Matching Auctions - Northwestern Universityfaculty.wcas.northwestern.edu/~apa522/MA-slides.pdf · Introduction Model Matching Auctions Truthful Bidding Pro–t Maximization Distortions

Introduction Model Matching Auctions Truthful Bidding Profit Maximization Distortions Endogenous processes Conclusions

Matching Auctions

Alessandro Pavan Northwestern University

Daniel Fershtman Tel Aviv University


Motivation

Mediated matching central to "sharing economy"

Most matching markets intrinsically dynamic – re-matching

- shocks to profitability of existing matching allocations

- gradual resolution of uncertainty about attractiveness

- preference for variety

Re-matching, while pervasive, largely ignored by matching theory


This paper

Dynamic matching

mediated (many-to-many) interactions

evolving private information

payments

capacity constraints

Applications

scientific outsourcing (Science Exchange)

lobbying

sponsored search

internet display advertising

lending (Prospect, LendingClub)

B2B

health-care (MEDIGO)

organized events (meetings.com)

Matching auctions

Dynamics under profit vv welfare maximization


Plan

Model

Matching auctions

Truthful bidding

Profit maximization

Distortions

Endogenous processes

Conclusions


Model

Profit-maximizing platform mediates interactions between 2 sides, A,B

Agents: NA = {1, ..., nA} and NB = {1, ..., nB }, nA , nB ∈N

Period-t match between agents (i , j) ∈ NA ×NB yields gross payoffs

vAijt = θAi · εAijt and vBijt = θBj · εBijt

θki : "vertical" type

εkijt : "horizontal" type (time-varying match-specific)

Agent i’s period-t (flow) type (i ∈ NA):

vAit = (vAi1t , v

Ai2t , ..., v

AinB t )

Agent i’s payoff (i ∈ NA):

UAi =∞

∑t=0

δt ∑j∈NB

vAijt · xijt∞

∑t=0

δtpAit

with xijt = 1 if (i , j)-match active, xijt = 0 otherwise.


Model








vAit = (vAi1t , v

Ai2t , ..., v

AinB t )


UAi =∞

∑t=0

δt ∑j∈NB

vAijt · xijt −∞

∑t=0

δtpAit



Model








vAit = (vAi1t , v

Ai2t , ..., v

AinB t )


UAi =∞

∑t=0

δt ∑j∈NB


∑t=0

δtpAit



Model








vAit = (vAi1t , v

Ai2t , ..., v

AinB t )


UAi =∞

∑t=0

δt ∑j∈NB


∑t=0

δtpAit



Model








vAit = (vAi1t , v

Ai2t , ..., v

AinB t )


UAi =∞

∑t=0

δt ∑j∈NB


∑t=0

δtpAit



Model

Platform’s profits:

∞

∑t=0

δt

(∑i∈NA

pAit + ∑j∈NB

pBjt − ∑i∈NA

∑j∈NB

cijt · xijt

)


Model

In each period t ≥ 1, each agent l ∈ Nk from each side k = A,B can bematched to at most mkl agents from side −k .- one-to-one matching: mkl = 1 all l = 1, ..., n

k , k = A,B

- many-to-many mathcing with no binding capacity constraints:mkl ≥ n−k , all l = 1, ..., nk , k = A,B

In each period t ≥ 1, platform can match up to M pairs of agents

- space, time, services constraint

- platform can delete previously formed matches and create new ones.Total number of existing matches cannot exceed M in all periods.


Model

In each period t ≥ 1, each agent l ∈ Nk from each side k = A,B can bematched to at most mkl agents from side −k .- one-to-one matching: mkl = 1 all l = 1, ..., n

k , k = A,B

- many-to-many mathcing with no binding capacity constraints:mkl ≥ n−k , all l = 1, ..., nk , k = A,B

In each period t ≥ 1, platform can match up to M pairs of agents

- space, time, services constraint

- platform can delete previously formed matches and create new ones.Total number of existing matches cannot exceed M in all periods.


