfy the same differences induced by

the virtual costs and that simultaneously satisfy the expected interim transfer constraints.

Hence, the coefficient matrix M as described in Section 5.4 will consist, for θ ∈ Θ

and each i ∈ Q(θ) of at most Q(θ) non-zero rows: Q(θ) − 1 correspond to the demand

equations3 and the remaining one corresponding to the expected transfer constraint. Note

that for given θ ∈ Θ, the demand equations are given by Eq. (C.9) where we replace Q by

Q(θ) and pQ by pQ(θ)(θ) in the left hand side. In the right hand side we replace prices pQ

by virtual costs vQ(θ)(θ).

For a given θ ∈ Θ, we denote by A(θ) the submatrix of M that contains the demand

constraints for θ, that is, A(θ) = (Mi(θ))i∈Q(θ)\ι(θ). Recall from Section 5.5, that the

demand constraints for both the Hotelling model and vertical model can be expressed as:

pϑθ(i)(θ)− pi(θ) = vϑθ(i)(θϑθ(i))− vi(θi) ∀θ ∈ Θ, i ∈ Q(θ), i 6= ι(θ). (C.10)

Therefore, we have that the ith row of A(θ) will consist of all zeros except for a 1 in column

3Note that if we can find prices pQ satisfying the constraints imposed by x1, . . . , x|Q|−1, then the last

constraint will also be satisfied as xQ = 1−∑|Q|−1j=1 xj .


ϑθ(i) and a −1 in column i for i = η(Q(p)) for all i ∈ Q(θ), i 6= ι(θ).4 The following claim

characterizes the matrix A(θ) for the general affine demand models defined in Section 5.5.3.

Claim 9. Let F = F (θ) = (DQ(θ))−1. Then, for every j ∈ Q(θ) and every i such that

1 ≤ i ≤ Q(θ), the coefficient for pj(θ) in equation i is given by:

A(θ)ij = −F ij +(1′Q(θ) · F ∗,j)(F i,∗ · 1Q(θ))

1′Q(θ)F1Q(θ)1Q(θ). (C.11)

The proof of the Claim is omitted, as it follows straightforward from the characterization

of demand given in Lemma 20.

C.5.2 Definitions and notation

We now state some definitions that we will use to prove the main theorem. Recall that θi

and θi denote the lowest and highest values in Θi. For each j ∈ N , let θuj = maxθj ∈Θj : θj ∈ Q(θj ,θ−j), that is, θuj is the maximum θj under which there exists a profile

θ = (θj ,θ−j) such that j ∈ Q(θ). We may assume that θj ≤ θuj for all agents j ∈ N , as

otherwise we can consider (w.l.o.g.) the reduced problem in which all agents for which the

condition is violated are removed.

Two profiles θ,θ′ ∈ Θ are defined to be adjacent if and only if θ and θ′ only differ in one

component and Q(θ) = Q(θ′). To illustrate, consider Example 3. There, profiles (2.5, 2.3)

and (2.5, 2) are adjacent, but profiles (2.5, 2) and (2.5, 1) are not. We define two profiles

θ,θ′ ∈ Θ to be connected if there exists a sequence of adjacent profiles such that one can

go from θ to θ′.

Definition 5 (Acceptable set). We say a subset of profiles Θ ⊆ Θ is an acceptable set if

the following conditions are simultaneously satisfied:

1. Q(θ) = N for every θ ∈ Θ.

4Alternatively, one could think of a matrix A(θ) in which the ith row has a 1 in column ϑθ(i) and a −1

in column i for i = η(Q(p)), and a 1 in column ϑθ(i) and a −2 in column i and a 1 in column %θ(i), for all

other i ∈ Q(θ), i 6= ι(θ). Note that both matrices will define the same solutions.


2. For each agent i, let Θi = θi ∈ Θi : ∃θi−1 such that (θi, θ−i) ∈ Θ. Then, for every

θi ∈ Θi such that min Θi ≤ θi ≤ max Θi we must have θi ∈ Θi. That is, each Θi must

be an interval.

3. For every profile θ such that θi ∈ Θi for all i ∈ N , we must have θ ∈ Θ. That is, any

two profiles in Θ can be connected through profiles in Θ and Θ must be maximal.

To illustrate, in Example 3 the set Θ = (1, 1), (2.5, 2) satisfies the first two conditions

but violates the third one as the profiles are not connected. The above definition of ac-

ceptable set will help us characterize sufficient conditions under which the optima of the

relaxed and original mechanisms agree. In particular, let a market be defined by the set of

suppliers, their product characteristics and cost distributions, as well as the demand model.

We define a relaxation-is-optimal market (RIOM) as follows.

Definition 6 (RIOM). We say a market is RIOM if there exists an acceptable set Θ under

which the following (additional) conditions are satisfied:

(4) For every i ∈ N we have |Θi| ≥ 3.

(5) Let θ ∈ Θ be a profile such that θi ≥ max Θi. Then, there exists a profile θ′ ∈ Θ such

that the profiles θ,θ′ are connected.

Intuitively, the above conditions can be satisfied when we require the difference in virtual

costs between adjacent points in the support to be small enough. To illustrate, we show

that the conditions of Theorem 12 imply that the market is RIOM. First, by condition (2)

in the statement of the theorem, a profile θ in which Q(θ) = N must exist. Furthermore,

|vi+1(θi) − vi(θi)| ≤ δ(ì+1 − ì)/2 for all i ∈ N . As vi(θj+1i ) − vi(θji ) ≤ δc∗

4 for all i ∈ N ,

and θji ∈ Θi, it follows that by letting θki denote θi we have Q(θk+2i ,θ−i) = Q(θk−2

i ,θ−i) =

N , provided these exist. As |Θi| ≥ 3 for all i ∈ N , we must have than an acceptable

Θ exists and |Θi| ≥ 3. Finally, the connectivity requirement follows from the fact that

vi(θj+1i )− vi(θji ) ≤ δc∗

4 for all i ∈ N , and θji ∈ Θi. Using the same arguments, it can be see

that the conditions of Theorem 14 imply the market is RIOM.


