1. Matching markets
with signals
A. Kushnir
Introduction
A simple example
Outline
Matching markets with signals Model
Almost aligned
preferences
The role of signals
Alexey Kushnir Conclusion
The Pennsylvania State University
January 20, 2009

2. Matching markets
Signaling with signals
A. Kushnir
Introduction
A simple example
Outline
Model
Almost aligned
preferences
I Signaling was introduced in job market for new Ph.D. The role of signals
economists in 2006 year. Conclusion
I Each candidate can send signals up to two departments.

3. Matching markets
What are signals needed for? with signals
A. Kushnir
Introduction
A simple example
Outline
Model
Almost aligned
preferences
I Reduce congestion (Roth, 2008) The role of signals
I Help students with atypical preferences Conclusion
I Insure high-ranked students

4. Matching markets
Congestion with signals
A. Kushnir
Introduction
A simple example
I Congestion is frictions in a decentralized market (…rms Outline
and workers). Model
Almost aligned
I Ideal world is a market without frictions. preferences
I Firms can make o¤ers following the deferred acceptance The role of signals
algorithm. Conclusion
I Real world
I limited period of time
I workers may begin accepting o¤ers from other
departments
I o¤ers may be costly

5. Matching markets
Related work with signals
A. Kushnir
Introduction
A simple example
Outline
Model
Almost aligned
preferences
I Coles and Niederle (2007)
The role of signals
Conclusion

6. Matching markets
Related work with signals
A. Kushnir
Introduction
A simple example
Outline
Model
Almost aligned
preferences
I Coles and Niederle (2007)
The role of signals
I Lee and Schwarz (2007a,b) Conclusion

7. Matching markets
Related work with signals
A. Kushnir
Introduction
A simple example
Outline
Model
Almost aligned
preferences
I Coles and Niederle (2007)
The role of signals
I Lee and Schwarz (2007a,b) Conclusion
I Abdulkadiroglu, Che, and Yasuda (2008)

8. Matching markets
Contribution of this paper with signals
A. Kushnir
Introduction
A simple example
Outline
I Private signals facilitate information asymmetry that Model
could harm decentralized matching markets. Almost aligned
preferences
I The introduction of signals in almost aligned The role of signals
preference environment Conclusion
I decreases number of matches,
I decreases total …rms’welfare,
I ambiguously e¤ects total workers’welfare.
compare to no signaling case.

9. Matching markets
A simple example with signals
A. Kushnir
I 3 …rms and 3 workers Introduction
A simple example
I θ f j = ( w1 , w2 , w3 )
Outline
I θ w i = (1 ε) θ 0 ε θ ai , i.i.d. Model
I ε << 1 Almost aligned
preferences
I θ 0 = (f1 , f2 , f3 ) - "typical"
The role of signals
I θ ai U (Θs ) - "atypical"
Conclusion
I No signals

10. Matching markets
A simple example with signals
A. Kushnir
Introduction
A simple example
I With signals Outline
Model
Almost aligned
preferences
The role of signals
Conclusion

11. Matching markets
Outline with signals
A. Kushnir
Introduction
A simple example
Outline
Model
1. Model. Almost aligned
preferences
2. Almost aligned preferences The role of signals
Conclusion
3. The role of signals: transmit information, facilitate
information asymmetry
4. Conclusion.

12. Matching markets
Model with signals
A. Kushnir
Introduction
A simple example
Outline
I M …rms, N workers (M N) Model
I Ordinal utility Almost aligned
preferences
I θ f 2 ΘF …rm f ’ preference list (strict)
s The role of signals
I θ w 2 ΘW worker w ’ preference list (strict)
s Conclusion
I θ f and θ w are i.i.d.; g (θ ) is their joint distribution
I Cardinal utility
I Assign any cardinal utility ua ( , θ a ) > 0 consistent with
θ a , ua (?, θ a ) = 0

13. Matching markets
Model with signals
A. Kushnir
Introduction
A simple example
Outline
Model
1. Each worker can send one signal to some …rm. The
Almost aligned
signals are private (observed only by …rms who receive preferences
them). The role of signals
Conclusion
2. Each …rm makes one o¤er to some worker. A …rm can
make an o¤er to any worker.
3. Workers choose o¤ers to accept. The realized matches
are observed.

14. Matching markets
Strategies with signals
A. Kushnir
Introduction
A simple example
Outline
Model
I 1 2
Worker w ’ strategy is a duple vw = vw , vw
s Almost aligned
preferences
I vw : Θ W ! F
1 The role of signals
I vw : ΘW 2F ! F
2
Conclusion
I Firm f ’ strategy vf : ΘF
s 2W ! W
I Perfect Bayesian equilibrium

15. Matching markets
Almost aligned preferences with signals
A. Kushnir
Introduction
A simple example
Outline
Model
I ˆ
8j, θ fj = θ = const (vf (h, θ f ) vf (h), h W) Almost aligned
preferences
I 8i, θ wi = (1 ε) θ 0 ε θ ai , i.i.d. The role of signals
I θ 0 = const - "typical" Conclusion
I θ ai A(Θs ) - "atypical"
I ˆ
Enumerate …rms and workers according to θ 0 and θ.

