Signaling in matching markets

Signaling in Matching Markets

Peter Coles Alexey Kushnir Muriel Niederle

Harvard Business School, The Pennsylvania State University,and Stanford University and NBER

October 2009

Motivation

Papers

"Signaling in Matching Markets"by Peter Coles, Alexey Kushnir, and Muriel Niederle

"Can Costless Signaling Be Harmful for Matching Markets?"Alexey Kushnir

Motivation

Signaling in practice

The entry-level market for clinical psychologists, Roth andXing (1994)

one-day markettransactions by telephone

College admissions, Avery and Levin (2009)

early action and early decisioncolleges want to admit students who are enthusiastic aboutattendingsignal enthusiasm

Electronic dating markets

electronic roses

Job market for new Ph.D. economists

each candidate can send signals up to two departmentssignals are private

Motivation

How signals can be helpful?Roth (2008) about the job market for new Ph.D. economists

1 Transmit information about Ph.D. candidate preferences

transmit candidate speci�c preferences to departments:a candidate wants to obtain position in Europe/ on West orEast U.S. coast/ in a speci�c city

2 Alleviate the coordination problem

Motivation

When signals can be helpful?Congestion

In congested markets

there is limited period of time

candidates may begin accepting o¤ers from other departments

o¤ers are costly

"...because of the arduous nature of the selection process, thehiring of one young professor can cost a school from $10,000to $15,000." (Mark Whitehouse, The Wall Street Journal,January 8, 2007)

Motivation

Main question

Understand the value of a signaling mechanism formarkets with various structures

Outline

1 Literature review2 A simple example3 Model4 Equilibrium analysis5 Signals and agent welfare6 Market structure and the value of a signaling mechanism

1 when is signaling most valuable?2 optimal number of signals3 many periods

Literature review

Costless signaling

Crawford and Sobel (1982)Sobel "Signaling games" (2009)

Costless signaling in centralized matching markets

Lee and Schwarz (2007)Abdulkadiroglu, Che, and Yasuda (2008)

Costless signaling in decentralized matching markets

Roth and Xing (1997)Avery and Levin (2009)

A simple example

2 �rms and 2 workers

Preferences of �rms are i.i.d. and

Pr(w1 �fj w2) = Pr( w2 �fj w1) =12

Preference of workers are i.i.d. and

Pr(f1 �wi f2) = Pr( f2 �wi f1) = 12

Cardinal utility of agent a

top choice ) 1second choice ) x , 1 > x > 0unmatched ) 0

A simple example

Markets with signals

Timing

1 Preferences are realized. Each worker sends up to one signalto one �rm. Workers send signals simultaneously.

2 Each �rm makes up to one o¤er to one worker. Firms makeo¤ers simultaneously.

3 Each worker chooses an o¤er to accept among available o¤ers.

Equilibrium concept: sequential equilibrium (with re�nement D1of Cho and Kreps)

A simple example

Markets with signalsWorkers strategies

Proposition

There are two types of symmetric sequential equilibria that satis�escriterion D1

Babbling equilibria

Equilibria where workers send their signals to their top �rms

A simple example

Markets with signalsFirms strategies

Firm�s tradeo¤: o¤er to better worker or o¤er to worker thatmore likely to accept

A simple example

Markets with signalsReduced game. Firm 1 receives a signal from its second choice

respond=o¤er to 2nd choiceignore=o¤er to 1st choice

Equilibria in pure strategies

(respond, respond) is alwaysan equilibrium

if �rm 2 is responding, �rm 1must respond!

(ignore, ignore) is also anequilibrium if x < 0.5

�rm 1n�rm 2 respond ignorerespond x xignore 0 1

A simple example

Welfare

(respond, respond)

uf =58 +

14 x , uw =

34 , µ = 7

4 (expected number of matches)

(ignore, ignore)

uf =34 , uw =

14 x , µ = 3

A simple example

Observations from our simple example

Firm strategies are strategic complements

if �rm 1 responds more to signals, then �rm 2 is weakly bettero¤ from responding more to signals

Equilibrium ranking

(ignore, ignore) �f (respond , respond)(respond , respond) �w (ignore, ignore)# of matches in (respond , respond) > # of matches in(ignore, ignore)

A simple example

Observations from our simple example

Game with signals versus game without signals

µsig � µno_sig

(uw )sig � (uw )no_sig(uf )sig 7 (uf )no_sig

F �rms, W workers

Ordinal preferences

θf 2 ΘF � �rm f �s preference list (strict)θw 2 ΘW � worker w�s preference list (strict)θf and θw are i.i.d.

