Matching Auctions Daniel Fershtman Alessandro Pavan Tel Aviv - - PowerPoint PPT Presentation

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

Matching Auctions

Alessandro Pavan Northwestern University Daniel Fershtman Tel Aviv University

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

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

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

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)

• rganized events (meetings.com)

Matching auctions Dynamics under profit vv welfare maximization

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

Plan

Model Matching auctions Truthful bidding Profit maximization Distortions Endogenous processes Conclusions

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

Model

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

Introduction 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

Introduction 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 v A

ijt = θA i · εA ijt

and v B

ijt = θB j · εB ijt

θk

i : "vertical" type

εk

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

Introduction 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 v A

ijt = θA i · εA ijt

and v B

ijt = θB j · εB ijt

θk

i : "vertical" type

εk

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

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

it = (v A i1t, v A i2t, ..., v A inB t)

Introduction 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 v A

ijt = θA i · εA ijt

and v B

ijt = θB j · εB ijt

θk

i : "vertical" type

εk

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

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

it = (v A i1t, v A i2t, ..., v A inB t)

Agent i’s payoff (i ∈ NA): UA

i = ∞

∑

t=0

δt ∑

j∈NB

v A

ijt · xijt − ∞

∑

t=0

δtpA

it

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

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

Model

Platform’s profits:

∞

∑

t=0

δt

• ∑

i∈NA

pA

it + ∑ j∈NB

pB

jt − ∑ i∈NA ∑ j∈NB

cijt · xijt

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

Model

In each period t ≥ 1, each agent l ∈ Nk from each side k = A, B can be matched to at most mk

l agents from side −k.

• one-to-one matching: mk

l = 1 all l = 1, ..., nk, k = A, B

• many-to-many mathcing with no binding capacity constraints:

mk

l ≥ n−k, all l = 1, ..., nk, k = A, B

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

Model

In each period t ≥ 1, each agent l ∈ Nk from each side k = A, B can be matched to at most mk

l agents from side −k.

• one-to-one matching: mk

l = 1 all l = 1, ..., nk, k = A, B

• many-to-many mathcing with no binding capacity constraints:

mk

l ≥ 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.

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

Model

Each θk

l drawn independently from (abs cont.) F k l

• ver Θk

l = [θk l , ¯

θk

l ]

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

Model

Each θk

l drawn independently from (abs cont.) F k l

• ver Θk

l = [θk l , ¯

θk

l ]

Period-t horizontal type εk

ijt drawn from cdf G k ijt(εk ijt | εk ijt−1)

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

Model

Each θk

l drawn independently from (abs cont.) F k l

• ver Θk

l = [θk l , ¯

θk

l ]

Period-t horizontal type εk

ijt drawn from cdf G k ijt(εk ijt | εk ijt−1)

Agents observe θk

i prior to joining, but learn (εk ijt) over time

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

Plan

Model Matching auctions Truthful bidding Profit maximization Distortions Endogenous processes Conclusions

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

Matching auctions

At t = 0 (i.e., upon joining the platform), each agent l ∈ Nk purchases membership status θk

l ∈ Θk l at price pk l (θ)

• higher status → more favorable treatment in subsequent auctions

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

Matching auctions

At t = 0 (i.e., upon joining the platform), each agent l ∈ Nk purchases membership status θk

l ∈ Θk l at price pk l (θ)

• higher status → more favorable treatment in subsequent auctions

At any t ≥ 1:

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

Matching auctions

At t = 0 (i.e., upon joining the platform), each agent l ∈ Nk purchases membership status θk

l ∈ Θk l at price pk l (θ)

• higher status → more favorable treatment in subsequent auctions

At any t ≥ 1:

agents bid bk

lt ≡ (bk ljt)j∈N−k , one for each partner from side −k

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

Matching auctions

At t = 0 (i.e., upon joining the platform), each agent l ∈ Nk purchases membership status θk

l ∈ Θk l at price pk l (θ)