C.5.3 Auxiliary Lemmas and Properties

We first state the following remark.

Remark 4. Suppose that vector of coefficients (a, b) is such that the equality given by

Eq. (C.5)) holds. If there exists θ−i such that Q(θi,θ−i) = i, then biθi (the coefficient

associated with row Ti(θi)) must be zero.

Note that the column corresponding to pi(θi,θ−i) will have exactly one non-zero element

located in row Ti(θi). Therefore, equality (C.5) will not hold unless the coefficient biθi is

zero. Next, we state and prove the following proposition.

Proposition 9. Suppose the coefficients (a, b) are such that equality in Eq. (C.5) holds.

For each i ∈ N and each θi ∈ Θi, let gi(θi) be defined as gi(θi) =biθifi(θi)

. Then for each

θ ∈ Θ, we must have ∑i∈Q(θ)

gi(θj)xi(θ) = 0 (C.12)

Proof. Fix θ ∈ Θ. We first show the result for the general affine demand model as described

in Section 5.5.3. Recall that the coefficients of the matrix corresponding to the demand

equations (that is, Eqs. (Mi(θ)) ) are as defined by Eq. (C.11). As the equality in Eq. (C.5)

holds, for each j ∈ Q(θ) we must have:

bjθjf(θ−j)xj(θ) +

Q(θ)−1∑i=1

aiθ

(−Fij + (1′Q(θ) · F ∗,j)(F i,∗ · 1Q(θ))

)= 0.

Therefore

∑j∈Q(θ)

bjθjf(θ−j)xj(θ) = −∑

j∈Q(θ)

Q(θ)−1∑i=1

aiθ

(−F ij +

(1′Q(θ) · F ∗,j)(F i,∗ · 1Q(θ))

1′Q(θ)F1Q(θ)1Q(θ)

)

= −Q(θ)−1∑i=1

aiθ

∑j∈Q(θ)

(−F ij +

(1′Q(θ) · F ∗,j)(F i,∗ · 1Q(θ))

1′Q(θ)F1Q(θ)1Q(θ)

)= −

Q(θ)−1∑i=1

aiθ

−F i,∗ · 1Q(θ) + F i,∗ · 1Q(θ)

∑j∈Q(θ)

(1′Q(θ) · F ∗,j)1′Q(θ)F1Q(θ)1Q(θ)

= −

Q(θ)−1∑i=1

aiθ(−F i,∗ · 1Q(θ) + F i,∗ · 1Q(θ)

)= 0


To complete the proof, note that∑

j∈Q(θ) bjθjf(θ−j)xj(θ) = f(θ)

(∑j∈Q(θ) gj(θj)xj(θ)

)=

0. Hence,∑

j∈Q(θ) gj(θj)xj(θ) = 0 as desired.

Next, we establish the result for the Hotelling and vertical models. In particular, we

show that whenever the coefficients (a, b) are such that equality (C.5) holds, then for each

θ ∈ Θ, we must have:

aiθ =∑

j∈Q(θ): j≤ibjθjf(θ−j)xj(θ) ∀ i ∈ Q(θ), i 6= ι(θ),

and, ∑j∈Q(θ)

bjθjf(θ−j)xj(θ) = 0

which implies the result.

Fix θ ∈ Θ. We show that aiθ =∑j∈Q(θ): j≤i b

jθjf(θ−j)xj(θ) by induction in the agents’

number. Consider the coefficients (a, b) involving i = η(Q(θ)), i.e., i is the leftmost vertex

agent in Q(θ). If Q(θ) = i is the leftmost active vertex, then biθi = 0, there is no such

coefficient aiθ and the result vacuously holds. Otherwise, we have that aiθ = biθif(θ−i)xi(θ),

which establishes the basis for the induction.

Suppose that the claim holds for every coefficient associated to the columns pj(θ) with

j ∈ Q(θ) and j < i. We show that it holds for the coefficients associated with pi(θ) with

i ∈ Q(θ). Consider the column associated to pi(θ). If i 6= ι(Q(θ)), then we need a%θ(i)θ −aiθ+

biθif(θ−i)xi(θ) = 0. By inductive hypothesis, a%θ(i)θ =

∑j∈Q(θ): j≤%θ(i) b

jθjf(θ−j)xj(θ),

and therefore aiθ =∑j∈Q(θ): j≤i b

jθjf(θ−j)xj(θ) as desired. Finally, if i = ι(θ), then

a%θ(i)+biθif(θ−i)xi(θ) = 0 together with the inductive hypothesis imply∑

j∈Q(θ) bjθjf(θ−j)xj(θ) =

0 as desired. To conclude, we note that the same result can be similarly obtained for the

alternative definition of A for the hotelling and vertical cases.

Let A = A(θ) for any θ ∈ Θ such that Q(θ) = N be as defined by Claim 9. We are

now going to prove two useful properties of A.

Remark 5. A is symmetric.

Note that, whenever Q(θ) = N , we have F = D−1 where F is as defined in Claim 9.

By assumption, D−1 is symmetric and positive definite. Therefore, A is also symmetric by

definition.


The second property is related to the rank of A. Note that we want to find prices

p such that x(p) = x(v(θ)), where v(θ) = (v1(θ), . . . , vn(θ)) is defined as the vector of

virtual costs. That is, we must have Ap = Av(θ). We now show that the dimension of

prices satisfying that is exactly one. In particular, we show that A has rank n− 1.

Claim 10. A has rank n− 1.

Proof. Let I denote the identity matrix of size n. Note that A = D1(−I + 1 1D−1

1D−11

).

Therefore,

rank(A) ≥ rank(D1) + rank

(−I + 1

1D−1

1D−11

)− n = rank

(−I + 1

1D−1

1D−11

),

as D−1 has full rank. In addition, we have5

rank

(−I + 1

1D−1

1D−11

)≥∣∣∣n− rank(1

1D−1

1D−11

) ∣∣∣ ≥ n− 1,

as the matrix 1 1D−1

1D−11has rank exactly one. The converse follows just from the definition of

A, as we know that one row must be redundant as all demands must some up to one.