Cardinal utility of agent a

ua(�, θa) > 0, consistent with θa, ua(?, θa) = 0for any permutation σ, ua(σ(θf ), σ(w)) = ua(θf ,w)

Block-correlated preferences

B blocks of �rms.

Firm preferences are uniformly distributed: θf � U(Θf ), i.i.d.

Workers�preferences are block uniform:1 For any b < b0, where b, b0 2 f1, ...,Bg, each worker prefersevery �rm in Fb to any �rm in Fb 0 .

2 Each worker�s preferences within block Fb are uniform anduncorrelated.

Agent strategies in the market with signals

No-signal (no o¤er) option NWorker�s strategy is

sw : Θw ! ∆(F [N )last stage: worker w accepts the best o¤er

Firm�s strategy

sf : ΘF � 2W ! ∆(W [N )

Equilibrium concept: sequential equilibrium (withre�nement D1 of Cho and Kreps)

Assumptions

De�nition Worker w�s strategy sw is anonymous (neutral):8σ 2 Σ, θw 2 Θw , σ(s(θw )) = s(σ(θw )).

De�nition Firm f �s strategy sf is anonymous (neutral):8σ 2 Σ, θf 2 Θf , h � W )σ(s(θf , h)) = s(σ(θf ), σ(h)).

where σ 2 Σ is some permutation of preference pro�les.

Equilibrium analysis

Block-symmetric sequential equilibria

De�nitionBlock-symmetric sequential equilibrium:

Firms that are within each block use the same anonymousstrategy and have the same beliefs.

All workers use the same anonymous strategy.

Characterization

Proposition

Let us consider some block-symmetric sequential equilibrium thatsatis�es criterion D1 of Cho and Kreps (1987). Then either

1 The equilibrium is a babbling equilibrium or2 Workers use top-�rm strategies and �rms have top-�rm beliefs

Babbling equilibria

Top-�rms equilibria

qb (pb) are the ex-ante probability of receiving an o¤er fromf 2 Fb , conditional on (not) sending a signal to �rm f .αb - the probability a worker sends her signal to block Fb .

Proposition

Let us consider some block-symmetric sequential equilibrium thatsatis�es criterion D1 of Cho and Kreps (1987). Then either

1 for any b 2 f1, ...,Bg, qb = pb or2 there exists b0 2 f1, ...,Bg such that qb0 > pb0 and

for any b : αb > 0, if a worker sends her signal to block Fb ,she sends her signal to her most preferred �rm within Fb andqb > pb .for any b0 : αb 0 = 0, the o¤-equilibrium beliefs of each �rm f2 Fb 0 are such that µf (Γjw � hf ) = 1, whereΓ = fθw 2 Θw : f = max

θw(f 0 2 Fb 0)g.

Worker strategies

We �x the worker strategies and �rms beliefs

Workers play a symmetric, top-�rm strategy (α1, ..., αB ).

αb is the probability of sending a signal to top �rm withinblock Fb

If �rm f receives worker w�s signal its on- and o¤- equilibriumbeliefs are that it is top worker w�s �rm within block F b .We now examine stage 2 behavior

Firm strategies

Each �rm chooses between TSW and TRW

Firm strategies: cuto¤ strategies

Cuto¤ strategy is a vector (j1, . . . , jW ) 2 [0,W ]WWe have a natural partial order of cuto¤ strategies:

j = (j1, . . . , jW ) � j 0 =�j 01, . . . , j 0W

�, 8w = 1, ...,W ,

jw � j 0w

De�nition

Strategy sf is a cuto¤ strategy for �rm f if 9j1, . . . , jW 2 [1,W ] :for any θf 2 Θf and h � W ,

s(θf , h) =�TSWf (θf ) if rankθf (TSWf (θf )) � jjhjTRWf (θf ) otherwise.

Proposition (Optimality of Cuto¤ Strategies)

For any strategy sf for �rm f , there exists a cuto¤ strategy whichprovides f weakly higher expected payo¤ than sf for anyanonymous strategies s�f of opponent �rms �f .

Existence

TheoremThere exists a block-symmetric sequential equilibrium where

1 workers play symmetric, top-�rm strategies and2 �rms play block-symmetric, anonymous, cuto¤ strategies.

Signals and agent welfare

Proposition (Strategic complements)

Suppose �rms �f use cuto¤ strategies. If �rm f 0 2 �f increasesits cuto¤s (responds more to signals), �rm f will also optimallyweakly increase its cuto¤s.

Welfare comparison

E (µ) No signaling � Signaling

E (W workers ) No signaling� Signaling

E (W �rms ) No signaling? Signaling

One block of �rms: additional welfare results

One block of �rmsAdditional welfare results

One block of �rms

Firm preferences, θf � U(Θf ), i.i.d.

