A Decentralized Mechanism Implementing in Nash Equilibria

Submitted to Operations Researchmanuscript

A Decentralized Mechanism Implementing in NashEquilibria the Optimal Centralized Solution of a

Supply-chain Problem

Shrutivandana SharmaDepartment of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, [email protected]

Volodymyr BabichMcDonough School of Business, Georgetown University, Washington, DC, [email protected]

Demosthenis TeneketzisDepartment of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, [email protected]

Mark Van OyenDepartment of Industrial and Operations Engineering, University of Michigan, Ann Arbor, [email protected]

We study the coordination of production decisions for multiple products among many manufacturers and

many suppliers, each with private information about its own objective and its own production capabilities.

Our methodology does not require a probabilistic model of the beliefs of each decision maker (manufacturer

or supplier) regarding asymmetric information. We discover a game form (decentralized mechanism) which

implements in Nash Equilibria the solution of the corresponding centralized supply chain problem. In our

mechanism the buyers and sellers in the supply chain submit bids/offers for the quantities they would like to

buy/sell and the prices they would like other buyers/sellers to pay/receive per unit of a product. We provide

rules for allocating orders and determining payments based on the bids/offers. We show that, in equilibrium,

the unit price for all firms supplying a particular product is equal to the unit price for all firms buying the

product. One could interpret this price as the clearing price for the product in the internal market among

firms or among divisions of the same firm. Unlike prior work on coordinating firm’s production and sales

decisions through internal markets, our model does not require a centralized planner to set the prices in the

internal markets. Instead, the clearing prices are part of the equilibrium outcome.

Key words : Supply-chain coordination, asymmetric information, competitive/selfish traders, mechanism

design, Nash implementation

Sharma et al.: Nash implementation of optimal trade in a supply-chainArticle submitted to Operations Research; manuscript no. 1

1. Introduction

We study the coordination of production decisions for multiple products among many decision

makers each with private information about its own objective and its own production capabilities.

We propose a constructive proof for the existence of Nash equilibria whose outcome is that of

the centralized system. This proof requires no assumptions on the distributions of beliefs of each

decision maker regarding asymmetric information.

One motivation for our research is the problem of supply chain coordination. If multiple suppliers

and manufacturers, each with private information, are involved in the trade of products, they can

agree a priori on the rules for allocation of orders and determination of prices. We provide such

rules and prove that strategic (competitive) suppliers and buyers voluntarily participate and follow

such rules, and achieve optimal trade of products.

Another motivation for our research is the coordination of cross-functional decisions within a

firm. Division managers of a large corporation (e.g. GE, GM, Apple) may have objectives that are

not aligned with those of the corporation and, by acting selfishly, they could harm the corporation

overall. The top corporate management would like to introduce the rules for allocation of orders

and profits among the divisions so that the strategic division managers voluntarily follow these

rules, the sum of the objectives of all divisions is maximized and the value for the corporation’s

shareholders is maximized.

The coordination problem has been studied extensively in the economics, operations manage-

ment, and other fields and a number of solutions to this problem have been suggested. Because

none of the solutions is the definitive one, the choice of the best solution depends on the partic-

ular problem. In this paper we describe what we believe to be a good solution to a large-scale,

multi-product, production-coordination problem, where many decision makers have very limited

information about each other and no one has a dominant position. The following is a review of

other alternatives for solving this type of coordination problem.

The naive solution approach to the coordination problem is to ask the decision makers to share

their private information and then act in the interest of the centralized system. However, without

Sharma et al.: Nash implementation of optimal trade in a supply-chain2 Article submitted to Operations Research; manuscript no.

proper incentives the decision makers will hoard the information and act selfishly. To counteract

this behavior one needs to introduce contracts that provide incentives for the decision makers to act

in the interest of the system. Cachon (2003) reviews research on contracting and coordination in

supply chains. The majority of papers in this literature focus on the coordination of systems where

all information is public. Furthermore, the majority of papers on supply chain coordination consider

systems with a single buyer and a single seller. In contrast, we allow for asymmetric information

and allow for multiple decision makers. Our work is related to research on allocation of supplier

capacity. Cachon and Lariviere (1999a,b,c) study a basic supply-chain model with a single supplier

and two retailers, who are monopolists in their respective markets, and investigate the effects

of the capacity allocation rules on the equilibrium retailers’ orders, the retailers’, the supplier’s

and the supply chain’s profits. Cachon and Lariviere (1999c) consider linear, proportional, and

uniform allocation rules and solve for pure-strategy Nash equilibria with respect to the retailers’

ordering decisions. They find that proportional and linear allocations lead to order inflation by

the retailers, which increases supplier’s profits, and can benefit the supply chain overall. Cachon

and Lariviere (1999a) study a two-period model where the retailers’ sales in the first period affect

second period’s capacity allocation. Finally, Cachon and Lariviere (1999b) study the model with

asymmetric information and compare allocation mechanisms that result in truth-telling equilibria

and that do not. They find that the mechanism resulting in truth-telling equilibria can lead to lower

profits for the supplier, the retailers, and the supply chain. In Cachon and Lariviere (1999a,b,c)

the contract (in particular, the wholesale price) between the supplier and the retailer is fixed.

Deshpande and Schwarz (2005) relax the restriction on the contract choice. They consider a model

similar to Cachon and Lariviere (1999b), but apply the principal-agent approach to obtain the

optimal capacity-allocation rule and pricing for the supplier.

Principal-agent framework is the canonical solution to the asymmetric information problem.

Examples of principal-agent analysis in economics are Myerson (1981, 1982), Grossman and Hart

(1983), Guesnerie and Laffont (1984), McAfee and McMillan (1986), Maskin and Tirole (1990,

1992). In addition to Deshpande and Schwarz (2005), examples of the principal-agent analysis in


operations management are Corbett and Tang (1999), Corbett and Groote (2000), Cachon and

Lariviere (2001), Iyer et al. (2005), Yang et al. (2009, 2008). A critical assumption underlying most

papers in the principal-agent literature is that all decision makers must know and agree upon the

distribution of asymmetric information (e.g. if there are two agent types, all decision makers must

agree on the probability of the nature drawing a particular type). Even when this assumption is

true, the principal-agent framework does not guarantee that the centralized solution outcome will

be achieved. For example, Ha (2001) shows that, for the principal-agent model where the marginal

production cost of the buyer is its private information, it is impossible for the supplier to achieve

the single-firm solution. Cakanyildirim et al. (2006) further show that coordination can be achieved

only with certain values for the reservation utilities of the players. In contrast, our approach

does not require the decision makers to have any information about the nature of asymmetric

information and guarantees the existence of Nash equilibria, whose outcome is the centralized

solution. Furthermore, in the principal-agent framework, one party (the principal) dominates the

contract design process. This abstraction of the actual negotiation process that happens between

firms in practice works well in many cases. However, it is even more likely that none of the firms in

the negotiation has the power to unilaterally dictate the terms of the contract. Our model captures

the practical situation that all decision makers have some power, but no one has the power to

dictate.

The papers Porteus and Whang (1991) and Kouvelis and Lariviere (2000) present principal-

agent problems that apply within the framework of a single firm. In the latter paper, the authors

study the coordination of cross-functional decisions within a firm (producing a single product)

using internal markets. The principal in their model is the top management of the company and

the agents are the division managers. Some of the divisions manufacture the product, while the

others distribute it. The actions of the managers in manufacturing divisions and individual outputs

from those divisions are not observable by the principal, but the convex manufacturing cost of each

division and the total output from manufacturing stage are public knowledge. The optimization

problems of each of the distribution divisions are public knowledge as well. The principal is the


market maker who determines the unit price at which the output of the manufacturing divisions

is sold to the internal market and the unit prices at which each of the distribution divisions can

buy products from the internal markets. To ensure participation of all divisions lump-sum side

payments are used to redistribute the system profit. All contracting takes place at time 0. The

authors show how the principal should set the prices of the internal market to induce actions from

each division manager that would be optimal for the centralized system.

Our model considers a general setting with many firms. It also applies to the case of a single

firm where the planner’s objective is to maximize the sum of utilities of the firm’s divisions (sup-

ply and manufacturing divisions or manufacturing and distribution divisions). The approach we

present in this paper is game theoretic where each firm is considered to be a self utility maximizer.

We formulate the supply-chain problem as a market problem of private goods exchange between

suppliers and manufacturers and investigate it within the framework of implementation theory

(Williams (2008), Hurwicz and Reiter (2006), Palfrey (2002), Maskin and Sjostrom (2002), Maskin

(1985), Jackson (2001)). Previous works on Nash implementation for private goods and Walrasian

economies can be found in Hammond (1979), Hurwicz (1979), Schmeidler (1980), Hurwicz and

Schmeidler (1975). For our work, we obtained inspiration from Hurwicz (1979). In the above paper

Hurwicz presents a Nash implementation mechanism for pure exchange economies where firms can

trade by exchanging products with any other firm. In return of the product exchange, each firm

makes/receives a payment which induces it to trade quantities that maximize the social welfare. In

this paper we consider a supply-chain system which consists of two types of firms: Suppliers who

only supply products and receive money for it, and manufacturers who only purchase products and

pay money for it. Because of these constraints on the exchange of products and money among the

firms, the supply-chain model and hence the supply-chain coordination problem we consider in this

paper is different from the pure exchange economy model and the decision problem addressed in

Hurwicz (1979). We consider a supply-chain consisting of selfish suppliers and manufacturers, where

each supplier has a production capacity constraint and each supplier/manufacturer obtains a util-

ity by supplying/purchasing products. For this model we investigate a supply-chain coordination


problem that aims at determining the supply quantities, purchase quantities, and corresponding

payments that, (i) satisfy the suppliers’ capacity constraints; (ii) balance the net supply and net

purchase; (iii) balance the net payment made by the manufacturers and the net payment received

by the suppliers; and (iv) maximize the sum of utilities of all suppliers and manufacturers (social

welfare). We present a decentralized mechanism for the above problem that induces all suppliers

and manufacturers to voluntarily participate in the mechanism, and induces a game among the

suppliers and manufacturers such that the abovementioned objectives (i)–(iv) are achieved at all

Nash equilibria of the induced game.

