Post on 29-Jun-2020
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)
organized 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
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.
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
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.
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
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.
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
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.
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
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.
Introduction Model Matching Auctions Truthful Bidding Profit Maximization Distortions Endogenous processes Conclusions
Model
Platform’s profits:
∞
∑t=0
δt
(∑i∈NA
pAit + ∑j∈NB
pBjt − ∑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 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.
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 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.
Introduction Model Matching Auctions Truthful Bidding Profit Maximization Distortions Endogenous processes Conclusions
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
Introduction Model Matching Auctions Truthful Bidding Profit Maximization Distortions Endogenous processes Conclusions
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
Introduction Model Matching Auctions Truthful Bidding Profit Maximization Distortions Endogenous processes Conclusions
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
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 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.
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 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.
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 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.
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 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.
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 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.
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 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.
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 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.
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 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.
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 ,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 )
Introduction Model Matching Auctions Truthful Bidding Profit Maximization Distortions Endogenous processes Conclusions
(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
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 = (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.
Introduction Model Matching Auctions Truthful Bidding Profit Maximization Distortions Endogenous processes Conclusions
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).
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 ,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.
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 ,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.
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
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)
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, 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
Introduction Model Matching Auctions Truthful Bidding Profit Maximization Distortions Endogenous processes 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
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 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
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 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
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 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
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!