Model

Each θkl drawn independently from (abs cont.) F kl over Θkl = [θ

kl , θ̄

kl ]

Period-t horizontal type εkijt drawn from cdf G kijt (εkijt | εkijt−1)

Agents observe θki prior to joining, but learn (εkijt ) over time


Model


kl , θ̄

kl ]




Model


kl , θ̄

kl ]




Plan

Model

Matching auctions

Truthful bidding

Profit maximization

Distortions


Conclusions


Matching auctions

At t = 0 (i.e., upon joining the platform), each agent l ∈ Nk purchasesmembership status θkl ∈ Θk

l at price pkl (θ)

- higher status → more favorable treatment in subsequent auctions

At any t ≥ 1:

agents bid bklt ≡ (bkljt )j∈N−k , one for each partner from side −k

each match (i , j) ∈ NA ×NB assigned score

Sijt ≡ βAi (θAi ) · bAijt + βBj (θ

Bj ) · bBijt − cijt

matches maximizing sum of scores s.t. individual and aggregate capacityconstraints implemented

unmatched agents pay nothing

matched agents pay pklt (θ, bt )

Full transparency - bids, payments, membership, matches all public.


Matching auctions


l at price pkl (θ)


At any t ≥ 1:










Matching auctions


l at price pkl (θ)


At any t ≥ 1:










Matching auctions


l at price pkl (θ)


At any t ≥ 1:










Matching auctions


l at price pkl (θ)


At any t ≥ 1:










Matching auctions


l at price pkl (θ)


At any t ≥ 1:










Matching auctions


l at price pkl (θ)


At any t ≥ 1:










Matching auctions


l at price pkl (θ)


At any t ≥ 1:










Payments (PST + BV)

Fixing weights β, weighted surplus:

wt ≡ ∑i∈NA

∑j∈NB

Sijt · χijt

w−i ,At = weighted surplus in absence of agent i ∈ NA (same as Wt , butwith SAijs = 0, all j ∈ NB ).

Period-t payments, t ≥ 1 :

ψAit = ∑j∈NB

bAijt · χijt −wt − w−i ,At

βAi (θAi )


(Horizontal) match quality under rule χ:

DAl (θ) ≡ Eλ[χ]|θ[

∞

∑t=1

δt ∑j∈NB

εAijtχijt

]

Period-0 membership fees:

ψAi0 = θAi DAi (θ)−

∫ θAi

θAiDAi (θ

A−i , y )dy −Eλ[χ]|θ0

[∞

∑t=1

δtψAit

]− LAi


Payments

Payments similar to GSPA for sponsored search but adjusted for

- dynamic externalities

- costs of information rents (captured by β)

- matches need not maximize true surplus


Plan

Model

Matching auctions

Truthful bidding

Profit maximization

Distortions


Conclusions


Truthful bidding

Definition

Strategy profile σ = (σkl )k=A,Bl∈N k truthful if each agent

- selects membership status corresponding to true vertical type- at each t ≥ 1, bids given by bkijt = v kijt = θkl · εkijt , all (i , j) ∈ NA ×NB ,k = A,B , irrespective of membership status selected at t = 0 and of past bids.Truthful equilibrium is an equilibrium in which strategy profile is truthful.


Truthful bidding

Theorem

Any matching auction in which Lkl large enough admits an equilibrium in whichall agents participate in each period and follow truthful strategies.Furthermore, such truthful equilibria are periodic ex-post (agents’strategies aresequentially rational, regardless of beliefs about other agents’past and currenttypes).


Plan

Model

Matching auctions

Truthful bidding

Profit maximization

Distortions


Conclusions


Profit maximization

Theorem

Let

βk ,Pl (θkl ) ≡ 1−1− F kl (θ

kl )

f kl (θkl )θ

kl

, all l ∈ Nk , k = A,B . (1)

Suppose Dkl (θ−l ,k , θkl ; β

P ) ≥ 0, all l ∈ Nk , k = A,B , and all θ−l ,k .Matching auctions with weights βP and payments s.t. Lkl = 0, all l ∈ Nk ,k = A,B , maximize platform’s profits across all possible mechanisms.