We conclude this section by noting that Claim 10 trivially holds for the hotelling and

vertical cases. Also, note that Remark 5 does not hold for the original definition of A(θ)

for the hotelling and vertical models, but it does hold for the alternative definition. We

highlight that this will not affect the proof: essentially, we require that for every i, j ∈ N ,

the coefficient of pj in the demand equation of i must be equation to the coefficient of pi in

the demand equation for j.

C.5.4 Main Theorem

We can now state and prove our main theorem. To avoid excessive notation, we assume that

we are working with the general affine demand model as defined in Section 5.5.3 but all steps

and calculations are also valid for the hotelling and vertical models, when the alternative

definition of the matrix is assumed. For completeness, we clarify using a footnote when the

validity of a step is not immediate.

5Matrix property: rank(A−B) ≥ |rank(A)− rank(B)|


Theorem 16. Consider the general setting in which agents have arbitrary costs distribu-

tions. If the market is RIOM, then OPT (P0) = OPT (P1).

Proof. To show OPT (P0) = OPT (P1), we show that the system of equations is consistent.

Let (a, b) be a vector of coefficients satisfying Eq. (C.5). Let gi(θi) be as defined in the

statement of Proposition 9. As the market is RIOM, we know that there exists a subset of

profiles Θ ⊆ Θ that satisfies conditions (1)-(5). The idea of the proof is as follows. First,

we show that if the market is RIOM all gi(θi) must be zero. To do so, we start by proving

that gi(θi) = 0 for all θi ∈ Θi, where Θi is as defined by condition (2). Then, we show that

this implies gi(θi) = 0 for all θi ∈ Θi. We conclude the proof by showing that the fact that

gi(θi) = 0 for all θi ∈ Θi implies that the system is consistent.

We now show that gi(θi) = 0 for all θi ∈ Θi. By assumption, Θ satisfies conditions

(1)-(5). Therefore, for every θ ∈ Θ for all i ∈ N we must have Q(θ) = N . Consider two

profiles θ = (θi,θ−i) and θ′ = (θ′i,θ−i) which only differ in agent i’s cost and such that

θ,θ′ ∈ Θ. By the definition of Θ, such pair of profiles exists (condition (4)). By Eq. (C.12),

we must have gi(θi)xi(θ) +∑

j 6=i gj(θj)xj(θ) = 0 and gi(θ′i)xi(θ

′) +∑

j 6=i gj(θj)xj(θ′) = 0.

Hence, by subtracting the second equality from the first one we obtain

gi(θi)xi(θ)− gi(θ′1)xi(θ′) =

∑j 6=i

gj(θj)[xj(θ

′)− xj(θ)].

For each j ∈ N , we must have xj(θ′)− xj(θ) = A(θ)j,i (vi(θ

′i)− vi(θi)), where we used the

fact that A(θ) = A(θ′) by definition (see Claim 9). Let A = A(θ), and note that this A

agrees with the one in Remark 5 and Claim 10. Hence, we can re-write the above equality

as:

gi(θi)xi(θ)− g1(θ′i)xi(θ′) =

(vi(θ

′i)− vi(θi)

)∑j 6=i

gj(θj)Aj,i

,

and therefore,

gi(θi)xi(θ)− gi(θ′i)xi(θ′)vi(θ′i)− vi(θi) =

∑j 6=i

gj(θj)Aj,i

. (C.13)

Fix an arbitrary j ∈ N with j 6= i and Aij 6= 0.6 Assume that j has cost θj in both θ and

θ′ as defined above. Let θ′j ∈ Θj be such that θ′j 6= θj and θ′j ∈ Θj . Define θ = (θi, θ′j ,θ−i,j)

6In the hotelling and vertical models, this implies that j = i− 1 or j = i+ 1.


and θ′

= (θ′i, θ′j ,θ−i,j). The only thing we assumed about θj was θj ∈ Θj . Therefore, the

above equality must also hold for any Θj . That is,

g1(θi)x1(θ)− g1(θ′i)x1(θ′)vi(θ′i)− vi(θi)

= gj(θ′j) +

∑k 6=i,j

gk(θk)Ak,i.

By subtracting the inequality when j has cost θj from the one when his cost is θ′j we get

gi(θ1)(xi(θ)− xi(θ)

)− gi(θ′1)

(xi(θ

′)− xi(θ′))

vi(θ′i)− vi(θi)= Aj,i

(gj(θ

′j)− gj(θj)

).

However, note that xi(θ)− xi(θ) = Ai,j

(vj(θ

′j)− vj(θj)

). Therefore,

Ai,jgi(θi)− gi(θ′i)vi(θ′i)− vi(θi)

= Aj,i

gj(θ′j)− gj(θj)

vj(θ′j)− vj(θj).

Recall that A is symmetric (Remark 5).7 Therefore, whenever Ai,j 6= 0 we must have:

gi(θi)− gi(θ′i)vi(θ′i)− vi(θi)

=gj(θ

′j)− gj(θj)

vj(θ′j)− vj(θj), ∀θi ∈ Θi, ∀θj ∈ Θj .

Furthermore, the above equality should hold for every i, j ∈ N as we can find a sequence

of agents l0 = i, . . . , lK = j such that Alk,lk+16= 0 for all 0 ≤ k < K.8

We now show that gi(θi) = 0 for all θi ∈ Θi. Suppose the numerator is zero for at

least one pair of gi(θi), gi(θ′i). Then, gj(θj) − gj(θ

′j) must be zero for every j ∈ N and

all pairs θj , θ′j ∈ Θj . We now show that gi(θi) = gj(θj) must hold for every θi ∈ Θi

and θj ∈ Θj and i, j ∈ N . This is trivial if i = j, as gi(θi) − gi(θ′i) must be zero for

every i ∈ N and all pairs θi, θ′i ∈ Θi. Otherwise, note that when gi(θi) = gi(θ

′i), we have

gi(θi)xi(θ) − gi(θ′i)xi(θ

′) = gi(θi)Ai,i (vi(θi)− vi(θ′i)). By Eq. (C.13) the above equality

reduces to ∑j∈N

gj(θj)Ai,j = 0, (C.14)

and this must be true for any i ∈ N . Let AR denote the submatrix of A consisting of (n−1)

linearly independent rows. By Claim 10, we know such matrix exists. Furthermore, we can

7Note that this also holds for the alternative definition in the hotelling and vertical cases.