The rest of the paper is organized as follows: In Section 2 we describe the supply-chain model

and present an equivalent centralized supply-chain problem. In Section 3.1 we model the supply-

chain problem in the framework of implementation theory. In Section 3.2 we present a game form

(decentralized mechanism) which induces a game such that all Nash equilibrium outcomes of the

game are solutions of the centralized supply-chain problem. For convenience, the proofs of all

theorems and results are presented in appendices. We conclude in Section 4.

Before we present the model in Section 2, we describe here the notation that we will use through-

out the paper.

Notation:

We use bold font to represent vectors as opposed to scalars. The elements of a vector are rep-

resented by a subscript on the vector symbol. A bold subscripted-symbol means that the vector

element is also a vector; e.g. in x = (x1,x2, . . . ,xN ), each xi, i = 1,2, . . . ,N, is a vector; and in

x = (x1, x2, . . . , xN), each xi, i = 1,2, . . . ,N, is a scalar. Unless otherwise stated, all vectors are

treated as column vectors. Bold 0 is treated as a zero vector of appropriate size determined by

the context. The notation (xi,x∗/i) (or (xi,x

∗/i)) is used to represent the following: (xi,x∗/i) (or

(xi,x∗/i)) is a vector of dimension same as that of x∗; the ith element of (xi,x

∗/i) (or (xi,x∗/i)) is

xi (or xi), all other elements of it are the same as the corresponding elements of x∗. We represent a

diagonal matrix of size N×N whose diagonal entries are elements of the vector x∈RN by diag(x).


2. The supply-chain problem

2.1. The model (M1)

We consider a supply chain consisting of NM manufacturers and NS suppliers where either NM ≥ 2

or NS ≥ 2 or both. We denote the manufacturers by j ∈ NM := {1,2, . . . ,NM}, and the suppliers

by i∈NS := {1,2, . . . ,NS}. The suppliers provide products to the manufacturers and the manufac-

turers either sell these products to the retailers or use them to produce new products. We assume

that there are L different types of products and we denote the set of products by L := {1,2, . . . ,L}.

Furthermore, each of the suppliers can supply some non-strict subset of the L types of products and

each of the manufacturers wishes to buy some non-strict subset of the L products. We represent the

bundle of L products supplied by supplier i, i ∈NS, by the vector xi := (xi1 , xi2 , . . . , xiL), and the

bundle of L products purchased by manufacturer j, j ∈NM , by the vector yj := (yj1 , yj2 , . . . , yjL).

We assume that xi ≥ 0 ∀ i ∈NS, and yj ≥ 0 ∀ j ∈NM . We also assume that each supplier i ∈NS

has a capacity constraint on the product quantity it can supply given by,

Aixi ≤ bi, i∈NS, (1)

where Ai ∈Rni×L+ , bi ∈Rni×1

+ , and for each i ∈NS, ni ∈ {1,2, . . . ,L}. The matrix Ai accounts for

the substitution effects or flexible production capacity among the L products supplied by supplier

i. These effects may arise because of the infrastructure constraints affecting simultaneous/parallel

production or storage of the products supplied by supplier i. For each of the ni bundles consisting

of products that are linked due to substitution/flexibility effects, the capacity constraint of the net

bundle supply by supplier i is given by the corresponding element in vector bi. We assume that,

Assumption 1 For each i∈NS, the matrix Ai and the vector bi are supplier i’s private informa-

tion, i.e., this information is known only to supplier i and nobody else in the system.

As discussed above, in this paper we assume linear capacity constraints for suppliers, and no

constraints on the purchase capacities of manufacturers. We make both these assumptions for

simplicity of presentation. We would like to emphasize that the mechanism proposed in this paper


can as well be used for models where the suppliers have any convex production capacity constraints,

and the manufacturers have any convex purchase capacity constraints. Furthermore, all the results

of this paper can be extended to these more general models.

We denote the payment vector from the manufacturers to the suppliers supplying their products

by (g,r) :=(g1, g2, . . . , gNM

, r1, r2, . . . , rNS

)∈RNM+NS

+ , where gj ∈R+ is the total payment given by

manufacturer j, j ∈NM , to all the suppliers, and ri ∈R+ is the total payment received by supplier

i, i∈NS, from all the manufacturers.

We denote the set of feasible transactions for supplier i, i∈NS, by DSi which is defined as follows,

DSi := {(ri,xi) | (ri,xi)∈R1+L

+ , Aixi ≤ bi}. (2)

Assumption 1 implies that the setDSi is user i’s private knowledge. Because a manufacturer does not

have any constraint associated with its purchase, the set of feasible transactions for manufacturer

j, j ∈NM , is defined as,

DMj := {yj | yj ∈RL

+}. (3)

There is a manager who governs the transaction between the suppliers and the manufacturers

according to an allocation mechanism. The allocation mechanism specifies the supply amounts

xi, i ∈NS, the purchase amounts yj , j ∈NM , and the payment vector (g,r). It is important that

a transaction does not lead to unclaimed products or money (not allocated to/claimed by any

supplier or manufacturer) in the system. To avoid this, the allocation mechanism that determines

the transaction must ensure that,

∑j∈NM

gj =∑i∈NS

ri, (4)

and∑i∈NS

xi =∑

j∈NM

yj . (5)

We call (4) and (5) the budget balance and product balance conditions respectively.


Every manufacturer or supplier has a preference associated with the trade of the products and

the corresponding monetary exchange. Supplier i’s, i ∈ NS, preference is quantified by a utility

function uSi :R1+L→R∪{−∞} defined as,

uSi (ri,xi) =

{ri− ci(xi), if (ri,xi)∈DS

i

−∞, otherwise.(6)

In (6), ci(xi) is the cost incurred by supplier i in the production of xi for (ri,xi)∈DSi . We assume

that,

Assumption 2 For each i ∈ NS, ci is a strictly convex function of xi over DSi with ci(0) = 0.

Furthermore, the function ci is supplier i’s private knowledge.

The preference of manufacturer j, j ∈ NM , is quantified by a utility function uMj : R1+L→ R ∪

{−∞} defined as,

uMj (gj,yj) =

{−gj + vj(yj), if yj ∈DM

j

−∞, otherwise.(7)

In (7) vj(yj) is the value manufacturer j obtains by making a purchase yj , yj ∈ DMj , from the

suppliers. We assume that,

Assumption 3 For each j ∈ NM , the function vj is strictly concave in yj with vj(0) = 0 and

∇yjvj(yj)≥ 0, ∀ yj ∈RL. Furthermore, vj is private knowledge of manufacturer j.

The assumptions of convex costs and concave values are standard in the operations research lit-

erature. In this paper we assume strict convexity of ci, i ∈NS, and strict concavity of vj, j ∈NM ;

such an assumption implies that there is a unique optimal supply-purchase vector for the supply-

chain problem we formulate in Section 2.2. We highlight the important fact that all the results

of this paper would also hold for any non strictly convex ci, i ∈NS, and any non strictly concave

vj, j ∈ NM . However, in the absence of strict convexity/concavity there may be multiple optimal

supply-purchase vectors; hence, different Nash equilibria of the game induced by the mechanism

proposed in Section 3.2 may result in different optimal supply-purchase vectors (nevertheless all


Nash equilibria will result in optimal supply-purchase vectors). The assumption of non negative

gradients of manufacturers’ value functions allows our model to include some well known examples

of value functions such as the newsvendor value function. 1

We assume that the preference of all the suppliers and manufacturers, in particular,

Ai,bi, ci, ∀ i∈NS, and vj ∀ j ∈NM , remain unchanged for the period of the settlement and imple-

mentation of the deal between the manufacturers and the suppliers.

Furthermore, we assume that all manufacturers and suppliers are selfish, i.e., they are all self

utility maximizers. On the other hand the manager that governs the transaction between the

suppliers and the manufacturers does not have any utility. It simply acts like an accountant that

facilitates the transfer of products/money between the suppliers and manufacturers.

In the following section we formulate the problem of optimal transaction determination for the

supply-chain model (M1).

