Decentralized Matching with Aligned Preferences · PDF file ·...

Matching Through Decentralized Markets

Decentralized Matching with Aligned Preferences

Muriel Niederle Leeat Yariv

May 7, 2011

Motivation

• Much of the matching literature has focused on centralizedmarkets

• Many real matching markets are decentralized: U.S. collegeadmissions, market for law clerks, junior economists, etc.

• One aspect of decentralized markets we will focus on is theinherent dynamic interaction

Motivation

The Goal

• Provide a framework to analyze a two-sided matching marketgame in which firms and workers interact over time.

• Identify conditions under which decentralized markets andcentralized markets produce identical outcomes

• Part of a general theoretical question - are therenon-cooperative foundations for cooperative solutions (e.g.,the core)?

The Goal

• Provide a framework to analyze a two-sided matching marketgame in which firms and workers interact over time.

• Identify conditions under which decentralized markets andcentralized markets produce identical outcomes

• Part of a general theoretical question - are therenon-cooperative foundations for cooperative solutions (e.g.,the core)?

Overview and Insights

• Main ingredients of market game:• preference distribution• information available

• Analyze equilibrium outcomes of this game

• Implementability: suffi cient preference richness allowsstability

• Uniqueness: complete information + aligned preferences +refinement

Related LiteratureEmpirical studies

• Avery, Jolls, Posner, and Roth (2001), Niederle and Roth(2003, 2007), Echenique and Yariv (2011), Fox (2010)

Analysis of dynamic games (mostly complete information,restricted strategy spaces)

• Outcomes: Blum, Roth, and Rothblum (1997), Haeringer andWooders (2009), Diamantoudi, Miyagawa, and Xue (2007)

• Implementation: Alcade (1996), Alcalde, Pérez-Castrillo, andRomero-Medina (1998), Alcalde and Romero-Medina (2000)

Strategic matching in markets with frictions

• Burdett and Coles (1997), Eeckhout (1999), Shimer andSmith (2000)

General Set Up

Economies and Preferences

• A market is a tripletM = (F ,W ,U)• Firms: F = {1, ...,F}• Workers: W = {1, ...,W }• Match utilities:

8><>:�ufij|{z}�,

firm i ′s utility from matching with j

�uwij|{z}�

worker j ′s utility from matching with i

• One-to-one matching with non-transferrable utilities

• Strict preferences, we say worker j is unacceptable to firm i ifufi∅ > u

fij . Similarly for workers.

• ufi∅, uw∅j > 0 for all i , j .

• An economy is a quadruplet (F ,W ,U ,G )• Firms: F = {1, ...,F}• Workers: W = {1, ...,W }• U is a finite collection of match utilities• G is a distribution over U

• One-to-one matching with non-transferrable utilities

• Strict preferences, we say worker j is unacceptable to firm i ifufi∅ > u

fij . Similarly for workers.

• ufi∅, uw∅j > 0 for all i , j .

• An economy is a quadruplet (F ,W ,U ,G )• Firms: F = {1, ...,F}• Workers: W = {1, ...,W }• U is a finite collection of match utilities• G is a distribution over U

Uniqueness

Assume every marketM = (F ,W ,U) has a unique stablematching µM (sidestep coordination).

General Set Up

Economies and PreferencesGame Structure

• Reminder: economy (F ,W ,U ,G )

• t = 0 : market is realized according to G• t = 1, 2, ... : two stages as follows

General Set Up

Economies and PreferencesGame Structure

• Reminder: economy (F ,W ,U ,G )• t = 0 : market is realized according to G• t = 1, 2, ... : two stages as follows

Game Structure

• t = 0 : market is realized according to G

• t = 1, 2, ... : two stages as follows

Stage 1: firms simultaneously decide whether and towhom to make an offer. Unmatched firm canhave at most one offer out.

Stage 2: each worker j who has received an offer from ican accept, reject, or hold the offer.

• Once an offer is accepted, worker j is matched to firm iirreversibly.