Plan

Model

Dynamic matching auctions

Truthful bidding

Profit maximization

Distortions


Conclusions


Welfare maximization

Theorem

Let βk ,Wl (θkl ) = 1, all θkl , l ∈ Nk , k = A,B.(i) Matching auctions with weights βW and payments with Lkl large enough, alll ∈ Nk , k = A,B , maximize ex-ante welfare over all possible mechanisms.(ii) Suppose Dkl (θ

−l ,k , θkl ; βW ) ≥ 0, all l ∈ Nk , k = A,B , and all θk−l .

Matching auctions with payment s.t. Lkl = 0, all l ∈ Nk , k = A,B, admitex-post periodic equilibria in which agents participate and follow truthfulstrategies at all histories. Furthermore, such auctions maximize the platform’sprofits over all mechanisms implementing welfare-maximizing matches andinducing the agents to join platform in period zero.


Distortions

Theorem

Assume horizontal types ε non-negative

(1) If none of capacity constraints binds

χPijt = 1 ⇒ χWijt = 1

(2) If only platform’s capacity constraint potentially binding

∑(i ,j)∈NA×NB

χWijt ≥ ∑(i ,j)∈NA×NB

χPijt

(3) If some of individual capacity constraints potentially binding,

∑(i ,j)∈NA×NB

χPijt > 0 ⇒ ∑(i ,j)∈NA×NB

χWijt > 0.

(*) Above conclusions can be reversed with negative horizontal types (upwarddistortions)


Plan

Model

Dynamic matching auctions

Truthful bidding

Profit maximization

Distortions


Conclusions


Endogenous Processes

Endogenous processes:

- when xijt−1 = 0, εkijt = εkijt−1 a.s.

- when xijt−1 = 1, kernel Gijt depends ont−1∑s=1

xijs

- costs cijt may also depend ont−1∑s=1

xijs

- example 1: experimentation in Gaussian world (εkijt = E[ωkij |(zkijs )s ])

- example 2: preference for variety

ε drawn independently across agents and from θ, given x


Endogenous Processes

Endogenous processes:

- when xijt−1 = 0, εkijt = εkijt−1 a.s.

- when xijt−1 = 1, kernel Gijt depends ont−1∑s=1

xijs

- costs cijt may also depend ont−1∑s=1

xijs

- example 1: experimentation in Gaussian world (εkijt = E[ωkij |(zkijs )s ])

- example 2: preference for variety

ε drawn independently across agents and from θ, given x


Index scores

Suppose that either Mt = 1 all t, or all capacity constraints arenon-binding

Auctions similar to those above but where at each t agents adjustmembership status to θklt ∈ Θk

l and scores given by following indexes

Sijt ≡ supτ

Eλij |θ0 ,θt ,bt ,x t−1[∑τs=t δs−t

(βAi (θ

Ai0) · bAijt + βBj (θ

Bj0) · bBijt − cijs (x s−1

)]Eλij |θ0 ,θt ,bt ,x t−1

[∑τs=t δs−t

]whereτ: stopping time

λij |θ0, θt , bt , x t−1: process over bids under truthful bidding, whenεkijt =

bkijtθkit

Same qualitative conclusions as for exogenous processes


Index scores




Sijt ≡ supτ


(βAi (θ




[∑τs=t δs−t



bkijtθkit



Index scores




Sijt ≡ supτ


(βAi (θ




[∑τs=t δs−t



bkijtθkit



Conclusions

Mediated (dynamic) matching- agents learn about attractiveness of partners over time- shocks to profitability of matching allocations

Matching auctions

- similar in spirit to GSPA for sponsored search BUT(i) richer externalities(i) costs of info rents

Ongoing work:- searching for arms/partners


Thank You!

Documents

Matching Auctions - Northwestern Universityfaculty.wcas.northwestern.edu/~apa522/MA-slides.pdf · Introduction Model Matching Auctions Truthful Bidding Pro–t Maximization Distortions