2.2. The decentralized Supply-Chain (SC) problem

For the supply-chain model (M1) we want to develop a mechanism for determining the amount of

supplies, purchases and the corresponding payments that works under the decentralized information

setting of the model and obtains a solution to the following centralized problem:

Problem (PC)

max(g,r,x,y)

∑i∈NS

uSi (ri,xi) +

∑j∈NM

uMj (gj,yj) (8)

s.t.∑

j∈NM

gj =∑i∈NS

ri (9)

and∑i∈NS

xi =∑

j∈NM

yj (10)

By substituting the utility functions uSi and uM

j from (6) and (7), Problem (PC) becomes equivalent

to,


max(g,r,x,y)∈D

∑i∈NS

−ci(xi) +∑

j∈NM

vj(yj)

where,

D :={(g,r,x,y) |

(g,r,x,y)∈R(1+L)(NS+NM )+ ; Aixi ≤ bi, i∈NS;

∑j∈NM

gj =∑i∈NS

ri;∑i∈NS

xi =∑

j∈NM

yj}

(11)

The optimization problem (11) is equivalent to (8) because for (g,r,x,y) /∈D, the objective function

in (8) is negative infinity by (6) and (7). Thus D is the set of feasible solutions of Problem (PC).

Because of Assumptions 2 and 3, the objective function in (11) is strictly concave in (x,y). There-

fore, there is a unique optimal supply-purchase vector (x∗,y∗) for Problem (PC). Furthermore,

since the objective function in (11) does not explicitly depend on (g,r), an optimal solution of

Problem (PC) must be of the form (g,r,x∗,y∗), where (g,r) is any feasible payment vector, i.e.

(g,r)∈R(NM+NS)+ and

∑j∈NM

gj =∑

i∈NSri.

2.3. Discussion

As described in Section 2.1, in the decentralized supply-chain model (M1) none of the firms (suppli-

ers or manufacturers) or the manager who governs the transaction between the firms has complete

information to solve Problem (PC) (Assumptions 1, 2, 3). Therefore, we must develop an allocation

mechanism that enables the firms and the manager to determine optimal solutions of Problem

(PC) via some communication with one another. Since the firms are assumed to be selfish, such a

mechanism must be robust to the selfish communication strategies of the firms.

A systematic approach to the development of resource allocation mechanisms for informationally

decentralized systems where individuals behave strategically, is provided by implementation theory

in Mathematical Economics. In the context of the SC problem, implementation theory can provide

guidelines for designing mechanisms that specify rules on: (i) how the firms should “communicate”

with one another and the manager, and (ii) how “the information communicated by the firms must

be used” to determine supply/purchase quantities and corresponding payments so as to induce the

selfish firms to communicate information that results in a system objective maximizing transaction.


In this paper we use an implementation theory-based approach for the solution of the SC prob-

lem presented in this section. Therefore, in the next section we provide a brief introduction to

implementation theory and set the preliminaries for our solution to the SC problem.

3. Solution of the SC problem

3.1. Embedding the SC problem for Model (M1) in the framework of implementation theory

In the implementation theory framework, a resource allocation problem is described by the triple

(E ,A, π): the environment space E , the action/allocation space A and the goal correspondence π.

The environment e of a resource allocation problem is the set of infrastructure available to

all the individuals, their utilities, and any other information available to them, taken together. 2

For the SC problem, the environment eSi of supplier i, i ∈NS, consists of its private knowledge 3

consisting of the matrix Ai, the vector bi and the function ci, as well as the common knowledge

about the time invariance of the suppliers’ and manufacturers’ utilities. Similarly, the environ-

ment eMj of manufacturer j, j ∈ NM , consists of its private knowledge of the function vj and

the abovementioned common knowledge. The environments of all the firms collectively define the

environment e := ((eSi )i∈NS, (eMj )j∈NM

) of the problem. The set of all possible environments eSi

of supplier i (respectively eMj for manufacturer j) defines its environment space ESi (respectively

EMj ), and those of all suppliers and manufacturers collectively define the environment space E :=

(∏

i∈NS(ESi )×

∏j∈NM

(EMj )) of the problem.

The action space A is the set of all resource allocations / transactions that are feasible in the

system. For the SC problem, A=D.

The goal correspondence π is a mapping from E to A which maps each environment e∈ E , to the

set of actions π(e) in A that are optimal according to some pre-specified system objective. For the

SC problem (PC), the system objective is the maximization of the sum utilities∑

i∈NSuSi (ri,xi) +∑

j∈NMuMj (gj,yj). In a centralized system a central agent who completely knows e can determine

the optimal transactions π(e)∈A by mathematical optimization methods.

On the other hand, in the supply-chain model (M1), which is a decentralized system, none of the


firms (suppliers or manufacturers) or the manager completely knows e. Therefore, it is not possible

for anyone to determine optimal centralized transactions, π(e), without some communication with

one another. If the firms are selfish, they have an incentive to misrepresent their private information

while communicating with one another so as to shift the transaction determined by the allocation

mechanism in their own favor. The firms may also choose not to participate in the communication

process if they know that the resulting transaction will make them worse off. Such a strategic

behavior of firms may defeat the goal of maximizing the system objective. Therefore, for the success

of a decentralized mechanism (in leading to desirable transactions) it is required that the mechanism

induces the firms to, (i) voluntarily participate in the communication and allocation process, and

(ii) communicate information that results in system objective maximizing transactions. The design

of such decentralized mechanisms is addressed by implementation theory as described next.

In implementation theory a decentralized resource allocation mechanism is formally described by

a game form (M, f). The first elementM :=∏

i∈NSMS

i ×∏

j∈NMMM

j is the message space which

specifies for each supplier i ∈NS and each manufacturer j ∈NM , the set of messages MSi , i ∈NS

and MMj , j ∈ NM , that they can broadcast to the system. The second element f is a function

from M to A called the outcome function, which specifies for each message profile m∈M,(m:=

((mSi )i∈NS

, (mMj )j∈NM

),mSi ∈MS

i ,mMj ∈MM

j

), the resulting transaction f(m) ∈ A. The game

form is assumed to be known to all the firms so they can participate in the decentralized mechanism.

The firms’ strategic behavior in such a mechanism is modeled by specifying games. A game

form (M, f) is said to induce a game (M, f,{uSi }i∈NS

,{uMj }j∈NM

) among firms when their utilities

are uSi , i ∈ NS and uM

j , j ∈ NM . 4 In this game the firms i ∈ NS and j ∈ NM are the players, the

message space MSi (respectively MM

j ) is the set of (communication) strategies of player i, i ∈NS

(respectively j, j ∈ NM), and uSi (f(m)), i ∈ NS, and uM

j (f(m)), j ∈ NM , are the players’ payoffs

resulting from the strategy/message profile m. A game analysis is useful if the game leads to some

equilibrium in the firms’ strategic behavior specified by various solution/equilibrium concepts. One

such solution concept is Nash Equilibrium (NE) which is defined as a message profile m∗ such that,

ui(f(m∗)) ≥ ui(f((mi,m∗/i))), ∀mi ∈Mi, ∀ i∈N ∪{0}. (12)


We desire for the equilibria obtained from a game to result in optimal centralized solutions

(solutions of Problem (PC) for the SC problem). To formally define this requirement, let

us first represent the set of all Nash equilibria of the game (M, f,{uSi }i∈NS

,{uMj }j∈NM

) by

NE(M, f,{uSi }i∈NS

,{uMj }j∈NM

) and let,

ANE(M, f,{uSi }i∈NS

,{uMj }j∈NM

) :={a∈A | a= f(m) for some m∈NE(M, f,{uS

i }i∈NS,{uM

j }j∈NM)},

(13)

that is, ANE is the set of transactions corresponding to all Nash equilibria of the game. Having

defined ANE we use it to define the notion of implementation in Nash equilibria.

Definition 1

Implementation in Nash equilibria: A goal correspondence π is said to be “implemented in

Nash equilibria” by the game form (M, f) if, 5

ANE(M, f,{uSi }i∈NS

,{uMj }j∈NM

) ⊂ π(e), ∀ e∈ E . (14)

Definition 1 says that when π is implemented by (M, f), then, for any given environment e=((eSi )i∈NS

, (eMj )j∈NM

)of the decentralized problem, the set of transactions resulting from the Nash

equilibria (through the outcome function f) of the game (M, f,{uSi }i∈NS

,{uMj }j∈NM

) is a subset of

the set of optimal centralized transactions π(e) corresponding to the problem specified by (e,A, π).

For a game form to implement a goal correspondence in NE, it is required that the firms partic-

ipate in the communication process specified by the game form. In order that the firms voluntarily

participate in this communication process, the game form must satisfy an additional property

known as individual rationality. Let the initial endowment of a firm be defined as the amount of

resources the firm has before participating in a game form. In the SC model (M1), the initial

endowment fSi

0of suppliers i, i ∈ NS, and fM

j

0of manufacturers j, j ∈ NM , are the amount of

products and money they have before the transaction, i.e., fSi

0= (r0i ,x

0i) = (0,0), i ∈ NS, and

fMj

0= (g0j ,y

0j) = (0,0). Then,


Definition 2

Individual rationality: A game form (M, f) is said to be individually rational if,

∀ i∈NS, uSi (f(m))≥ uS

i (fSi

0), ∀m∈NE(M, f,{uS

i }i∈NS,{uM

j }j∈NM) (15)

∀ j ∈NM , uMj (f(m))≥ uM

j (fMj

0), ∀m∈NE(M, f,{uS

i }i∈NS,{uM

j }j∈NM). (16)

Definition 2 says that, at any NE transaction the utility of every firm is at least as much as its

utility without participating in the game form. Thus, an individually rational game form makes

sure that a firm finds it beneficial to participate if other firms also participate in the game form.

For the SC problem, this condition in particular implies that a supplier (respectively manufacturer)

benefits from participation in the game form if at least two manufacturers (respectively suppli-

ers) participate, or at least one other supplier and manufacturer participate. Thus, under above

conditions, every firm voluntarily participates in the game form.