• higher status → more favorable treatment in subsequent auctions

At any t ≥ 1:

agents bid bk

lt ≡ (bk ljt)j∈N−k , one for each partner from side −k

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

i (θA i ) · bA ijt + βB j (θB j ) · bB ijt − cijt

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

Matching auctions

At t = 0 (i.e., upon joining the platform), each agent l ∈ Nk purchases membership status θk

l ∈ Θk l at price pk l (θ)

• higher status → more favorable treatment in subsequent auctions

At any t ≥ 1:

agents bid bk

lt ≡ (bk ljt)j∈N−k , one for each partner from side −k

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

i (θA i ) · bA ijt + βB j (θB j ) · bB ijt − cijt

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

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

Matching auctions

At t = 0 (i.e., upon joining the platform), each agent l ∈ Nk purchases membership status θk

l ∈ Θk l at price pk l (θ)

• higher status → more favorable treatment in subsequent auctions

At any t ≥ 1:

agents bid bk

lt ≡ (bk ljt)j∈N−k , one for each partner from side −k

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

i (θA i ) · bA ijt + βB j (θB j ) · bB ijt − cijt

matches maximizing sum of scores s.t. individual and aggregate capacity constraints implemented unmatched agents pay nothing

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

Matching auctions

At t = 0 (i.e., upon joining the platform), each agent l ∈ Nk purchases membership status θk

l ∈ Θk l at price pk l (θ)

• higher status → more favorable treatment in subsequent auctions

At any t ≥ 1:

agents bid bk

lt ≡ (bk ljt)j∈N−k , one for each partner from side −k

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

i (θA i ) · bA ijt + βB j (θB j ) · bB ijt − cijt

matches maximizing sum of scores s.t. individual and aggregate capacity constraints implemented unmatched agents pay nothing matched agents pay pk

lt(θ, bt)

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

Matching auctions

At t = 0 (i.e., upon joining the platform), each agent l ∈ Nk purchases membership status θk

l ∈ Θk l at price pk l (θ)

• higher status → more favorable treatment in subsequent auctions

At any t ≥ 1:

agents bid bk

lt ≡ (bk ljt)j∈N−k , one for each partner from side −k

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

i (θA i ) · bA ijt + βB j (θB j ) · bB ijt − cijt

matches maximizing sum of scores s.t. individual and aggregate capacity constraints implemented unmatched agents pay nothing matched agents pay pk

lt(θ, bt)

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

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

Payments (PST + BV)

Fixing weights β, weighted surplus: wt ≡ ∑

i∈NA ∑ j∈NB

Sijt · χijt w −i,A

t

= weighted surplus in absence of agent i ∈ NA (same as Wt, but with SA

ijs = 0, all j ∈ NB ).

Period-t payments, t ≥ 1 : ψA

it = ∑ j∈NB

bA

ijt · χijt − wt − w −i,A t

βA

i (θA i )

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

(Horizontal) match quality under rule χ: DA

l (θ) ≡ Eλ[χ]|θ

• ∞

∑

t=1

δt ∑

j∈NB

εA

ijtχijt

• Period-0 membership fees:

ψA

i0 = θA i DA i (θ) −

θA

i

θA

i

DA

i (θA −i, y)dy − Eλ[χ]|θ0

• ∞

∑

t=1

δtψA

it

• − LA

i

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

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

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

Plan

Model Matching auctions Truthful bidding Profit maximization Distortions Endogenous processes Conclusions

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

Truthful bidding

Definition Strategy profile σ = (σk

l )k=A,B l∈N k

truthful if each agent

• selects membership status corresponding to true vertical type
• at each t ≥ 1, bids given by bk

ijt = v k ijt = θk l · εk ijt, 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.

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

Truthful bidding

Theorem Any matching auction in which Lk

l large enough admits an equilibrium in which

all agents participate in each period and follow truthful strategies. Furthermore, such truthful equilibria are periodic ex-post (agents’ strategies are sequentially rational, regardless of beliefs about other agents’ past and current types).