8Here we are implicitly assuming that matrix A has only one block. If A has more than one block, then

we can use the same argument for each block.


assume that those are the n−1 demand equations that appear in the matrix coefficient M .

Let g = (g1, . . . , gn) denote the vector of coefficients gi = gi(θi) for θ ∈ Θ. By Eq. (C.14),

the vector g must be in the nullspace of AR. However, as AR ∈ R(n−1)×n has dimension

(n−1) the dimension of its nullspace is at most 1. We will show that 1 is in Null(A), which

implies that all gi with i ∈ N must be equal.

Consider Ai,∗, that is, row i of the coefficient matrix A. We will show that Ai,∗ · 1 = 0.

Note that

Ai,∗ · 1 =∑j

(−Aij +

(1′Q(θ) ·A∗,j)(Ai,∗ · 1Q(θ))

1′Q(θ)A1Q(θ)1Q(θ)

)= −Ai,∗ · 1 +Ai,∗ · 1 = 0,

as desired. Therefore, 1 is in Null(A) and gi(θi) = gj(θj) for all i, j ∈ N , θi ∈ Θi, θj ∈ Θj .

Using that gi(θi) = gj(θj) for all θi ∈ Θi and θj ∈ Θj , we now show that gi(θi) = 0 for

all i ∈ N and all θi ∈ Θi which implies biθi = 0 for all θi ∈ Θi. If gi(θi) = 0, for some i ∈ Nand θi ∈ Θi, we are done. Otherwise, suppose that gi(θi) = k 6= 0 for all i ∈ N and all

θi ∈ Θi. By Proposition 9 we have:

0 =∑

j∈Q(θ)

gj(θj)xj(θ) = k

∑j∈Q(θ)

xj(θ)

= k,

which is a contradiction.

Now suppose that there exists a pair gi(θi), gi(θ′i) such that

gi(θi)−gi(θ′i)vi(θ′i)−vi(θi)

= k 6= 0, and

rewrite gi(θi) = gi(θ′i) + k[vi(θ

′i) − vi(θi)]. Let θi, θ

′i, θ′′i ∈ Θi and let θ−i ∈ Θ−i. Then, we

must have

(vi(θ

′i)− vi(θi)

)∑j 6=iAjigj(θj) = gi(θi)xi(θ)− gi(θ′i)xi(θ′)

=(gi(θ

′i) + k[vi(θ

′i)− vi(θi)]

)xi(θ)− gi(θ′i)xi(θ′)

= gi(θ′i)(xi(θ)− xi(θ′)

)+ k[vi(θ

′1)− vi(θi)]xi(θ)

= gi(θ′i)Aii

(vi(θi)− vi(θ′i)

)+ k[vi(θ

′i)− vi(θi)]xi(θ)

By dividing on both sides by vi(θ′i)− vi(θ) we obtain:

∑j 6=iAjigj(θj) = −gi(θ′i)Aii + kxi(θ)


In addition, since θ′′i ∈ Θi, we havegi(θ

′′i )−gi(θ′i)

vi(θ′i)−vi(θ′′i )= k and thus:∑

j 6=iAjigj(θj) = −gi(θ′i)Aii + kxi(θ

′′)

which is a contradiction as the virtual costs are strictly increasing and therefore xi(θ) 6=xi(θ

′′).

Next, we show that gj(θj) = 0 for the remaining cases, that if, whenever θj < min Θi

or θj > max Θj . For θj < min Θj consider a profile θ = (θj ,θ−j) such that θi ∈ Θi for all

i 6= j. By the definition of Θj , we must have have xj(θ) > 0. By Proposition ?? we have

0 =∑i∈Q(θ)

gi(θi)xi(θ) = gj(θj)xj(θ).

and therefore gj(θj) = 0 for all θj < min Θj and all j ∈ N . For θj > max Θj , let θ =

(θj ,θ−j) be a profile such that j ∈ Q(θ). We may assume that θ is such that θi ≥ min Θi

for all i ∈ N , as otherwise we can increase the θi < min Θi to satisfy this condition and j will

still be active. By the definition of Θ, θ = (θj ,θ−j) must be connected to a profile θ′ ∈ Θ.

That means, that there exists a sequence of adjacent profiles θ0 = θ′, . . . ,θK = θ. Given

that θ′ ∈ Θ, we must have that gi(θ′i) = 0 for all i ∈ N . Let k be the component in which

θ0 and θ1 differ. By Proposition 9 we have∑

i∈N gi((θ1)i)xi(θ1) = 0. As θ′ and θ1 only

differ in the kth component, we must have gk((θ1)k) = 0. We can inductively repeat this

argument to show that all the g’s corresponding to a profile in the path between θ′ and θ

must be zero, which implies gj(θj) = 0. Therefore, we have gj(θj) = 0 for all i ∈ N and all

θi ∈ Θi which implies biθi = 0 for all i ∈ N and all θi ∈ Θi.

To conclude the proof, we show that biθi = 0 for all i ∈ N and all θi ∈ Θi implies that

the system is consistent. To that end, consider a vector (a,0) satisfying Eq. (C.5). For

each θ ∈ Θ, we have

|Q(θ)|−1∑i=1

aiθ

∑j∈Q(θ)

A(θ)i,jvj(θj)

=∑

j∈Q(θ)

vj(θj)

|Q(θ)|−1∑i=1

aiθA(θ)i,j

= 0,

as (a,0) satisfying Eq. (C.5) implies∑|Q(θ)|−1

i=1 aiθA(θ)i,j = 0. Hence, we have shown that

(a,0) also satisfies Eq. (C.6). Therefore, the system is consistent and OPT (P1) = OPT (P0)

as desired.