In the sequel, we ensure that the desirable properties of implementation in Nash equilibria

and individual rationality made precise in Definitions 1 and 2 are achieved in our design of the

decentralized mechanism for the SC problem presented in Section 2.2. More precisely, our goal is:

The goal:

To design an individually rational, budget balanced and product balanced game form (M, f)

for the SC problem presented in Section 2.2 that implements in NE the goal correspondence π

corresponding to Problem (PC).

It is important to clarify the rationale behind choosing NE as the solution concept for the SC

problem. Note that because of assumptions 1, 2 and 3 in Model (M1), the environment of the

SC problem is one of incomplete information. Therefore, one may speculate the use of Bayesian

Nash or dominant strategy as appropriate solution concepts for this problem. However, because

the firms in Model (M1) do not possess any prior beliefs about the utility functions and capacity

constraints of other firms, we cannot use Bayesian Nash as a solution concept for Model (M1).

Furthermore, because of impossibility results for the existence of non-parametric efficient domi-

nant strategy mechanisms in classical private good environments (Groves and Ledyard (1987)), we


do not know if it is possible to design such mechanisms for Model (M1) that has characteristics

similar to private goods environments. The well known Vickrey-Clarke-Groves (VCG) mechanisms

that achieve incentive compatibility and efficiency with respect to non-numeraire goods, do not

guarantee budget balance (Groves and Ledyard (1987)). Hence, they are inappropriate for our SC

problem as budget balance is one of the desirable properties in this problem. VCG mechanisms

are also unsuitable for our SC problem because they are direct mechanisms (Groves and Ledyard

(1987), Hurwicz and Reiter (2006)) and require infinite message space to communicate the generic

continuous (and concave) utility functions of firms in Model (M1). For the above reasons, and the

known existence results for non-parametric, individually rational, budget-balanced Nash imple-

mentation mechanisms for classical private goods environments (Groves and Ledyard (1987)), we

choose Nash as the solution concept for our SC problem.

We would also like to clarify the interpretation of NE in the context of our SC problem. Note

that NE in general describe the strategic behavior of individuals in games of complete information.

This can be seen from (12) where, to define NE, it requires complete information of all firms’ utility

functions. Because the firms in Model (M1) do not know each other’s utilities, for any profile of the

firms’ utilities the resulting game is not one of complete information. Therefore, to use NE as the

solution concept for our SC problem, we adopt the interpretation of Reichelstein and Reiter (1988)

and (Groves and Ledyard 1987, Section 4). Specifically, by quoting (Reichelstein and Reiter 1988,

page 664), “we interpret our analysis as applying to an unspecified (message exchange) process

in which firms grope their way to a stationary message and in which the Nash property (12) is a

necessary condition for stationarity.” Alternatively, by quoting (Groves and Ledyard 1987, Section

4, page 69), “we do not suggest that each firm knows all of system environment when it computes its

message. We do suggest, however, that the complete information Nash game-theoretic equilibrium

messages may be the possible stationary messages of some unspecified dynamic message exchange

process.”

In the next section we present a game form for the SC problem that achieves the abovementioned

desirable properties of Nash implementation, individual rationality, and budget balance.


3.2. A game form for the SC problem

In this section we present a game form for the SC problem, and we specify its elements, the message

space and the outcome function.

The message space:

Since we are interested in determining the quantities of supply and purchase and the payments for

the firms in SC, the communication among the firms and the manager should contain information

that is helpful in determining the optimal amounts of each of these. Each supplier i∈NS broadcasts

to the system (including the manager and other firms) a message mSi ∈MS

i := RL × RL+ of the

following form:

mSi := (xi,p

Si ); xi ∈RL, pS

i ∈RL+. (17)

The two elements of message mSi are: xi = (xi1 , xi2 , . . . , xiL), which can be interpreted as the

amount of each of the L products that supplier i, i∈NS, proposes to supply to the manufacturers,

and pSi = (pSi1 , p

Si2, . . . , pSiL), which can be interpreted as the unit price proposed by supplier i, i∈NS,

for each of the L products.

Similarly, each manufacturer j ∈NM broadcasts to the system (including the manager and other

firms) a message mMj ∈MM

j :=RL×RL+ of the following form:

mMj := (yj ,p

Mj ); yj ∈RL, pM

j ∈RL+. (18)

The two elements of message mMj are: yj = (yj1 , yj2 , . . . , yjL), which can be interpreted as the

amount of each of the L products that the manufacturer j, j ∈ NM , proposes to purchase from

the suppliers, and pMj = (pMj1 , p

Mj2, . . . , pMjL), which can be interpreted as the unit price proposed by

manufacturer j, j ∈NM , for each of the L products.

Outcome function:

Before we present the outcome function for the SC problem, we would like to emphasize that the

specification of the outcome function, in particular the payment function, is the most important and

challenging task in the construction of a game form. Since the designer of the mechanism cannot


alter the firms’ utility functions uSi , i∈NS, and uM

j , j ∈NM , the only way it can achieve the desir-

able properties of Nash implementation, individual rationality and budget balance is through the

provision of appropriate payments/incentives that induce strategic firms to follow the mechanism’s

operational rules.

For the SC problem the outcomes are determined (by the manager or the firms themselves) based

on the broadcast message profile m = ((mSi )i∈NS

, (mMj )j∈NM

). We designate the outcomes with

a “hat”: the supply quantity xi, i ∈ NS, the purchase quantity yj , j ∈ NM , the total payment ri

received by supplier i∈NS, and the total payment gj made by manufacturer j ∈NM . The outcome

function is given below:

xi(m) = xi−1

(NS − 1)

∑k∈NSk 6=i

xk +1

NS

∑j∈NM

yj (19)

yj(m) = yj −1

(NM − 1)

∑k∈NMk 6=j

yk +1

NM

∑i∈NS

xi(m) (20)

ri(m) = pS−i(m)

Txi−

(pSi −pS

−i(m))T (pSi −pS

−i(m))

(21)

where pS−i(m) :=

1

(NS − 1 +NM)

( ∑k∈NSk 6=i

pSk +

∑j∈NM

pMj

)(22)

gj(m) = pM−j(m)

Tyj +

(pMj −pM

−j(m))T (pMj −pM

−j(m))

(23)

where pM−j(m) :=

1

(NM − 1 +NS)

( ∑k∈NMk 6=j

pMk +

∑i∈NS

pSi

). (24)

The game form defined by (17)–(24) together with the firms’ utility functions in (6) and (7)

induces a game. The strategy of supplier i, i ∈ NS, (respectively manufacturer j, j ∈ NM ,) in this

game is its message mSi (respectively mM

j ). Note that the message mSi of supplier i, i ∈ NS,

(respectively message mMj of manufacturer j, j ∈ NM ,) is allowed to take any value (which can

be unboundedly large) in the space RL × RL+, and it is not restricted to lie in DS

i (respectively

DMj ). A Nash equilibrium of the above game is a message profile m∗ from which no firm wants to

unilaterally deviate (see (12)) even when arbitrary deviations are possible by unbounded magnitude

of messages.


We next establish that the above game form possesses the properties of Nash implementation,

individual rationality, and budget balance. We first provide an intuitive explanation on how the

structure of the above game form leads to achieving these properties; then, we present theorems

that formalize the results.

3.3. Intuition behind the construction of the game form

As stated above, we want to establish three properties of the game form: (i) Nash implementation;

(ii) individual rationality; and (iii) budget balance. We begin with a discussion on achieving Nash

implementation. Nash implementation requires that the transactions obtained at all NE must be

optimal centralized solutions (solutions to Problem (PC)). To see how the proposed game form

leads to this property, we first discuss how this game form obtains feasbile solutions of (PC) at all

NE. Using the property of NE we then argue that the feasible solutions obtained by the game form

must be optimal solutions to (PC).

To see the feasibility of NE transactions, let us intuitively discuss equation (19) which determines

the supply vector of suppliers. Note from (19) that the quantity of each product supplied by

supplier i equals the average demand per supplier for that product (the third term in (19)), plus an

increment equal to the difference between the supply proposal of supplier i and the average supply

proposal of other suppliers for the same product (first and second terms in (19)). Thus, supplier i’s

message xi can be interpreted as the supply proposal that makes its net supply xi(m) match its

desired value through appropriate adjustment of the abovementioned increment. Here, the desired

value of xi(m) means one that lies within DSi and maximizes the utility of supplier i. Because the

message space (17) gives each supplier the flexibility to submit any supply proposal in RL, each

supplier i∈NS can bring its supply vector xi(m) within DSi by unilateral deviation. Thus, at a NE

all supply vectors xi(m), i ∈NS, must lie within the feasible transaction region. Comparing (20)

with (19) it can be seen that the purchase vectors are determined from manufacturers’ messages in

a similar way as the supply vectors are determined from suppliers’ messages. Therefore, as discussed

in the previous paragraph, through an appropriate purchase proposal in RL each manufacturer


j ∈ NM can bring its purchase vector yj(m) within DMj by unilateral deviation. It can also be

seen from equations (19) and (20) that for all message profiles, the net product transfer from the

suppliers to manufacturers is balanced, i.e. xi(m), i∈NS, and yj(m), j ∈NM , satisfy the product

balance condition (5). Since this condition is satisfied for all message profiles, it is also satisfied at

all NE.