Payoffs

• Firm i matched to worker j at time t → payoffs δtufij andδtuwij , where δ ≤ 1 is the market discount factor. Unmatchedagents receive 0.

• To ease getting stable matching: focus on high δ

Payoffs

• Firm i matched to worker j at time t → payoffs δtufij andδtuwij , where δ ≤ 1 is the market discount factor. Unmatchedagents receive 0.

• To ease getting stable matching: focus on high δ

General Set Up

Economies and PreferencesGame StructureInformation

• t = 0 : underlying structure (particularly G ) is commonknowledge. Two information structures:

• Complete Information: all agents are informed of realized U.• Private Information: each agent is informed of their ownrealized match utilities.

General Set Up

Market Monitoring

• Firms and workers observe receival, rejection, and deferral onlyof own offers. When an offer is accepted, the whole market isinformed of the match. Similarly, when there is market exit.

• Equilibrium notion: Bayesian Nash equilibrium.

Market Monitoring

• Firms and workers observe receival, rejection, and deferral onlyof own offers. When an offer is accepted, the whole market isinformed of the match. Similarly, when there is market exit.

• Equilibrium notion: Bayesian Nash equilibrium.

Setup Summary

• Strategic dynamic game: Two important components• Preference distribution (unique stable outcome)• Information: complete or private

• Assumptions making stability easier to achieve:• In any market, unique stable matching• High discount factor

Setup Summary

Complete Information

When information is complete, all agents can compute the stablematching.

Proposition 1: For any economy in the market game there existsa Nash equilibrium in strategies that are not weakly dominatedthat generates the unique stable matching.

Intuition:

• t = 1 : each firm i makes offer to µM (i).

• t = 1 : each worker j accepts firm µM (j) or more preferred, orexits if no offers.

But there can be other (unstable) equilibrium outcomes...

Intuition:

Example: Multiplicity

F1 : W 2�W1� W 3F2 : W 1�W2 � W 3F3 : W 1 � W 2 �W3

,W1: F1� F3 �F2W2: F2�F1� F3W3 : F1 � F3 � F2

�F1 F2 F3W 1 W 2 W 3

�, µ̃ =

�F1 F2 F3W 2 W 1 W 3

µM unique stable matching, can implement µ̃.

Example: Multiplicity

F1 : W 2�W1� W 3F2 : W 1�W2 � W 3F3 : W 1 � W 2 �W3

,W1: F1� F3 �F2W2: F2�F1� F3W3 : F1 � F3 � F2

�F1 F2 F3W 1 W 2 W 3

�, µ̃ =

�F1 F2 F3W 2 W 1 W 3

µM unique stable matching, can implement µ̃.

In “sub-market”without (F3, W 3), multiple stable matchings:

F1 : W 2 � W 1F2 : W 1 � W 2 ,

W1 : F1 � F2W2 : F2 � F1 .

�F1 F2 F3W1 W2 W 3

�, µ̃ =

�F1 F2 F3W2 W1 W 3

Stage 1 : F3 and W 3 match, Stage 2: follow µ̃.

µ̃ induces the firm preferred stable matching in stage 2.

In “sub-market”without (F3, W 3), multiple stable matchings:

F1 : W 2 � W 1F2 : W 1 � W 2 ,

W1 : F1 � F2W2 : F2 � F1 .

�F1 F2 F3W1 W2 W 3

�, µ̃ =

�F1 F2 F3W2 W1 W 3

Stage 1 : F3 and W 3 match, Stage 2: follow µ̃.

µ̃ induces the firm preferred stable matching in stage 2.

Aligned Preferences

Aligned preferences: [Today] uwij = αufij for some α > 0 for i , jmutually acceptable

Implications:

• For any submarket (F ′,W ′,U ′), there exists a top match,where participants are with their favorite option in thesubmarket.

• When preferences are aligned, there is a unique stablematching µM (cf. Clark, 2006).