Value of δOptimal ChileCompra

award avg. low price award equation strat. low[1fH

(θH − θL),∞)

split(fH/2+fL(1−x))θH+(x−1/2)θL

fL/2+fHx

where x =1/fH (θH−θL)+δ

2δ

splitθH

[(θH − θL), 1

fH(θH − θL)

]

single θL+fHθH1+fH

[(θH−θL)

2+fH, (θH − θL)

]θL+fHθH+δ

1+fH[fL2

(θH − θL),(θH−θL)

2+fH

)single

θH − δ[fHfL(θH−θL)

12

(1+fH )2+fHfL, fL

2(θH − θL)

]θL + δ 1+fH

fL[0,

fHfL(θH−θL)12

(1+fH )2+fHfL

]no BNE -

Table C.1: Comparison betweent te optimal mechanism and ChileCompra mechanism with

reserve price θH . In all cases, the expected price for an item of cost θH is θH .

C.6 Supplement to Section 5.6

C.6.0.1 Optimal bidding strategies for the agents under the ChileCompra

mechanism with reserve price θH .

We can analytically calculate the optimal bidding strategies for the agents under the Chile-

Compra mechanism with reserve price θH . Using standard arguments, it is straightforward

to verify that the equilibrium bid for a high-type agent is θH . The following proposition

characterizes the bid for the low type. In Table C.1, we compare the equilibrium bidding

strategy for the low-type agent in ChileCompra with reserve price θH9 to the average price

per unit payed to a supplier of type θL in the optimal mechanism.10

C.6.0.2 Ex-Ante Restricted-Entry Mechanism.

We analyze what happens if competition for the market is induced by restricting entry

before bids are placed. Suppose that we decide how many agents will be in the menu before

observing the bids and then run a FPA type mechanism to decide the prices. In our two-

9Note that for low-values of δ a BNE does not exist for the same reasons a BNE does not typically exist

in first price auctions with discrete types [Krishna, 2009].

10The prices given by the optimal mechanism are not unique. Therefore, we calculate the average price

per unit payed to a supplier of type low as T (θL)/X(θL).


agent model, this amounts to deciding when does choosing a single winner using a FPA

outperforms ChileCompra’s mechanism.

Recall that, in general, a FPA does not have an equilibrium in pure strategies when

types are discrete. However, by allowing equilibria in mixed strategies, expected payments

in the FPA are given by θH − f2L(θH − θL).11 By adding the transportation cost, the total

expected cost faced by a designer who chooses to run a FPA is θH − f2L(θH − θL) + δ

2 .

Using these analytical expressions, we can characterize the set of parameters for which the

FPA outperforms ChileCompra. To illustrate, for fixed θL = 10 and θH = 12, the relative

performance of ChileCompra and FPA as a function of parameters (fL, δ) can be seen in

Figure C.1.12

As it can be observed, FPA may or may not improve over ChileCompra, depending on the

combination of parameters. In particular, ChileCompra outperforms the FPA mechanisms

when both fL and the differentiation cost δ are relatively small (the white area). As the

differentiation cost increases beyond θH − θL but fL remains small, the FPA is still worse

than ChileCompra. In that region (light gray area), the equilibrium strategy for the low-

type in ChileCompra mechanism is to bid θH , which agrees with the bid a low-type agent

will place if there was no competition. However, the designer cannot improve by switching

to a FPA; in the light gray area, the reduction in purchasing costs that results from the

price competition cannot compensate for the large transportation cost, even when bids in

the ChileCompra mechanism are as high as possible. On the other hand, as fL increases, it

is profitable to restrict the entry using a FPA even if that implies a higher transportation

cost (gray area);13 this is due to the fact that a FPA is able to obtain much lower (expected)

bids from the low-type.

11This follows from standard arguments. For completeness, the proof is provided in the companion ap-

pendix.

12The black area is omitted from the analysis, as no equilibrium in pure strategies exists in ChileCompra’s

mechanism.

13We note that the non-convexity of the areas FPA and ChileCompra is due to the fact that,

in ChileCompra, the equilibrium bidding strategy as a function of δ is decreasing in the interval[fL2

(θH − θL), 12+fH

(θH − θL)].


ChileCompra low type bid = θH

ChileCompra

FPA

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

fL

Diff

eren

tiati

on

cost

Figure C.1: For θL = 10, θH = 12, we show when it is profitable to restrict the entry using a

FPA as a function of fL and δ. The black area is omitted from the analysis, as no equilibrium

in pure strategies exists in the ChileCompra mechanism. ChileCompra outperfoms the FPA

mechanisms only in the white area. The single-winner FPA is better in dark gray area. In

the light-gray area, ChileCompra has the highest possible low-type bid, but it is still better

than a single-winner FPA.


C.7 Proofs of Equilibria

Lemma 21. The unique PSNBE for the Chilecompra mechanism with reserve price R = θH

are as given by the following table:

Value of δ Eq. str. bL Award Expected procurement cost

[(θH − θL),∞) θH split θH + δ4[

(θH−θL)2+fH

, (θH − θL)]

fHθH+δ+θL1+fH

splitfLθL+fH (4−fL)θH

(1+fH )2− fHfL(θH−θL)2

2δ(1+fH )2+ δ

(1+fH )2+ fL

4δ[

fL2

(θH − θL),(θH−θL)

2+fH

)θH − δ single θH + δ

4− fL(1+fL)

2δ[

fHfL(θH−θL)12

(1+fH )2+fHfL,fL(θH−θL)

2

]θL + δ 1+fH

fLsingle f2

HθH + fL(1 + fH)θL +17−10fL−2f2L

4δ[

0,fHfL(θH−θL)

12

(1+fH )2+fHfL

]No PSBNE - -

Proof. Let Π(b, (bL, bH) denote the best response function when a player’s type is θL, his

adversary plays (bL, bH = θH) and his bid is b. We have three different cases depending on

the value of bL. We denote the cases by I, II or III depending on whether bL ∈ [θH−δ, θH ],

bL ∈ [θH − 2δ, θH − δ], or, bL ∈ [θL, θH − 2δ] respectively.