Having intuitively discussed the properties of supply-purchase vectors at NE, to establish that

the NE transactions are feasible it remains to show the following: (i) the individual payments,

ri(m), i ∈ NS, and gj(m), j ∈ NM , are all non-negative; and (ii) the payment vectors satisfy the

budget balance condition (4). To establish these properties let us intuitively discuss the payment

function (21) (the analysis of (23) follows similarly). Note that the payment of supplier i, i ∈NS,

consists of two terms: a “price taker” term pS−i(m)

Txi which induces supplier i to be a price taker at

all NE, and a “penalty” term −(pSi −pS

−i(m))T (pSi −pS

−i(m))

which induces supplier i to propose,

at all NE, the same price vector as the average price proposal of all other firms. Note that the price

proposal pSi of supplier i does not affect its supply xi(m), but imposes a payment penalty on it if

pSi is different from the average price proposal pS

−i(m) of other firms. Therefore, the NE strategy

of supplier i that maximizes its payment and hence its utility is to propose pSi = pS

−i. Because

every firm follows a similar NE strategy, it follows that at all NE, all firms propose a common

price vector 6 p∗ = pS−i(m

∗) = pM−j(m

∗), ∀ i ∈ NS, ∀ j ∈ NM . This common price proposal p∗ can

be interpreted as the competitive price of the L products at which the transaction is agreed upon.

Because of the abovementioned NE strategy, the penalty terms vanish from the firms’ payments

at NE. Therefore, the NE payment of the firms is ri(m∗) = pS

−i(m∗)

Txi(m

∗) for supplier i, i∈NS,

and gj(m∗) = pM

−j(m∗)

Tyj(m

∗) for manufacturer j, j ∈ NM . For each i ∈ NS, the NE payment

ri(m∗) must be non-negative because pS

−i(m∗) is an average of the non-negative price proposals

of firms other than supplier i, and xi(m∗) is a non-negative supply vector in DS

i as discussed

before. Similarly, the NE payment gj(m∗) of each manufacturer j ∈NM must also be non-negative.

This establishes property (i) of the two properties that were required to show the feasibility of NE

payments. To establish property (ii), note from above that for each of the L products, the NE price


of the product is the same for all firms. Therefore, the NE payment, p∗T xi(m∗) for supplier i∈NS,

and p∗T yj(m∗) for manufacturer j ∈NM , is proportional to the supply/purchase of the respective

supplier/manufacturer. Because the product transfer from suppliers to manufacturers is balanced

at all NE (established earlier), by above arguments the net money transfer from manufacturers

to suppliers must also be balanced at NE. This establishes property (ii) of the NE payments and

hence, the feasibility of NE transaction.

We now argue that the feasible NE transactions are in fact optimal solutions of Problem (PC).

As shown above, the NE payment for supplier i, i∈NS, is pS−i(m

∗)Txi(m

∗). Because the NE price

pS−i(m

∗) is not controlled by supplier i’s own message, the only way supplier i can influence its NE

utility is through the allocation xi(m∗). In other words, supplier i must behave as a price taker at

NE, and given the NE price pS−i(m

∗), it must choose a strategy (message) so as to control its NE

allocation xi(m∗) and maximize its own utility. Because supplier i has the flexibility to submit

any supply proposal in RL, for any given price pS−i(m

∗) it can choose a message (x∗i ,pSi

∗) that

brings its allocation xi(m∗) to a value that maximizes its utility at price pS

−i(m∗). Note from (19)

that if supplier i’s supply proposal for some product is higher than the average supply proposal for

the same product by other suppliers, then supplier i supplies more (positive increment) than the

average demand per supplier for that product and vice versa. This allows the individual incentives

of suppliers to be aligned with the system objective. Similarly, each manufacturer j ∈NM can also

choose a message that maximizes its utility given its NE price pM−j(m

∗). Because at all NE the price

of a given product is the same for all the firms, the abovementioned individual utility maximization

of each firm at the common NE price leads to the maximization of the system objective function

at NE.

Following the above argument it can also be seen that at all NE, each supplier and manufacturer

can guarantee a non-negative utility for itself. This is because under any situation, each firm can

unilaterally change its message (supply/purchase proposal and price proposal) so as to: (i) make its

allocated supply/purchase quantity equal to zero; and (ii) make its payment penalty equal to zero.

Such a message would also make the firm’s payment and hence its utility equal to zero because in


the absence of payment penalty, the payment of each firm is proportional to its supply/purchase

quantity. Thus each firm can obtain zero utility by unilaterally changing its message. Therefore,

for a message profile to be a NE, it must provide each firm a non-negative utility, and this makes

the game form individually rational.

In the next section we present theorems that formalize all the above intuitive arguments.

3.4. Optimality of the game form

The main results of this paper are summarized by Theorems 1 and 2, which assert that the game

form proposed in Section 3.2 achieves the goal stated in Section 3.1.

Theorem 1 Let m∗ be a NE of the game induced by the game form presented in Section 3.2 and

the firms’ utility functions (6) and (7). Let (g(m∗), r(m∗), x(m∗), y(m∗)) =: (g∗, r∗, x∗, y∗) be the

transaction at m∗ determined by the game form. Then,

(a) All firms weakly prefer (g∗, r∗, x∗, y∗) to the initial allocation (0,0,0,0). Mathematically,

uSi (r∗i , x

∗i ) ≥ uS

i (0,0), ∀ i∈NS,

uMj (g∗j , y

∗j ) ≥ uM

j (0,0) ∀ j ∈NM .

(b) (g∗, r∗, x∗, y∗) is an optimal solution of the centralized problem (PC).

Furthermore, all NE of the game result in the same optimal transaction vector, i.e., if m is any

other NE, then, x(m) = x(m∗) and y(m) = y(m∗).

Theorem 2 Let (x∗, y∗) be the optimum supply and purchase vector for Problem (PC). Then,

(a) There exists a price vector p∗ such that,

∀ i∈NS, x∗i = arg max{xi| xi∈RL

+,Aixi≤bi}p∗T xi− ci(xi), (25)

∀ j ∈NM , y∗j = arg max{yj | yj∈RL

+}−p∗T yj + vj(yj). (26)

(b) There exists at least one NE m∗ of the game induced by the game form of Section 3.2 and

the firms’ utilities specified by (6) and (7) such that, (x(m∗), y(m∗)) = (x∗, y∗). Furthermore,


if r∗i := p∗T x∗i , i ∈ NS, and g∗j := p∗T y∗j , j ∈ NM , the set of all NE m∗ defined as mSi

∗:=

(x∗i ,pSi

∗), i ∈NS, and mM

j

∗:= (y∗j ,p

Mj

∗), j ∈NM , that result in (g∗, r∗, x∗, y∗) are character-

ized by the solution of the following set of equations:

x∗i −1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

∑j∈NM

y∗j = x∗i , i∈NS, (27)

y∗j −1

(NM − 1)

∑k∈NMk 6=j

y∗k = y∗j −1

NM

∑i∈NS

x∗i , j ∈NM , (28)

pSi

∗= pM

j

∗= p∗, ∀ i∈NS, ∀ j ∈NM . (29)

Because Theorem 1 is stated for an arbitrary NE m∗ of the game induced by the game form

presented in Section 3.2 and the firms’ utility functions (6) and (7), the assertion of the theorem

holds for all NE of this game. Thus, part (a) of Theorem 1 establishes the individual rationality

property of the game form presented in Section 3.2.

Part (b) of Theorem 1 asserts that all NE of the game induced by the game form presented in

Section 3.2 and the firms’ utility functions (6) and (7) result in optimal centralized transactions

(solutions of Problem (PC)). Thus, the set of NE transactions is a subset of the set of optimal

centralized transactions. This establishes that the game form presented in Section 3.2 implements in

NE the goal correspondence π defined by Problem (PC) (see Section 3.1). Because of this property,

the game form guarantees to provide an optimal centralized transaction irrespectively of which NE

is achieved in the game induced by the game form.

The assertion of Theorem 1 that establishes the above two properties of the game form is based

on the assumption that there exists a NE of the game induced by the game form of Section 3.2

and the firms’ utility functions (6) and (7). However, Theorem 1 does not say anything about

the existence of NE. Theorem 2 proves that NE exist in the above game, and provides condi-

tions that characterize the set of all NE that result in optimal centralized transactions of the

form (g∗, r∗, x∗, y∗) = ((p∗T y∗j )j∈NM, (p∗T x∗i )i∈NS

, x∗, y∗), where (x∗, y∗) is the optimal centralized

supply-purchase vector.


4. Conclusion

We constructed a decentralized mechanism for supply-chain coordination. The mechanism leads

to a game among suppliers and manufacturers, all Nash Equilibria of which result in production

decisions that are optimal for the corresponding centralized supply chain system. Our results

apply to general multi-product procurement problems where all firms behave selfishly, possess

private information about both their production capabilities (expressed as linear constraints) as

well as their objective functions (general concave utility functions), and do not share common

beliefs about the private information of other firms. In fact, unlike the canonical contract theory

approach (Laffont and Martimort (2002)) which is based on the Bayesian Equilibrium concept,

we make no a priori assumptions on the beliefs of any of the participants. Thus, our approach is

an appealing modeling framework for describing complex supply chain systems, where common

beliefs assumptions are not always justifiable or where specifying a priori beliefs might be difficult.