Intuition: Construct stable match recursively:• top match of entire market must be part of stable match• then top match of remaining market must be part of stablematch

• etc.

Aligned Preferences

Implications:

• etc.

Aligned Preferences

Implications:

• etc.

Aligned Preferences

Implications:

• etc.

Aligned Preferences —Uniqueness

Proposition 2 (Complete Information - Alignment): Withcomplete information, when all supported preferences are aligned,the stable matching of each realized market is the unique Nashequilibrium outcome surviving iterated elimination of weaklydominated strategies.

Complete Information - Interim Summary

• Stable matching is always an equilibrium outcome

• Aligned Preferences: All equilibria surviving iteratedelimination of weakly dominated strategies yield stability.

• In general: There may be equilibria that yield unstableoutcomes.

Centralized clearinghouse with complete information: AllNash equilibria in weakly undominated strategies yield the stableoutcome.

In decentralized markets: Firms can condition their secondround offers on the first period matches, and more outcomes canbe achieved in equilibrium.

Economies with Uncertainty

• Incomplete Information: economy (F ,W ,U ,G ) , eachagent informed of own match utilities only.

• Need to find the stable matching, then implement it.• Transmission of information:

• Match formation or market exit• Making offers• Reacting to offers: acceptance, rejection, or holding

• For the rest of the talk, assume preferences are aligned.

• Need to find the stable matching, then implement it.

• Transmission of information:• Match formation or market exit• Making offers• Reacting to offers: acceptance, rejection, or holding

Aligned Economies: No Frictions

Suppose agents follow deferred acceptance strategies.

• Firms make offers to workers according to their ordinalpreferences.

• Firms exit when all acceptable workers rejected them or exited.• Workers hold most preferred acceptable offer, accept an offerfrom most preferred unmatched firm.

• Workers exit as soon as no acceptable firm is unmatched.

Proposition 3: Suppose preferences are aligned, and δ = 1.Deferred acceptance strategies constitute a Bayesian Nashequilibrium in weakly undominated strategies and yield the stablematching µM .

Note: Alignment — In every period some information becomespublic.

Suppose agents follow deferred acceptance strategies.• Firms make offers to workers according to their ordinalpreferences.

Aligned Economies: Adding Frictions

Will agents use deferred acceptance strategies even withdiscounting (frictions)?

Example: one market economy

U1 =W 1 W 2

F1 3 6F2 4 5

• F2 knows W 1 will accept an offer immediately.• F2 will not make an offer to W 2.

Will agents use deferred acceptance strategies even withdiscounting (frictions)?Example: one market economy

U1 =W 1 W 2

F1 3 6F2 4 5

Will agents use deferred acceptance strategies even withdiscounting (frictions)?Example: one market economy

U1 =W 1 W 2

F1 3 6F2 4 5

Incentive Issues with Alignment

In general, this sort of skipping can lead to economies in which noequilibrium implements the stable matching.

Example: Suppose all prefer to be matched over unmatched,uwij = u

p : U1 =W 1 W 2

F1 3 6F2 4 7

, 1-p : U2 =W 1 W 2

F1 3 6F2 4 5

• Firm 1 and Worker 1 cannot tell U1 and U2 apart.

• Suppose all follow deferred acceptance, with appropriateskipping.

In general, this sort of skipping can lead to economies in which noequilibrium implements the stable matching.Example: Suppose all prefer to be matched over unmatched,uwij = u

p : U1 =W 1 W 2

F1 3 6F2 4 7

, 1-p : U2 =W 1 W 2

F1 3 6F2 4 5

p : U1 =W 1 W 2

F1 3 6F2 4 7

, 1-p : U2 =W 1 W 2

F1 3 6F2 4 5

p : U1 =W 1 W 2

F1 3 6F2 4 7

, 1-p : U2 =W 1 W 2

F1 3 6F2 4 5

p : U1 =W 1 W 2

F1 3 6F2 4 7

, 1-p : U2 =W 1 W 2

F1 3 6F2 4 5

• Firm 1 makes an offer to Worker 2, then Worker 1

• Firm 2 makes an offer to Worker 2 in U1, to Worker 1 in U2• Firm 1 can try to speed up the process by making an offer toWorker 1 in period 1