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

Plan

Model Matching auctions Truthful bidding Profit maximization Distortions Endogenous processes Conclusions

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

Profit maximization

Theorem Let βk,P

l

(θk

l ) ≡ 1 − 1 − F k l (θk l )

f k

l (θk l )θk l

, all l ∈ Nk, k = A, B. (1) Suppose Dk

l (θ−l,k, θk l ; βP ) ≥ 0, all l ∈ Nk, k = A, B, and all θ−l,k.

Matching auctions with weights βP and payments s.t. Lk

l = 0, all l ∈ Nk,

k = A, B, maximize platform’s profits across all possible mechanisms.

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

Plan

Model Dynamic matching auctions Truthful bidding Profit maximization Distortions Endogenous processes Conclusions

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

Welfare maximization

Theorem Let βk,W

l

(θk

l ) = 1, all θk l , l ∈ Nk, k = A, B.

(i) Matching auctions with weights βW and payments with Lk

l large enough, all

l ∈ Nk, k = A, B, maximize ex-ante welfare over all possible mechanisms. (ii) Suppose Dk

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

Matching auctions with payment s.t. Lk

l = 0, all l ∈ Nk, k = A, B, admit

ex-post periodic equilibria in which agents participate and follow truthful strategies at all histories. Furthermore, such auctions maximize the platform’s profits over all mechanisms implementing welfare-maximizing matches and inducing the agents to join platform in period zero.

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

Distortions

Theorem Assume horizontal types ε non-negative (1) If none of capacity constraints binds χP

ijt = 1 ⇒ χW ijt = 1

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

∑

(i,j)∈N A×N B

χW

ijt ≥

∑

(i,j)∈N A×N B

χP

ijt

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

∑

(i,j)∈N A×N B

χP

ijt > 0 ⇒

∑

(i,j)∈N A×N B

χW

ijt > 0.

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

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

Plan

Model Dynamic matching auctions Truthful bidding Profit maximization Distortions Endogenous processes Conclusions

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

Endogenous Processes

Endogenous processes:

• when xijt−1 = 0, εk

ijt = εk ijt−1 a.s.

• when xijt−1 = 1, kernel Gijt depends on

t−1

∑

s=1

xijs

• costs cijt may also depend on

t−1

∑

s=1

xijs

• example 1: experimentation in Gaussian world (εk

ijt = E[ωk ij|(zk ijs)s])

• example 2: preference for variety

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

Endogenous Processes

Endogenous processes:

• when xijt−1 = 0, εk

ijt = εk ijt−1 a.s.

• when xijt−1 = 1, kernel Gijt depends on

t−1

∑

s=1

xijs

• costs cijt may also depend on

t−1

∑

s=1

xijs

• example 1: experimentation in Gaussian world (εk

ijt = E[ωk ij|(zk ijs)s])

• example 2: preference for variety

ε drawn independently across agents and from θ, given x

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

Index scores

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

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

Index scores

Suppose that either Mt = 1 all t, or all capacity constraints are non-binding Auctions similar to those above but where at each t agents adjust membership status to θk

lt ∈ Θk l and scores given by following indexes

Sijt ≡ sup

τ

Eλij |θ0,θt,bt,x t−1 ∑τ

s=t δs−t

βA

i (θA i0) · bA ijt + βB j (θB j0) · bB ijt − cijs(xs−1

Eλij |θ0,θt,bt,x t−1 ∑τ

s=t δs−t

where τ: stopping time λij|θ0, θt, bt, xt−1: process over bids under truthful bidding, when εk

ijt = bk

ijt

θk

it

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

Index scores

Suppose that either Mt = 1 all t, or all capacity constraints are non-binding Auctions similar to those above but where at each t agents adjust membership status to θk

lt ∈ Θk l and scores given by following indexes

Sijt ≡ sup

τ

Eλij |θ0,θt,bt,x t−1 ∑τ

s=t δs−t

βA

i (θA i0) · bA ijt + βB j (θB j0) · bB ijt − cijs(xs−1

Eλij |θ0,θt,bt,x t−1 ∑τ

s=t δs−t

where τ: stopping time λij|θ0, θt, bt, xt−1: process over bids under truthful bidding, when εk

ijt = bk

ijt

θk

it

Same qualitative conclusions as for exogenous processes

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

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

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

Thank You!