In our mechanism, the buyers and sellers in the supply chain submit bids/offers for the quantities

they would like to buy/sell and the prices they would like other buyers/sellers to pay/receive per

unit of each product. We provide rules for allocating orders and determining payments based on

the bids/offers. We show that, in equilibrium, the unit price received by all the firms supplying a

particular product is equal to the unit price paid by all the firms buying that product. One could

interpret this price as the clearing price for the product in the internal market among firms or

among divisions of the same firm. Unlike prior work on coordinating firm’s production and sales

decisions through internal markets, our model does not require a centralized planner to set the

prices in the internal markets. Instead, the clearing prices are part of the equilibrium outcome.

Our results are very general, so a number of assumptions were necessary. Our model would not

apply to a simple supply chain with a single buyer and single seller. In order for the proposed

mechanism to work, the system must have at least two buyers or at least two sellers. We believe this

is not a restrictive assumption for many real procurement situations, because several suppliers are

likely to offer their products and several buyers are likely to place bids with the suppliers. In our


model, we focused on a one-time exchange of goods and money between the buyers and the sellers.

Therefore, there is no possibility of observing random events and sequential contracting based

upon the outcomes of those events. Similarly, we do not allow contracts that are contingent on the

realization of any random variables. This might be restrictive in some cases, for example, when

suppliers might want to take an “equity” position in the buyer’s sales, and receive payments for the

products based on the realized buyer demand. However, this is also a strength of our model, because

the buyer’s demand might not be contractible, or even observable in many practical settings. The

models with sequential contracting and contingent contracts have important applications, but the

theory to handle these features under asymmetric information is still under development. We defer

the analysis of such models to future research. Finally, the approach presented in this paper is a

constructive one which proves the existence of a decentralized mechanism and a resulting game

such that all its Nash equilibria lead to optimal trade of products. We do not have an algorithm

for the computation of these equilibria. For our problem, orthogonal search algorithms do not

guarantee convergence to Nash equilibria because the games corresponding to the proposed game

form are not, in general, supermodular. The development of algorithms that guarantee convergence

to Nash equilibria for the games constructed in this paper remains an important open problem. The

development of mechanisms/game forms that lead to supermodular games is another important

open problem.

Acknowledgments: This work was supported in part by NSF grants NSF DMII-0457445 and

DMII-0539348 for Babich, CCR-0325571 for Teneketzis, and DMI-0542063 for Van Oyen.

Notes

1The newsvendor value function is given by min(quantity purchased, demand).

2The environment is the set of circumstances that cannot be changed either by the agents in the system or by the

designer of the resource allocation mechanism.

3Note that the knowledge of Ai,bi and ci imply the knowledge of the utility function uSi .

4The utilities are specified by the system environment e∈ E .

5Note that the utility uSi (respectively uM

j ) is specified by eSi , i ∈ NS (respectively eM

j , j ∈ NM ). Therefore, the

left hand side of (14) depends also on e := ((eSi i∈NS ), (eM

j )j∈NM ).


6The common price vector p∗ may be different at different NE. However, at any given NE, all firms must propose

the same price vector p∗.

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Appendix A: Proof of Theorem 1

In this section we present the proof of Theorem 1. We divide the proof into several claims to organize

the presentation. Throughout the proof, we represent the NE allocations as follows for simplicity of

notation: xi(m∗) =: x∗i , yj(m

∗) =: y∗j , ri(m∗) =: r∗i , gj(m

∗) =: g∗j , pS−i(m

∗) =: pS−i∗, pM−j(m

∗) =:

pM−j∗.

Claim 1 If m∗ is a NE of the game specified by the game form presented in Section 3.2 and the

firms’ utility functions (6) and (7), then, the NE payments r∗i to the suppliers i ∈ NS are of the

form, r∗i = (pS−i∗)T x∗i , and the NE payments g∗j by the manufacturers j ∈ NM are of the form,

g∗j = (pM−j∗)T y∗j .

Proof:

Let m∗ be a NE described in Claim 1. Then, for each i∈NS,

uSi

(ri(m

Si ,m

∗/i), xi(mSi ,m

∗/i))≤ uS

i (r∗i , x∗i ), ∀mS

i ∈MSi . (30)

Substituting mSi = (x∗i ,p

Si ), pS

i ∈RL+, in (30) and using (19) implies that,

uSi

(ri((x∗i ,p

Si ),m∗/i

), x∗i

)≤ uS

i

(r∗i , x

∗i

), ∀ pS

i ∈RL+. (31)

Since uSi is increasing in ri (see (6)), (31) implies that,

ri((x∗i ,p

Si ),m∗/i

)≤ r∗i , ∀ pS

i ∈RL+. (32)

Substituting (21) in (32) implies that,

(pS−i∗)Tx∗i − (pS

i −pS−i∗)T (pS

i −pS−i∗) ≤ (pS

−i∗)Tx∗i − (pS

i

∗−pS−i∗)T (pS

i

∗−pS−i∗),

or (pSi −pS

−i∗)T (pS

i −pS−i∗) ≥ (pS

i

∗−pS−i∗)T (pS

i

∗−pS−i∗), ∀ pS

i ∈RL+. (33)

Since (33) must hold for all pSi ≥ 0, ∀ i∈NS, it implies that,

pSi

∗= pS

−i∗, ∀ i∈NS,


or (NS − 1 +NM)pSi

∗=∑k∈NSk 6=i

pSk

∗+∑i∈NM

pMi

∗, ∀ i∈NS,

or (NS +NM)pSi

∗=∑k∈NS

pSk

∗+∑

j∈NM

pMj

∗, ∀ i∈NS. (34)

Following the same steps as in (30)–(34) and using the manufacturers’ utilities uMj , j ∈NM , we also

get

(NM +NS)pMj

∗=∑i∈NS

pSi

∗+∑

k∈NM

pMk

∗, ∀ j ∈NM . (35)

It follows from (34) and (35) that at any NE m∗,

pSi

∗= pM

j

∗= p∗, ∀ i∈NS, ∀ j ∈NM . (36)

Substituting (36) in (21) we obtain that the NE payment to the suppliers must be of the form

r∗i = (pS−i∗)Tx∗i = (p∗)

Tx∗i , ∀ i∈NS. (37)

Similarly, substituting (36) in (23) we obtain that the NE payment by the manufacturers must be

of the form

g∗j = (pM−j∗)Ty∗j = (p∗)

Ty∗j , ∀ j ∈NM . (38)

Claim 2 If m∗ is a NE of the game specified by the game form presented in Section 3.2 and the

firms’ utility functions (6) and (7), then the allocation (g∗, r∗, x∗, y∗) at m∗ is a feasible solution

of Problem (PC), i.e., (g∗, r∗, x∗, y∗)∈D.

Proof:

By the design of the outcome function (20),

∑j∈NM

yj(m) =∑i∈NS

xi(m) ∀m∈M. (39)


Since (39) holds for all messages m ∈M, it also holds for the NE m∗. 7 Furthermore, Claim 1

implies that

∑i∈NS

r∗i −∑

j∈NM

g∗j =∑i∈NS

(pS−i∗)Tx∗i −

∑j∈NM

(pM−j∗)Ty∗j = (pS

−i∗)T( ∑

i∈NS

x∗i −∑

j∈NM

y∗j

)= 0. (40)

The second equality in (40) follows from the fact that at NE m∗, pSi

∗= pM

j

∗ ∀ i∈NS, j ∈NM , and

hence, pS−i∗

= pM−j∗, ∀ i∈NS, j ∈NM . The last equality in (40) follows from (39).

We complete the proof of the claim by contradiction as follows. Suppose that the NE transaction

(r∗i , x∗i ) /∈DS

i for some i∈NS. Then by (6), uSi (r∗i , x

∗i ) =−∞. Consider mS

i = (xi,pSi ) where pS

i =

pS−i∗ ∈RL

+, and xi ∈RL is such that

xi(mSi ,m

∗/i) =xi−1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

∑j∈NM

y∗j = 0. (41)

The construction of mSi implies that

ri(mSi ,m

∗/i) = (pS−i∗)Txi(m

Si ,m

∗/i) −(pSi −pS

−i∗)T (

pSi −pS

−i∗)

= 0, (42)

and (41) and (42) imply that

uSi

(ri(m

Si ,m

∗/i), xi(mSi ,m

∗/i))

= 0>uSi

(r∗i , x

∗i

). (43)

Thus supplier i will find it profitable to deviate to mSi . This means that m∗ is not a NE which is

a contradiction. Therefore we must have that for each i∈NS, (r∗i , x∗i )∈DS

i .

Next suppose that the NE purchase y∗j /∈DMj for some j ∈NM . Then by (7), uM

j (g∗j , y∗j ) =−∞.

Consider mMj = (yj ,p

Mj ) where pM

j = pM−j∗ ∈RL

+, and yj ∈RL is such that

yj(mMj ,m∗/j)

=yj −1

(NM − 1)

∑k∈NMk 6=j

y∗k +1

NM

∑i∈NS

(x∗i −

1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

(yj +

∑k 6=j

k∈NM

y∗k))

= 0. (44)

The construction of mMj implies that

gj(mMj ,m∗/j) = (pM

−j∗)Tyj(m

Mj ,m∗/j) −

(pMj −pM

−j∗)T (

pMj −pM

−j∗)

= 0, (45)


and (44) and (45) imply that

uMj

(gj(m

Mj ,m∗/j), yj(m

Mj ,m∗/j)

)= 0>uM

j

(g∗j , y

∗j

). (46)

Thus manufacturer j will find it profitable to deviate to mMj . This means that m∗ is not a NE

which is a contradiction. Therefore we must have that for each j ∈NM , y∗j ∈DMj , i.e. y∗j ≥ 0. With

the result of Claim 1 it also implies that

g∗j = (pM−j∗)Ty∗j ≥ 0. (47)

It follows from contradictions (43), (46) and relations (47), (39) and (40) that the NE transaction

(g∗, r∗, x∗, y∗)∈D.