Case bL Best response function

I [θH − δ, θH ] ΠI(b, (bL, bH)) =

(b− θL)

(fHθH+fLbL+δ−b

2δ

)if b ∈ [θH − δ, θH ]

(b− θL)(fH + fL

bL+δ−b2δ

)if if b ∈ [bL − δ, θH − δ]

(b− θL) otherwise

II [θH − 2δ, θH − δ] ΠII(b, (bL, bH)) =

(b− θL)(fH

θH+δ−b2δ

)if b ∈ [bL + δ, θH ]

(b− θL)(fHθH+fLbL+δ−b

2δ

)if b ∈ [θH − δ, bL + δ]

(b− θL)(fH + fL

bL+δ−b2δ

)if if b ∈ [bL − δ, θH − δ]


III [θL, θH − 2δ] ΠIII(b, (bL, bH)) =

(b− θL)(fH

θH+δ−b2δ

)if b ∈ [θH − δ, θH ]

(b− θL)fH if b ∈ [bL + δ, θH − δ](b− θL)

(fH + fL

bL+δ−b2δ

)if if b ∈ [bL − δ, bL + δ]


Case δ ∈ [(θH − θL),∞). We claim that (bH , bL) = (θH , θH) is a PSBNE. For a player of

type θL, the best response function is as defined in case I. However, since δ ≥ (θH−θL) the

only meaningful case is the first one, that is: Π(b, (b∗, θH)) = (b− θL)(fHθH+fLb

∗+δ−b2δ

)for

b ∈ [θL, θH ]. We now focus on finding a symmetric equilibrium b∗. By the FOCs we must

have fHθH+fLb∗+δ−2b+θL2δ = 0, or equivalently, (1 + fH)b∗ = fHθH + δ + θL. However, as

δ ≥ θH − θL we obtain b∗ ≥ θH . Hence, the best response for a player of type θL is b = θH .


Furthermore, the same argument shows that θH is the unique symmetric equilibrium. ♦

Case δ ∈[

(θH−θL)2+fH

, (θH − θL)]. We claim that (bH , bL) =

(θH ,

fHθH+δ+θL1+fH

)is the unique

PSBNE. Note that bL ∈ [θH − δ, θH ] and therefore the best response function is as defined

by case I. It can be verified that ∂∂b

((b− θL)

(fH + fL

bL+δ−b2δ

))is positive at θH−δ for all δ

in the considered interval. Therefore, the best response must be in the interval [θH − δ, θH ],

and by deriving the function ΠI in that interval we can see that fHθH+δ+θL1+fH

is indeed a best

response. To check uniqueness, we divide it into two cases: b < θH − δ and b ≥ θH − δ. If

b ≥ θH − δ, the best response function is Π(b, (b∗, θH)) = (b − θL)(fHθH+fLb

∗+δ−b2δ

)and it

can be seen that bL as defined above is the unique b for which the FOCs are satisfied. If

b∗ < θH − δ, the best response function is Π(b, (b∗, θH)) = (b− θL)(fH + fL

b∗+δ−b2δ

). Then,

b∗ can never be a symmetric equilibrium as ∂Π∂b > 0 at b = b∗ for any b∗ < θH − δ. ♦

Case δ ∈[fL2 (θH − θL), (θH−θL)

2+fH

]. We claim that (bH , bL) = (θH , θH − δ) is a PSNE. In

this case, the best response function is a particular case of case I. It suffices to show that

the left derivative of the best response function is positive in θH−δ and the right derivative

is negative in θH − δ. The right derivative at θH − δ is ∂Π∂b (θH − δ) = −θH+θL+(2+fH)δ

2δ , which

cannot be positive as long as δ ≤ 12+fH

(θH − θL). On the other hand, the left derivative is

∂Π∂b (θH − δ) = fh + fL(−θH+θL+2δ)

2δ which is non-negative as long δ ≥ fL2 (θH − θL). Therefore

θH − δ is a best response.

To show uniqueness, suppose there exists a different symmetric equilibrium with bL = b∗

with b∗ 6= θH − δ. First, consider the case in which b∗ > θH − θL. In that case, the BR

function is Π(b, (b∗, θH)) = (b − θL)(fHθH+fLb

∗+δ−b2δ

)for b ∈ [θH − δ, θH ]. By imposing

symmetry, the FOCs are fHθH + δ + θL = (1 + fH)b∗ which implies fHθH + δ + θL > (1 +

fH)(θH−δ) as b∗ ∈ (θH−δ, θH ] by assumption. However, this reduces to (2+fH)δ > θH−θLwhich is a contradiction. Next, consider the case b∗ < θH −δ. The best response function is

Π(b, (b∗, θH)) = (b−θL)(fH + fL

b∗+δ−b2δ

)for b < θH − δ. The FOCs are fH +fL

b∗+δ−2b+θL2δ .

By imposing symmetry, we must have (1 + fH)δ + fLθL = fLb∗ < fL(θH − δ) which is

possible only if 2δ < fL(θH − θL), which is a contradiction. ♦


Case δ ∈[

fHfL(θH−θL)12

(1+fH)2+fHfL, fL2 (θH − θL)

]. We claim that (bH , bL) =

(θH , θL + δ 1+fH

fL

)is a

PSNE. Note that bL ≤ θH − δ for all values of δ considered.

