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 and
Xing (1994)
one-day market
transactions by telephone
College admissions, Avery and Levin (2009)
early action and early decision
colleges want to admit students who are enthusiastic about
attending
signal enthusiasm
Electronic dating markets
electronic roses
Job market for new Ph.D. economists
each candidate can send signals up to two departments
signals 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 or
East 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, the
hiring of one young professor can cost a school from $10,000
to $15,000." (Mark Whitehouse, The Wall Street Journal,
January 8, 2007)

Motivation

Main question

Understand the value of a signaling mechanism for
markets with various structures

Outline

Outline

1 Literature review
2 A simple example
3 Model
4 Equilibrium analysis
5 Signals and agent welfare
6 Market structure and the value of a signaling mechanism
1 when is signaling most valuable?
2 optimal number of signals
3 many periods

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)

A simple example

A simple example

2 …rms and 2 workers
Preferences of …rms are i.i.d. and
Pr (w1 w2 ) = Pr( w2 w1 ) = 1
fj fj 2
Preference of workers are i.i.d. and
Pr (f1 1
w i f2 ) = Pr( f2 w i f1 ) = 2
Cardinal utility of agent a
top choice ) 1
second choice ) x, 1 > x > 0
unmatched ) 0

A simple example

Markets with signals

Timing
1 Preferences are realized. Each worker sends up to one signal
to one …rm. Workers send signals simultaneously.
2 Each …rm makes up to one o¤er to one worker. Firms make
o¤ers simultaneously.
3 Each worker chooses an o¤er to accept among available o¤ers.
Equilibrium concept: sequential equilibrium (with re…nement D1
of Cho and Kreps)

A simple example

Workers strategies

Proposition
There are two types of symmetric sequential equilibria that satis…es
criterion D1
Babbling equilibria
Equilibria where workers send their signals to their top …rms

A simple example

Firms strategies

Firm’ tradeo¤: o¤er to better worker or o¤er to worker that
s
more likely to accept

A simple example

Reduced game. Firm 1 receives a signal from its second choice

Equilibria in pure strategies
(respond, respond) is always
an equilibrium
respond=o¤er to 2nd choice if …rm 2 is responding, …rm 1
ignore=o¤er to 1st choice must respond!
(ignore, ignore) is also an
equilibrium if x < 0.5

…rm 1n…rm 2 respond ignore
respond x x
1
ignore 0 2 1

A simple example


Welfare
(respond, respond)
uf = 5 + 1 x, uw = 3 , µ = 7 (expected number of matches)
8 4 4 4
(ignore, ignore)
uf = 3 , uw =
4
1
2
1
+ 4 x, µ = 3
2

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 better
o¤ 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

Model

Model

Model

Model

Model

F …rms, W workers
Ordinal preferences
θ f 2 ΘF …rm f ’ preference list (strict)
s
θ w 2 ΘW worker w ’ preference list (strict)
s
θ f and θ w are i.i.d.
Cardinal utility of agent a
ua ( , θ a ) > 0, consistent with θ a , ua (?, θ a ) = 0
for any permutation σ, ua (σ(θ f ), σ(w )) = ua (θ f , w )

Model
Block-correlated preferences


Model


B blocks of …rms.
Firm preferences are uniformly distributed: θ f U (Θf ), i.i.d.
Workers’preferences are block uniform:
1 For any b < b 0 , where b, b 0 2 f1, ..., B g, each worker prefers
every …rm in Fb to any …rm in Fb 0 .
2 Each worker’ preferences within block Fb are uniform and
s
uncorrelated.

Model

Agent strategies in the market with signals

No-signal (no o¤er) option N
Worker’ strategy is
s
sw : Θw ! ∆(F [ N )
last stage: worker w accepts the best o¤er
Firm’ strategy
s
sf : Θ F 2W ! ∆(W [ N )
Equilibrium concept: sequential equilibrium (with
re…nement D1 of Cho and Kreps)

Model

Assumptions

De…nition Worker w ’ strategy sw is anonymous (neutral):
s
8σ 2 Σ, θ w 2 Θw , σ(s (θ w )) = s (σ(θ w )).
De…nition Firm f ’ strategy sf is anonymous (neutral):
s
8σ 2 Σ, θ f 2 Θf , h W )
σ(s (θ f , h)) = s (σ(θ f ), σ(h)).
where σ 2 Σ is some permutation of preference pro…les.

Equilibrium analysis




Block-symmetric sequential equilibria

De…nition
Block-symmetric sequential equilibrium:
Firms that are within each block use the same anonymous
strategy and have the same beliefs.
All workers use the same anonymous strategy.