Worker preferences, θw � U(Θw ),i.i.d.

Equilibrium existence

TheoremThere exists a symmetric sequential equilibrium in pure strategieswhere

each worker send a signal to her top �rm and

�rms employ symmetric strategies.

Welfare ranking of symmetric equilibria

Proposition (Welfare ranking of symmetric equilibria)

In a symmetric equilibrium with greater cuto¤s:

the expected number of matches is higher

workers have higher expected payo¤s

�rms have lower expected payo¤s.

Extensions

Many-to-many markets with many signals

Workers can send several signalsFirms have several positions to �ll

additive valuations for any h � W ,uf (h, θf ) = ∑w2h uf (w , θf )

Workers occupy several positions (ex. positions=interviews)

Market structure and the value of a signaling mechanism

Market structure and the value of a signalingmechanism

Market structure andthe value of a signaling mechanism

Market structure and the value of a signalingmechanism

Questions:

Large vs small markets: when is signaling most valuable?

Many periods of interactions

What is the optimal number of signals?

Pure coordination model

One block of �rms, B = 1

Agents care only about obtaining amatch

for any w 2 W , f 2 F ,uw (f , θw ) = uw > 0for any w 2 W , f 2 F ,uf (w , θf ) = uf > 0

Balanced markets

The value of a signaling mechanism

D(F ,W ) - the expected increase in the number of matches fromthe introduction of the signaling mechanism

0 10 20 30 40 500.0

0 100 200 300 400 5000

0 10 20 30 40 500.00.51.01.5

0 100 200 300 400 5000

Balanced markets

The value of a signaling mechanism for large markets

Proposition (Large markets)

D(F ,W ) is "almost" a homogeneous of degree one

D(F ,W ) = Fα(WF ) +OF (1)

D(F ,W ) = W β( FW ) +OW (1)

where OF (1) and OW (1) are functions that are smaller than aconstant for large F and W correspondingly.

Proposition (Balanced markets)

For �xed W , D(F ,W ) attains its maximum value atF ' 1.0121W +OW (1).For �xed F , D(F ,W ) attains its maximum value atW ' 1.8842F +OF (1).

Matching markets with many periods

There are L periods of interactions.

Agents observe agents that match and leave the market.

Period 0. Workers send their signals.

Each worker sends one signal to some �rm.

Periods 1, ...,L. Each period consists of two stages:

Each �rm makes one o¤er to some worker.Each worker can accept one o¤er from the set of o¤ers shereceives.

The symmetric sequential equilibrium in the o¤er game withsignals

each worker sends her signal to her top �rms;each worker accepts the best available o¤er immediatelyeach �rm always responds to signals

Proposition

In a market with F �rms and W workers, the value of a signalingmechanism D(F ,W ) decreases with the number of periods ofinteractions.

Matching markets with many periodsSimulation results

Simulation results:For the markets with F = W :

D is decreasing over L.

D decreases by 55% forL = 2

D decreases by 95% forL = 4

Matching markets with I interviews and K signals

Matching markets with I interviews and K signalsSimulation results

Markets with I = 1 (solid), 2 (dashed), and 3(dot-dashed)interviews, W = 100 workers and F = I � 100 �rms.

Summary

1 Model of decentralized matching markets where signals

transmit informationalleviate coordination problem

2 We show that the introduction of a signaling mechanism

1 increases the expected number of matches2 increases the expected welfare of workers

3 We analyze the value of a signaling mechanism for variousmarket structures

1 Several signals2 Several periods of interaction3 Several �rms�positions4 Several interviews

Summary

Thank you

Motivating example for "Can private costless signaling be harmful for matching markets?"

Example

Motivating example for"Can costless signaling be harmful for

matching markets?"

3 �rms and 3 workers

Preferences

Firms�preferences are the same and publicly known

θfj = (w1,w2,w3)

Workers�preferences

θwi = (1� ε) θ0 � ε θai , i .i .d .θ0 = (f1, f2, f3) the same and publicly known ("typical")θai � U(Θw ) ("atypical")

1 Preferences are realized. Each worker can send one signal toone �rm.

2 Each �rm can make one o¤er to one worker.3 Each worker chooses an o¤er to accept among available o¤ers.

Matching market without signals

Matching market with signals

Observations from example

In the o¤er game with signals

Some workers received better matches

Some workers (and �rms) are unmatched

Firms, conditional on receiving a signal, obtain weakly bettermatches

Observations from example

Game with signals versus game without signals

µsig � µno_sig

(uw )sig 7 (uw )no_sig(uf )sig 7 (uf )no_sig

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