We first show that bL satisfies the FOCs. As usual, consider b∗ ≤ θH − δ. The best

response function is Π(b, (b∗, θH)) = (b − θL)(fH + fL

b∗+δ−b2δ

)for b ≥ θH − δ. The FOCs

are fH + fLb∗+δ−2b+θL

2δ . By imposing symmetry, we must have (1 + fH)δ + fLθL = fLb∗

, or equivalently, bL = b∗ = θL + δ 1+fHfL

as desired. In addition, we must show that the

agent cannot benefit by deviating to bL − δ. To that end, note that the expected profit

at bL is (bL − θL)(fH + fL/2) = δ (1+fH)2

2fLand the expected profit at bL − δ is δ 2fH

fL. As

(1+fH)2

2fL> 2fH

fL, the deviation is not profitable. Furthermore, if bL ≤ θH − 2δ, we must also

guarantee that a deviation in the interval [bL + δ, θH − δ] is not profitable. In that case,

the best response function is as described by Case III. Note that the best response function

is strictly increasing in the interval [bL + δ, θH − δ] and therefore we need to compare the

max in the interval [bL + δ, θH − δ] with that in the interval [θL, bL + δ] to obtain the global

maximum and thus the best response. If δ ≤ 13(θH − θL), the maximum of the interval

[bL + δ, θH − δ] will be in θH − δ as the right derivative at that point is negative. Since

δ ≤ fL2+fL

(θH − θL) (as otherwise we are in the previous case) and fL2+fL

≤ 13 , we conclude

that the maximum in the interval [bL + δ, θH − δ] is achieved at θH − δ and the expected

revenue is fH(θH − δ− θL). Note that for fH(θH − δ− θL) < (bL− θL)(fH + fL/2) we must

have δ ≥ fHfL(θH−θL)12

(1+fH)2+fHfL.

To show uniqueness, we show that there cannot be an equilibrium with b∗ > θH − δ. In

that case, the FOCs are fHθH + δ+ θL = (1 + fH)b∗ and fHθH + δ+ θL > (1 + fH)(θH − δ)only if (2 + fH)δ > θH − θL, which is a contradiction. ♦

Case δ < fHfL(θH−θL)12

(1+fH)2+fHfL. The lack of equilibria follows from the arguments in the previous

case.

Proposition 10. Let rL and rU be defined as:

rL =fL(θH − θL)− (1 + 3fH)δ − 2

√δ(1 + fH)fH(2δ − fL(θH − θL))

f2L

,


rU =fL(θH − θL)− (1 + 3fH)δ + 2

√δ(1 + fH)fH(2δ − fL(θH − θL))

f2L

.

Then, for every C in the (possibly empty) intervals indicated below, θH − C is the unique

equilibrium bidding strategy for the low-type.

Valu

eofδ

Inte

rval

ofC

[ ( 2+fHfL

+√ (1

+fH

)fH

(2+fL

+f2 L

)) (θH−θL

)

2(1

+fH

),f2 L

(θH−θL

)

fL−

2fH

(fL>

2 3)]

[ f L 2+fL

(θH−θ L

),m

in

( δ−(θH−θL

)fL

,fH

(θH−θL

)

1+fH−fL

(θH−θL

)δ

)][ (θ

H−θ L

),2+fHfL

+√ (1

+fH

)fH

(2+fL

+f2 L

)

2(1

+fH

)(θH−θ L

)][ f L 2

+fL

(θH−θ L

),r U

][

2fL

(2+fL

)(1+fH

)(θH−θ L

),θ H−θ L

][ m

ax( f L 2

+fL

(θH−θ L

),rL

) ,m

in( δ,

max( r U

,(θH−θL

)−δ

1+fH

))][ f L 2

(θH−θ L

),2fL

(2+fL

)(1+fH

)(θH−θ L

)][ m

ax( (θ

H−θ L

)−

1+fH

fL

δ,r L

) ,m

in( δ,

max( r U

,(θH−θL

)−δ

1+fH

))]


Proof. We must consider δ ≥ fL2 (θH − θL), as otherwise we know that the equilibrium

bidding strategy for an agent of type low is smaller than θH−δ and, therefore, smaller than

θH − C. We first show that, under the stated conditions, θH − C is an equilibrium. The

best response function is:

Π(b, (θH − C, θH)) =

π1(b) = fH2

(θH − θL) if b = θH

π2(b) = fH (b− θL) θH−b+δ2δ

+ fL (b− θL) θH−C−b+δ2δ

if b ∈ (θH − C, θH)

π3(b) = fH (b− θL) + fL (b− θL) θH−C−b+δ2δ

if b ∈ (θH − 2C, θH − C]

π4(b) = b− θL otherwise

For θH−C to be an equilibrium, we need θH−C to be a maximizer of Π(b, (θH−C, θH)).

The following conditions are then necessary (and sufficient):

(a) ∂π3(θH−C)∂b ≥ 0.

(b) π3(θH − C) ≥ π4(θH − 2C)

(c) π3(θH − C) ≥ maxb∈(θH−C,θH ] π2(b)

We now derive conditions under which (a)− (c) hold:

Condition for (a): ∂π3(b)∂b = fH + fL

θH+θL−C−2b+δ2δ . Then, ∂π3(θH−C)

∂b = fH +

fLθH+θL−C−2(θH−C)+δ

2δ = fH + fLθL−θH+C+δ

2δ and it is non-negative whenever C ≥ (θH −θL)− 1+fH

fLδ.

Condition for (b): π3(θH−C) ≥ π4(θH−2C) is equivalent to(fH + fL

2

)(θH − C − θL) ≥

(θH−θL−2C) which occurs if and only if(

1 + fL2

)C ≥ fL

2 (θH − θL) or C ≥ fL2+fL

(θH − θL).

Condition for (c): We consider the case where the maximum is in (θH − C, θH ]. Note

that ∂π2(b)∂b = ∂

∂b

((b− θL) θH−b+δ2δ − fL (b− θL) C

2δ

)= θH+θL−2b+δ

2δ − fL C2δ .