Claim 3 The game form presented in Section 3.2 is individually rational, i.e., if m∗ is a NE of

the game induced by this game form and the firms’ utility functions (6) and (7), then, the NE

allocation (g∗, r∗, x∗, y∗) is preferred by each firm over its initial allocation. Mathematically,

uSi (r∗i , x

∗i ) ≥ uS

i (0,0), ∀ i∈NS,

uMj (g∗j , y

∗j ) ≥ uM

j (0,0) ∀ j ∈NM .

Proof:

From Claim 1 we know the form of NE payment. Substituting that from (37) into (30) we obtain

sthat for each i∈NS,

uSi

(ri((xi,p

Si ), m∗/i

), xi

((xi,p

Si ), m∗/i

))≤ uS

i

((pS−i∗)Tx∗i , x

∗i

), ∀mS

i = (xi,pSi )∈MS

i . (48)

In particular, choosing mSi = (xi,p

Si

∗), substituting for ri and xi in (48) from (21) and (19), and

using (34) we obtain

uSi

((pS−i∗)T (xi−

1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

∑j∈NM

y∗j),(xi−

1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

∑j∈NM

y∗j))


≤ uSi


∗i

), ∀xi ∈RL. (49)

Substituting xi− 1(NS−1)

∑k∈NSk 6=i

x∗k + 1NS

∑j∈NM

y∗j = xi in (49) implies that

uSi

((pS−i∗)Txi, xi

)≤ uS

i


∗i

), ∀ xi ∈RL. (50)

Since (50) is satisfied for any xi ∈RL, letting xi = 0 we get from (50) that

uSi (0,0)≤ uS

i


∗i

). (51)

For j ∈NM we have

uMj

(gj((yj ,p

Mj ), m∗/j

), yj

((yj ,p

Mj ), m∗/j

))≤uM

j

((pM−j∗)Ty∗j , y

∗j

), ∀mM

j = (yj ,pMj )∈MM

j .(52)

In particular, choosing mMj = (yj ,p

Mj

∗), substituting for gj and yj in (52) from (23) and (20), and

using (35) we obtain

uMj

((pM−j∗)T(yj −

1

(NM − 1)

∑k∈NMk 6=j

y∗k +1

NM

∑i∈NS

(x∗i −

1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

(yj +

∑k∈NMk 6=j

y∗k)))

,

(yj −

1

(NM − 1)

∑k∈NMk 6=j

y∗k +1

NM

∑i∈NS

(x∗i −

1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

(yj +

∑k∈NMk 6=j

y∗k))))

≤ uMj


∗j

), ∀ yj ∈RL. (53)

Substituting yj− 1(NM−1)

∑k∈NMk 6=j

y∗k + 1NM

∑i∈NS

(x∗i− 1

(NS−1)

∑k∈NSk 6=i

x∗k + 1NS

(yj +∑

k∈NMk 6=j

y∗k))

= yj

in (53) implies that

uMj

((pM−j∗)Tyj , yj

)≤ uM

j


∗j

), ∀ yj ∈RL. (54)

For yj = 0, (54) implies that

uMj (0,0)≤ uM

j


∗j

), (55)

and (51) and (55) together complete the proof of Claim 3.


Claim 4 A NE transaction (g∗, r∗, x∗, y∗) is an optimal solution of the centralized problem (PC).

Proof:

From (50) we have that, for each i∈NS,

x∗i = arg maxxi∈RL

uSi

((pS−i∗)Txi, xi

)= arg max{xi| xi∈RL

+,Aixi≤bi}(pS−i∗)Txi− ci(xi). (56)

By Assumption 2 the objective function in (56) is concave. Therefore, KKT conditions are necessary

and sufficient for x∗i to be the maximizer in (56). The KKT condition for (56) says that, ∃λSi ∈RL

+

and νi ∈Rni+ such that x∗i ,λ

Si and νi satisfy

−pS−i∗

+∇xici(xi) |xi=x∗

i−λS

i +ATi νi = 0 (57)

(λSi )

Tx∗i = 0 (58)

νiT(Aix

∗i − bi

)= 0 (59)

x∗i ≥ 0 (60)

Aix∗i ≤ bi. (61)

From (54) we also have that, for each j ∈NM ,

y∗j = arg maxyj∈RL

uMj

((pM−j∗)Tyj , yj

)= arg max

yj∈RL+

−(pM−j∗)Tyj + vj(yj). (62)

Since vj is concave, KKT conditions are necessary and sufficient for y∗j to be the maximizer in

(62). The KKT condition for (62) says that ∃λMj ∈RL

+ such that y∗j and λMj satisfy

pM−j∗−∇yj

vj(yj) |yj=y∗j−λM

j = 0 (63)

(λMj )

Ty∗j = 0 (64)

y∗j ≥ 0. (65)

Let us now analyze the solution of Problem (PC). Since ci, i∈NS, are convex, and vj, j ∈NM , are

concave, and D is a compact set, Problem (PC) has a unique solution in (x,y) which is characterized


by the KKT conditions. The solution in (g,r) on the other hand trivially exists. Re-writing the

optimization problem (PC) only for (x,y) we have

max(x,y)∈DX ,Y

−∑i∈NS

ci(xi) +∑

j∈NM

vj(yj) (66)

where,

DX ,Y := {(x, y) | (x, y)∈R(NS+NM )L+ , Aixi ≤ bi, i∈NS,

∑i∈NS

xi =∑

j∈NM

yj}. (67)

Suppose (x∗, y∗) is the optimal transaction corresponding to Problem (PC). Then, (x∗, y∗) along

with some λSi ∈ RL

+,νi, i ∈ NS, λMj ∈ RL

+, j ∈ NM , and µ ∈ RL must satisfy the KKT conditions

given below:

µ+∇xici(xi) |xi=x∗

i−λS

i +ATi νi = 0, i∈NS, (68)

−µ−∇yjvj(yj) |yj=y∗

j−λM

j = 0, j ∈NM , (69)

(λSi )

Tx∗i = 0 (70)

νiT(Aix

∗i − bi

)= 0 (71)

x∗i ≥ 0 (72)

Aix∗i ≤ bi (73)

(λMj )

Ty∗j = 0 (74)

y∗j ≥ 0 (75)

µT (∑i∈NS

x∗i −∑

j∈NM

y∗j ) = 0. (76)

Since the Nash equilibrium prices satisfy pS−i∗

= pM−j∗, ∀ i ∈NS, ∀ j ∈NM , by taking µ=−pS

−i∗

=

−pM−j∗, ∀ i ∈NS, ∀ j ∈NM , the relations (57)–(61) and (63)–(65) imply (68)–(76). Thus the Nash

equilibrium transaction (x∗, y∗) satisfies the KKT conditions for Problem (PC). Furthermore, since

the NE transaction also satisfies

r∗i = (pS−i∗)Tx∗i ≥ 0,

g∗j = (pM−j∗)Ty∗j ≥ 0,


and∑i∈NS

r∗i =∑

j∈NM

g∗j , (77)

all the feasibility conditions of Problem (PC) are satisfied by the NE transaction (g∗, r∗, x∗, y∗).

This proves that the NE transaction is an optimum solution of the centralized problem (PC).

Since the NE m∗ we analyzed in Claims 1–4 was arbitrarily chosen, the results of Claims 1–4

hold true for all NE of the game induced by the game form of Section 3.2 and the firms’ utilities

(6) and (7). Therefore, all NE corresponding to the aforementioned game form result in an optimal

solution of Problem (PC). Furthermore, since Problem (PC) has a unique solution in (x,y), all NE

result in the same optimal supply and purchase vectors, i.e., if m is a NE other than m∗, then,

x(m) = x(m∗) and y(m) = y(m∗). This completes the proof of Theorem 1.

Theorem 1 shows that if there exists a NE of the game induced by the game form of Section 3.2,

then the transaction at the NE is an optimum centralized transaction (optimum solution of Problem

(PC)). However, Theorem 1 does not guarantee the existence of a NE; in other words, it does not

guarantee that the optimum centralized transaction is attainable through NE. This is guaranteed

by Theorem 2 which is proved next.

Appendix B: Proof of Theorem 2

We prove Theorem 2 in two steps. In the first step we show that there exists a price vector such

that when the suppliers and manufacturers individually maximize their respective utilities taking

this price as given, they obtain as their optima the optimal centralized supply and purchase

vectors x∗i , i ∈NS, and y∗j , j ∈NM . In the second step of the proof we show that the above price

vector and the optimum supply and purchase vector (x∗, y∗) can be used to construct message

profiles that are NE of the game induced by the game form of Section 3.2 and the firms’ utility

functions (6) and (7).


