Signaling in matching markets
Transcript of Signaling in matching markets
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
Signaling in Matching Markets
Motivation
Papers
"Signaling in Matching Markets"by Peter Coles, Alexey Kushnir, and Muriel Niederle
"Can Costless Signaling Be Harmful for Matching Markets?"Alexey Kushnir
Signaling in Matching Markets
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
Signaling in Matching Markets
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
Signaling in Matching Markets
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)
Signaling in Matching Markets
Motivation
Main question
Understand the value of a signaling mechanism formarkets with various structures
Signaling in Matching Markets
Outline
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
Signaling in Matching Markets
Literature review
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)
Signaling in Matching Markets
A simple example
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
Signaling in Matching Markets
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)
Signaling in Matching Markets
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
Signaling in Matching Markets
A simple example
Markets with signalsFirms strategies
Firm�s tradeo¤: o¤er to better worker or o¤er to worker thatmore likely to accept
Signaling in Matching Markets
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
2�1
Signaling in Matching Markets
A simple example
Markets with signals
Welfare
(respond, respond)
uf =58 +
14 x , uw =
34 , µ = 7
4 (expected number of matches)
(ignore, ignore)
uf =34 , uw =
12 +
14 x , µ = 3
2
Signaling in Matching Markets
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)
Signaling in Matching Markets
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
Signaling in Matching Markets
Model
Model
Model
Signaling in Matching Markets
Model
Model
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)
Signaling in Matching Markets
Model
Block-correlated preferences
Block-correlated preferences
Signaling in Matching Markets
Model
Block-correlated preferences
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.
Signaling in Matching Markets
Model
Block-correlated preferences
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)
Signaling in Matching Markets
Model
Block-correlated preferences
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.
Signaling in Matching Markets
Equilibrium analysis
Equilibrium analysis
Equilibrium analysis
Signaling in Matching Markets
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.
Signaling in Matching Markets
Equilibrium analysis
Block-symmetric sequential equilibria
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
Signaling in Matching Markets
Equilibrium analysis
Block-symmetric sequential equilibria
Babbling equilibria
Signaling in Matching Markets
Equilibrium analysis
Block-symmetric sequential equilibria
Top-�rms equilibria
Signaling in Matching Markets
Equilibrium analysis
Block-symmetric sequential 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.
Signaling in Matching Markets
Equilibrium analysis
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
Signaling in Matching Markets
Equilibrium analysis
Firm strategies
Each �rm chooses between TSW and TRW
Signaling in Matching Markets
Equilibrium analysis
Firm strategies: cuto¤ strategies
Signaling in Matching Markets
Equilibrium analysis
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
Signaling in Matching Markets
Equilibrium analysis
Firm strategies: cuto¤ strategies
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.
Signaling in Matching Markets
Equilibrium analysis
Firm strategies: cuto¤ strategies
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 .
Signaling in Matching Markets
Equilibrium analysis
Existence
TheoremThere exists a block-symmetric sequential equilibrium where
1 workers play symmetric, top-�rm strategies and2 �rms play block-symmetric, anonymous, cuto¤ strategies.
Signaling in Matching Markets
Signals and agent welfare
Signals and agent welfare
Signals and agent welfare
Signaling in Matching Markets
Signals and agent welfare
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.
Signaling in Matching Markets
Signals and agent welfare
Welfare comparison
E (µ) No signaling � Signaling
E (W workers ) No signaling� Signaling
E (W �rms ) No signaling? Signaling
Signaling in Matching Markets
Signals and agent welfare
One block of �rms: additional welfare results
One block of �rmsAdditional welfare results
One block of �rms: additional welfare results
Signaling in Matching Markets
Signals and agent welfare
One block of �rms: additional welfare results
One block of �rms
One block of �rms
Firm preferences, θf � U(Θf ), i.i.d.
Worker preferences, θw � U(Θw ),i.i.d.
Signaling in Matching Markets
Signals and agent welfare
One block of �rms: additional welfare results
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.
Signaling in Matching Markets
Signals and agent welfare
One block of �rms: additional welfare results
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.
Signaling in Matching Markets
Extensions
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)
Signaling in Matching Markets
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
Signaling in Matching Markets
Market structure and the 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?
Signaling in Matching Markets
Market structure and the value of a signaling mechanism
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
Signaling in Matching Markets
Market structure and the value of a signaling mechanism
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
0.5
1.0
1.5
F
D
W=10
0 100 200 300 400 5000
5
10
15
F
D
W=100
0 10 20 30 40 500.00.51.01.5
W
D
F=10
0 100 200 300 400 5000
5
10
15
W
D
F=100
Signaling in Matching Markets
Market structure and the value of a signaling mechanism
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).
Signaling in Matching Markets
Market structure and the value of a signaling mechanism
Matching markets with many periods
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.
Signaling in Matching Markets
Market structure and the value of a signaling mechanism
Matching markets with many periods
Matching markets with many periods
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
Signaling in Matching Markets
Market structure and the value of a signaling mechanism
Matching markets with many periods
Matching markets with many periods
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.
Signaling in Matching Markets
Market structure and the value of a signaling mechanism
Matching markets with many periods
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
Signaling in Matching Markets
Market structure and the value of a signaling mechanism
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.
Signaling in Matching Markets
Summary
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
Signaling in Matching Markets
Summary
Thank you
Signaling in Matching Markets
Motivating example for "Can private costless signaling be harmful for matching markets?"
Example
Motivating example for"Can costless signaling be harmful for
matching markets?"
Signaling in Matching Markets
Motivating example for "Can private costless signaling be harmful for matching markets?"
Model
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")
Signaling in Matching Markets
Motivating example for "Can private costless signaling be harmful for matching markets?"
Markets with signals
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.
Signaling in Matching Markets
Motivating example for "Can private costless signaling be harmful for matching markets?"
Matching market without signals
Matching market without signals
Signaling in Matching Markets
Motivating example for "Can private costless signaling be harmful for matching markets?"
Matching market with signals
Matching market with signals
Signaling in Matching Markets
Motivating example for "Can private costless signaling be harmful for matching markets?"
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
Signaling in Matching Markets
Motivating example for "Can private costless signaling be harmful for matching markets?"
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