• Suppose 1 accepts a period 1 offer (add more markets...).

p : U1 =W 1 W 2

F1 3 6F2 4 7

, 1-p : U2 =W 1 W 2

F1 3 6F2 4 5

• Firm 2 makes an offer to Worker 2 in U1, to Worker 1 in U2

• Firm 1 can try to speed up the process by making an offer toWorker 1 in period 1

p : U1 =W 1 W 2

F1 3 6F2 4 7

, 1-p : U2 =W 1 W 2

F1 3 6F2 4 5

p : U1 =W 1 W 2

F1 3 6F2 4 7

, 1-p : U2 =W 1 W 2

F1 3 6F2 4 5

p : U1 =W 1 W 2

F1 3 6F2 4 7

, 1-p : U2 =W 1 W 2

F1 3 6F2 4 5

When Firm 1 observes (3, 6) ,

• Follows deferred acceptance ⇒ payoff: 6(1− p) + 3pδ

• Deviate to an immediate offer to W 1 ⇒ payoff:6(1− p)δ+ 3p

• If p > 2/3 the deviation is profitable.

• Can add markets so that no equilibrium (mixed or pure)generates the stable match always.

Main Issue: The timing of offers in and of itself is informative

p : U1 =W 1 W 2

F1 3 6F2 4 7

, 1-p : U2 =W 1 W 2

F1 3 6F2 4 5

p : U1 =W 1 W 2

F1 3 6F2 4 7

, 1-p : U2 =W 1 W 2

F1 3 6F2 4 5

Simply Use Gale-Shapley?

• Two potential problems:1. How do workers know when to accept an offer (the marketends)?• Alignment helps - there always exists an agent who exits, or atop match.

2. Do agents have incentives to operate in order of theirpreference list (that leads to stability...)?• The example illustrated one incentive issue: the incentive tospeed up matches.

• Another issue: the incentive to alter final match.

• For ‘deferred acceptance’to be incentive compatible, learningmust be limited:• ‘Rich’economies...

Rich Economies

An economy is rich if:

• All ordinal aligned preference constellations are in the supportof the economy

• The support of match utilities is countable (say, integers);• Generation by a two-stage randomization.

Rich Economies

• The support of match utilities is countable (say, integers);

• Generation by a two-stage randomization.

Rich Economies

• The support of match utilities is countable (say, integers);• Generation by a two-stage randomization.

Proposition 4: In a rich economy, for suffi ciently high δ deferredacceptance strategies constitute a Bayesian Nash equilibrium instrategies that are not weakly dominated.

Corollary: In a rich economy, for suffi ciently high δ, the stablematch is implementable through a Bayesian Nash equilibrium instrategies that are not weakly dominated.

In general: Can define ‘learning free’economies that rule outpossibility to speed up or alter matches using deferredacceptance-type of strategies.

How Alignment Helps

• At every stage some information becomes public.

• No incentive to reject a firm in order to trigger a chain leadingto a superior offer.

Conclusions

• With complete information, the unique stable match isalways implementable

• generally not uniquely

• With incomplete information,• Without frictions (δ = 1), can always implement the stablematching

• With frictions, implementability for suffi ciently high δ when theeconomy is rich enough (learning from timing per-se is limited)

Conclusions

• With complete information, the unique stable match isalways implementable

• generally not uniquely

• With incomplete information,• Without frictions (δ = 1), can always implement the stablematching

• With frictions, implementability for suffi ciently high δ when theeconomy is rich enough (learning from timing per-se is limited)

Extensions

Some market attributes that make achieving stability more diffi cult:

• General preferences

• Wages

• Exploding offers

Extensions

• Wages

Extensions

• Wages

T H E E N D

Decentralized Matching with Aligned Preferences · PDF file ·...

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