Claim 5 If (x∗, y∗) is an optimal solution of Problem (PC), there exists a price vector p∗ such

that,

∀ i∈NS, x∗i = arg max{xi| xi∈RL

+,Aixi≤bi}p∗T xi− ci(xi), (78)

∀ j ∈NM , y∗j = arg max{yj | yj∈RL

+}−p∗T yj + vj(yj). (79)

Proof:

As mentioned in the proof of Claim 4, Problem (PC) has a unique solution in (x,y) since it

involves maximization of a concave function in (x,y) over a compact set D in (x,y). Suppose

(x∗, y∗) is the optimal transaction corresponding to Problem (PC). Then it must satisfy the KKT

conditions. KKT conditions imply that there exist λSi ∈ RL

+, νi ∈ Rni+ for i ∈ NS, λ

Mj ∈ RL

+ for

j ∈NM , and µ∈RL such that these parameters along with (x∗, y∗) satisfy (68)–(76). From (69) it

can be seen that only µ≤ 0 can satisfy the equality since ∇yjvj(yj) |yj=y∗

j≥ 0 and λM

j ≥ 0. Thus

the solution to (68)–(76) must have µ≤ 0.

Let us define p∗ := −µ. By the above argument p∗ ≥ 0. By substituting −p∗ for µ in (68)

and combining it with (70) – (73) we obtain (78). Similarly, substituting p∗ for −µ in (69) and

combining it with (74) – (75) we obtain (79). This completes the proof of Claim 5.

Claim 6 Let (x∗, y∗) be the optimal transaction corresponding to Problem (PC), let p∗ be the

price vector defined in Claim 5, and let r∗i := p∗T x∗i , i ∈NS, and g∗j := p∗T y∗j , j ∈NM . Let mSi

∗:=

(x∗i ,pSi

∗), i∈NS, and mM

j

∗:= (x∗j ,p

Mj

∗), j ∈NM , be a solution to the following set of relations:

x∗i −1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

∑j∈NM

y∗j = x∗i , i∈NS, (80)

y∗j −1

(NM − 1)

∑k∈NMk 6=j

y∗k = y∗j −1

NM

∑i∈NS

x∗i , j ∈NM , (81)

pSi

∗= pM

j

∗= p∗, ∀ i∈NS, ∀ j ∈NM . (82)

Then, m∗ :=((mS

i

∗)i∈NS

, (mMj

∗)j∈NM

)is a NE of the game induced by the game form defined in


Section 3.2. Furthermore, for each i ∈ NS, xi(m∗) = x∗i and ri(m

∗) = r∗i , and for each j ∈ NM ,

yj(m∗) = y∗j and gj(m

∗) = g∗j .

Proof:

Note that equations (80) and (81) are conditions that any NE m∗ of the game induced by the

game form of Section 3.2 must satisfy so as to result in allocation (g∗, r∗, x∗, y∗). (This follows

from (19), (20) and (34)). Therefore, the set of solutions of (80)–(82), if one exists, is a superset of

the set of all NE that result in (g∗, r∗, x∗, y∗). Below we show that the solution set of (80)–(82) is

in fact exactly the set of the NE that result in (g∗, r∗, x∗, y∗).

First note that equations (80) and (81) do have a solution since it is a set of (NS + NM)L

independent equations in (NS + NM)L variables. The existence of a solution of (82) is trivial.

We now show that the set of solutions m∗ of (80)–(82) is the set of NE that result in the given

centralized transaction. From Claim 5, (78) can be equivalently written as,


uSi (p∗T xi, xi), i∈NS. (83)

A change of variable xi =xi− 1(NS−1)

∑k∈NSk 6=i

x∗k + 1NS

∑j∈NM

y∗j in (83) gives


uSi

(p∗T

(xi−

1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

∑j∈NM

y∗j

),

xi−1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

∑j∈NM

y∗j

).

(84)

From (82) we have that, pS−i(m

∗) = pSi

∗= p∗ ∀ i∈NS. Therefore, (84) also implies that

(x∗i ,pSi

∗) = arg max

(xi,pSi)∈RL×RL

+

uSi

(pS−i(m

∗)T(xi−

1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

∑j∈NM

y∗j

)

−(pSi −pS

−i(m∗))T (pSi −pS

−i(m∗)), xi−

1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

∑j∈NM

y∗j

).

(85)

Next, we can equivalently write (79) as,


uMj (p∗T yj , yj), j ∈NM . (86)


A change of variable yj = yj − 1(NM−1)

∑k∈NMk 6=j

y∗k + 1NM

∑i∈NS

(x∗i − 1

(NS−1)

∑k∈NSk 6=i

x∗k + 1NS

(yj +∑

k∈NMk 6=j

y∗k))

, j ∈NM , in (86) gives


uMj

(p∗T

(yj −

1

(NM − 1)

∑k∈NMk 6=j

y∗k

+1

NM

∑i∈NS

(x∗i −

1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

(yj +

∑k∈NMk 6=j

y∗k)))

,

yj −1

(NM − 1)

∑k∈NMk 6=j

y∗k +1

NM

∑i∈NS

(x∗i −

1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

(yj +

∑k∈NMk 6=j

y∗k)))

.

(87)

Since by construction (82), p−j(m∗) = pM

j

∗= p∗ ∀ j ∈NM , it is also implied from (87) that,

(y∗j ,pMj

∗) = arg max

(yj ,pMj

)∈RL×RL+

uMj

(pM−j(m

∗)T(yj −

1

(NM − 1)

∑k∈NMk 6=j

y∗k +1

NM

∑i∈NS

(x∗i

− 1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

(yj +

∑k∈NMk 6=j

y∗k +(pMj −pM

−j(m∗))T (pMj −pM

−j(m∗)))))

,

yj −1

(NM − 1)

∑k∈NMk 6=j

y∗k +1

NM

∑i∈NS

(x∗i −

1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

(yj +

∑k∈NMk 6=j

y∗k)))

.

(88)

Eq. (85) and (88) imply that, if the message exchange and the determination of transaction are

done according to the game form defined in Section 3.2, then supplier i, i ∈ NS, (respectively

manufacturer j, j ∈ NM) maximizes its utility at mSi

∗(respectively mM

j

∗) given that all other

agents use their respective messages constructed from (80)–(82). This implies that a message profile

m∗ that is a solution to (80)–(82) is a NE of the game induced by the game form of Section 3.2.

Furthermore, it follows from (80)–(82) that the allocation at m∗ is,

xi(m∗) = x∗i −

1

(NS − 1)

∑k∈NSk 6=i

x∗k +1

NS

∑j∈NM

y∗j = x∗i , i∈NS, (89)

yj(m∗) = y∗j −

1

(NM − 1)

∑k∈NMk 6=j

y∗k +1

NM

∑i∈NS

xi(m∗) = y∗j , j ∈NM , (90)

pS−i(m

∗) =1

(NS − 1 +NM)

( ∑k∈NSk 6=i

pSk

∗+∑

j∈NM

pMj

∗)

= p∗, i∈NS, (91)


pM−j(m

∗) =1

(NM − 1 +NS)

( ∑k∈NMk 6=j

pMk

∗+∑i∈NS

pSi

∗)

= p∗, j ∈NM , (92)

ri(m∗) = pS

−i(m∗)

Tx∗i −

(pSi

∗−pS−i(m

∗))T (pSi

∗−pS−i(m

∗))

= p∗T x∗i = r∗i , i∈NS, (93)

gj(m∗) = pM

−j(m∗)

Ty∗j +

(pMj

∗−pM−j(m

∗))T (pMj

∗−pM−j(m

∗))

= p∗T y∗j = y∗j , j ∈NM . (94)

Eq. (89)–(94) imply that the set of solutions to (80)–(82) is exactly the set of all NE that result

in the transaction (g∗, r∗, x∗, y∗). This completes the proof of Claim 6 and hence the proof of

Theorem 2.

Having proven Theorem 2, we would like to point out a special feature of the game form pre-

sented in Section 3.2. Recall that Theorem 1 asserts that the game form presented in Section 3.2

implements in NE the goal correspondence π corresponding to Problem (PC), i.e., the set of NE

transactions corresponding to the game form of Section 3.2 is a subset of the set of optimal cen-

tralized transactions (solutions to Problem (PC)) characterized by π. Now suppose we restrict our

attention to Walrasian equilibrium transactions corresponding to Model (M1). A Walrasian equi-

librium transaction for Model (M1) is a transaction (g∗, r∗, x∗, y∗) such that there exists a price

vector p∗ ≥ 0 which along with (g∗, r∗, x∗, y∗) satisfies

x∗i = arg maxxi

ri− ci(xi)

s.t. xi ≥ 0

Aixi ≤ bi

−p∗T xi + ri ≤ 0, i∈NS,

(95)

and

y∗j = arg maxyj

−gj + vj(yj)

s.t. yj ≥ 0

p∗T yj − gj ≤ 0, j ∈NM .

(96)

The conditions −p∗T xi + ri ≤ 0 and p∗T yj − gj ≤ 0 in (95) and (96) are the budget constraints.

Because we do not assume any initial endowment for any supplier or manufacturer in Model (M1),

the right hand side of both the above constraints are 0. If we substitute these budget constraints


into the objective functions in (95) and (96), we get (78) and (79) of Claim 5. Thus, the assertion

of Claim 6 holds for all Walrasian equilibrium transactions (g∗, r∗, x∗, y∗), and this leads to the

following.

Let us define a goal correspondence π′ : E → D that maps every e ∈ E to the set of Walrasian

transactions in D. Then, Theorem 2 implies that the goal correspondence π′ is fully implemented

in NE by the game form presented in Section 3.2, i.e., the set of NE transactions corresponding to

this game form is exactly the set of Walrasian transactions characterized by π′.

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