First, note that if ∂π2∂b (θH − C) ≤ 0, condition (c) is automatically satisfied as π3(θH −

C) ≥ π2(θH − C). Hence, condition (c) holds whenever C ≤ (θH−θL)−δ1+fH

. Next, consider

the case in which maxb∈(θH−C,θH ] π2(b) is achieved at θH − C < b∗ < θH . Then, we must

have ∂π2∂b (θH) < 0, or equivalently, δ − (θH − θL) < fLC. In that case, we must have


π3(θH − C) ≥ π2(b∗), or equivalently(fH + fL

2

)(θH − C − θL) ≥ ((θH−θL+δ)−CfL)2

8δ . Note

that this quadratic constraint imposes both a lower and upper bound on C. Finally, if the

maximum is achieved at θH , we must have ∂π2∂b (θH) ≥ 0, therefore, δ − (θH − θL) ≥ fLC.

In addition, we must have fH (θH − θL) ≥(

1 + fH − fL(θH−θL)δ

)C.

We can summarize the conditions (a)−(c) by requiring C ∈ C, where the set C is defined

as follows:

C =

C : (1) max(

(θH − θL)− 1+fHfL

δ, fL2+fL

(θH − θL))≤ C ≤ δ and either

(2A) δ − (θH − θL) < fLC and(fH + fL

2

)(θH − C − θL) ≥ ((θH−θL+δ)−CfL)2

8δ , or,

(2B) δ − (θH − θL) ≥ fLC and fH (θH − θL) ≥(

1 + fH − fL(θH−θL)δ

)C, or,

(2C) C ≤ (θH−θL)−δ1+fH

Constraint (1) groups the constraints imposed (a) and (b) plus requiring C ≤ δ. Con-

straints (2A)− (2C) represent the (disjoint) constraints imposed in (c). By using algebraic

manipulations we can obtain the intervals in Table ??. In particular, rL and rU correspond

to the roots of the quadratic equation given in (2A).

As the designer is utilitity-maximizer, we are concerned with the biggest C under which

we can have an equilibrium. This yields 3 different cases:

Value of δ Best low-type bid[1

2+fH(θH − θL),

(1+fH )(√

2+√fH )2

(2+fL)(θH − θL)

]θH − rU[

fL2

(θH − θL), 12+fH

(θH − θL)]

θH − δ

Table C.2: Case 1:(fL ≥ 1

6

(1− 23

(181+24√

78)1/3 + ((181 + 24√

78)1/3)≈ 0.8641

)

Value of δ Best low-type bid[2+fHfL+

√(1+fH )fH (2+fL+f2

L)

2(1+fH )(θH − θL),

f2LfL−2fH

(θH − θL)

]θH − fH (θH−θL)

1+fH−fL(θH−θL)

δ[1

2+fH(θH − θL),

2+fHfL+√

(1+fH )fH (2+fL+f2L

)

2(1+fH )(θH − θL)

]θH − rU[

fL2

(θH − θL), 12+fH

(θH − θL)]

θH − δ

Table C.3: Case 2: 2/3 < fL ≤ 16

(1− 23

(181+24√

78)1/3 + ((181 + 24√

78)1/3)≈ 0.8641


Value of δ Best low-type bid[2+fHfL+

√(1+fH )fH (2+fL+f2

L)

2(1+fH )(θH − θL),∞

)θH − fH (θH−θL)

1+fH−fL(θH−θL)

δ[1

2+fH(θH − θL),

2+fHfL+√

(1+fH )fH (2+fL+f2L

)

2(1+fH )(θH − θL)

]θH − rU[

fL2

(θH − θL), 12+fH

(θH − θL)]

θH − δ

Table C.4: Case 3: fL ≤ 2/3

To derive Case 1, we know that rU >fL

2+fL(θH − θL) only if

δ ≤(2 + fH)(1 + fH) + 2

√2√fH(4− f2

L)2

(2 + fL)2(θH − θL) =

(1 + fH)(√

2 +√fH)2

(2 + fL)(θH − θL).

Note that, whenever (1+fH)(√

2+√fH)2

(2+fL) (θH−θL) ≤ 2+fHfL+√

(1+fH)fH(2+fL+f2L)

2(1+fH) (θH − θL)

(equivalently, fL > l1 = 16

(1− 23

(181+24√

78)1/3 + ((181 + 24√

78)1/3)

or fL ≈ 0.8641. In

addition, we highlight that (1+fH)(√

2+√fH)2

(2+fL) (θH − θL) > (θH − θL) whenever fH ≥√

2 −√2√

2− 1 ≈ 0.062. Therefore, if fL > 0.938, we have that our mechanism will not work

better than the original for δ ≥ (θH − θL). Case 2 is derived by the fact that we have an

upper bound on the largest interval only if fL > 2/3.

We highlight that, even if a δ > C > 0 exists, it might not be profitable for the designer

to commit to this strategy, as choosing such a C implies single-award which (of course)

yields a higher transportation cost.

We show uniqueness (except in border cases) by contradiction. Suppose there exists

a symmetric equilibrium strategy b∗ that is an equilibrium. First, it is easy to see that

b∗ < θH − C is not possible unless δ ≤ fL2 (θH − θL). Second, we argue that b∗ cannot be

θH . The profit when both players select θH is θh−θL2 ; by deviating to θH − C the profit is

(θH − C − θL), which is bigger provided C < θH−θL2 . However, note that C ≤ rU for the

appropriate δ and rU as a function of δ is concave, achieves its max at 1+fH2 (θH − θL) and

the max value is θH−θL2 . Therefore, whenever rU is binding, C ≤ rU < θH−θL

2 . as desired.

For δs for which fH(θH−θL)

1+fH− fL(θH−θL)

δ

is binding, note that we must δ ≥ (θH − θL) and therefore

the condition is satisfied. Finally, for the cases in which b∗ ∈ (θH − C, θH), we have that

b∗ = fHθH+δ+θL1+fH

(must satisfy the first order conditions) and hence δ ≤ (θH−θL). However,

the reader can verify that θH − C is a profitable deviation for the appropriate values of C.


In particular, this holds for C = rU .

Documents

Design and Analysis of Matching and Auction Markets · Design and Analysis of Matching and Auction ... Design and Analysis of Matching and Auction Markets Daniela Saban ... The outcome