Characterization

Proposition
Let us consider some block-symmetric sequential equilibrium that
satis…es criterion D1 of Cho and Kreps (1987). Then either
1 The equilibrium is a babbling equilibrium or
2 Workers use top-…rm strategies and …rms have top-…rm beliefs



Babbling equilibria



Top-…rms equilibria


q b (p b ) are the ex-ante probability of receiving an o¤er from
f 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 that
satis…es criterion D1 of Cho and Kreps (1987). Then either
1 for any b 2 f1, ..., B g, q b = p b or
2 there exists b0 2 f1, ..., B g such that q b0 > p b0 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 and
qb > pb .
for any b 0 : αb 0 = 0, the o¤-equilibrium beliefs of each …rm f
2 Fb 0 are such that µf (Γjw hf ) = 1, where
Γ = fθ w 2 Θw : f = max (f 0 2 Fb 0 )g.
θw


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 within
block Fb
If …rm f receives worker w ’ signal its on- and o¤- equilibrium
s
beliefs are that it is top worker w ’ …rm within block F b .
s
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 ]W
We have a natural partial order of cuto¤ strategies:
j = (j1 , . . . , jW ) j 0 = j1 , . . . , jW , 8w = 1, ..., W ,
0 0
jw 0
jw



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

TSWf (θ f ) if rankθ f (TSWf (θ f )) jjh j
s (θ f , h) =
TRWf (θ f ) otherwise.



Proposition (Optimality of Cuto¤ Strategies)
For any strategy sf for …rm f , there exists a cuto¤ strategy which
provides f weakly higher expected payo¤ than sf for any
anonymous strategies s f of opponent …rms f .


Existence

Theorem
There exists a block-symmetric sequential equilibrium where
1 workers play symmetric, top-…rm strategies and
2 …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 increases
its cuto¤s (responds more to signals), …rm f will also optimally
weakly 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 …rms
Additional welfare results



One block of …rms

One block of …rms
Firm preferences, θ f U (Θf ), i.i.d.
Worker preferences, θ w U ( Θw ),
i.i.d.


Equilibrium existence

Theorem
There exists a symmetric sequential equilibrium in pure strategies
where
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

Extensions

Many-to-many markets with many signals
Workers can send several signals
Firms have several positions to …ll
additive valuations for any h W,
uf (h, θ f ) = ∑w 2h uf (w , θ f )
Workers occupy several positions (ex. positions=interviews)

Market structure and the value of a signaling mechanism

Market structure and the value of a signaling
mechanism

Market structure and
the value of a signaling mechanism


Market structure and the value of a signaling
mechanism

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 a
match
for any w 2 W , f 2 F ,
uw (f , θ w ) = uw > 0
for any w 2 W , f 2 F ,
uf (w , θ f ) = uf > 0

Balanced markets

The value of a signaling mechanism

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

W=10 W=100
1.5 15
D D
1.0 10

0.5 5

0.0 0
0 10 20 30 40 50 0 100 200 300 400 500
F F

F=10 F=100
1.5 15
D 1.0 D
10
0.5 5
0.0 0
0 10 20 30 40 50 0 100 200 300 400 500
W W

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 ) = F α( W ) + OF (1)
F
F
D (F , W ) = W β( W ) + OW (1)
where OF (1) and OW (1) are functions that are smaller than a
constant for large F and W correspondingly.

Proposition (Balanced markets)
For …xed W , D (F , W ) attains its maximum value at
F ' 1.0121W + OW (1).
For …xed F , D (F , W ) attains its maximum value at
W ' 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 she
receives.



The symmetric sequential equilibrium in the o¤er game with
signals
each worker sends her signal to her top …rms;
each worker accepts the best available o¤er immediately
each …rm always responds to signals



Proposition
In a market with F …rms and W workers, the value of a signaling
mechanism D (F , W ) decreases with the number of periods of
interactions.


Simulation results

Simulation results:
For the markets with F = W :
D is decreasing over L.
D decreases by 55% for
L=2
D decreases by 95% for
L=4

Matching markets with I interviews and K signals

Matching markets with I interviews and K signals
Simulation results

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

Summary

Summary

1 Model of decentralized matching markets where signals
transmit information
alleviate coordination problem
2 We show that the introduction of a signaling mechanism
1 increases the expected number of matches
2 increases the expected welfare of workers
3 We analyze the value of a signaling mechanism for various
market structures
1 Several signals
2 Several periods of interaction
3 Several …rms’positions
4 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?"


Model

3 …rms and 3 workers
Preferences
Firms’preferences are the same and publicly known
θ f j = (w 1 , w 2 , w 3 )
Workers’preferences
θ w i = (1 ε) θ 0 ε θ ai , i .i .d .
θ 0 = (f1 , f2 , f3 ) the same and publicly known ("typical")
θ ai U (Θw ) ("atypical")



1 Preferences are realized. Each worker can send one signal to
one …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 without signals


Matching market with 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 better
matches


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

Signaling in matching markets

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