16. Matching markets
Assumptions. with signals
A. Kushnir
Introduction
A simple example
Outline
Model
Almost aligned
Assumption. 0 < ε << 1. preferences
Assumption. A(Θs ) has a full support
The role of signals
Conclusion
Assumption (Positive Role of Signals). 8f 2 F , w 2 W ,
h W vf (hnw ) = w ) vf (h [ w ) = w .

17. Matching markets
Assumptions. with signals
A. Kushnir
Introduction
A simple example
Outline
Model
De…nition. h W is reached for …rm f when agents
Almost aligned
follow v = (vw , vf ) if Pr (hf (θ ) = hjv ) > 0 preferences
The role of signals
De…nition. Firm f responds to worker w ’ signal if
s Conclusion
I 9h W : hnw and h [ w are reached for …rm f and
I vf (hnw ) 6= vf (h [ w ).

18. Matching markets
Benchmark. No signals with signals
A. Kushnir
Introduction
A simple example
Outline
I The only equilibrium is the assortative matching.
Model
Almost aligned
preferences
The role of signals
Conclusion

19. Matching markets
Signals. The set of equilibria. with signals
A. Kushnir
Introduction
A simple example
Outline
Model
De…nition. Almost aligned
preferences
I An equilibrium is babbling if no …rm responds to any The role of signals
signal. Conclusion
I An equilibrium is informative if there is at least one
…rm that responds to some worker’ signal.
s

20. Matching markets
Babbling equilibria with signals
A. Kushnir
Introduction
A simple example
Outline
Model
Almost aligned
preferences
Theorem The role of signals
The only possible match in a babbling equilibrium is the Conclusion
assortative match.

21. Matching markets
Informative equilibria with signals
A. Kushnir
Introduction
A simple example
Theorem Outline
If each …rm responds to all signals from workers better or Model
equal to the corresponding one, the set of strategies Almost aligned
preferences
- vw i (θ w i ) = maxθ wi (f 0 2 F : f 0 θ 0 fi ) The role of signals
( Conclusion
maxθ fj (w : w 2 h) if h fw1 , ..., wj g 6= 0
- vfj (h) =
wj + 1 if h fw1 , ..., wj g = 0
and set of …rms’beliefs consistent with agents’strategies
constitute a unique equilibrium.
Illustration

22. Matching markets
Welfare measure. with signals
A. Kushnir
Introduction
A simple example
Outline
I EMatches (v ) = ∑θ g (θ )m (v (θ ), θ ) Model
Almost aligned
I m (v ( θ ), θ ) is the ex-post number of matches for preferences
v = (vw , vf ) and θ 2 ΘW ΘF . The role of signals
Conclusion
I EW…rms (v ) =
∑f ∑θ g (θ )Prf (hf (θ ))uf (vf (θ )jv f (θ ), hf (θ ), θ f )
I EWworkers (v ) =
∑w ∑θ g (θ )uw (vw (θ w ) jv w (θ w ), θ w )

23. Matching markets
Welfare comparison. with signals
A. Kushnir
Introduction
A simple example
Outline
Model
EMatches No signaling Signaling (informative) Almost aligned
preferences
The role of signals
EW workers No signaling 7 Signaling (informative) Conclusion
EW …rms No signaling Signaling (informative)

24. Matching markets
Welfare comparison. with signals
A. Kushnir
Introduction
Theorem A simple example
Outline
If …rm fj 1 and at least one …rm among ffj +1 , ..., fM g
Model
responds to worker wj 1 ’ signal in an informative
s
Almost aligned
equilibrium, then preferences
The role of signals
Conclusion
EMatches fj No signaling > Signaling (informative)
EW fj No signaling > Signaling (informative)

25. Matching markets
Uniform distribution (Coles and Niederle, 2007) with signals
VS almost aligned preferences A. Kushnir
Introduction
A simple example
Outline
Model
Welfare comparison Almost aligned
preferences
Preferences No signals EMatch EWwork EW…rm The role of signals
Conclusion
Almost aligned 0
Uniform distr. 0 + +

26. Matching markets
Uniform distribution (Coles and Niederle, 2007) with signals
VS almost aligned preferences A. Kushnir
Introduction
A simple example
Outline
Role of signals Model
Almost aligned
preferences
Transmit Facilitate information
Preferences The role of signals
information asymmetry Conclusion
Almost aligned Small Large
Uniform distr. Large Small

27. Matching markets
Conclusion with signals
A. Kushnir
Introduction
A simple example
Outline
I Matching model of the Economics job Model
Almost aligned
market with signals preferences
I there is a "realistic" environment when signals The role of signals
!!do harm!! the matching market. Conclusion
I The roles of private signals in matching markets
I transmit information (positive)
I facilitate information asymmetry (negative)

28. Matching markets
with signals
A. Kushnir
Introduction
A simple example
Outline
Model
Almost aligned
Thank you preferences
The role of signals
Conclusion

29. I Workers’strategies Matching markets
with signals
A. Kushnir
Introduction
A simple example
Outline
Model
Almost aligned
preferences
The role of signals
Conclusion
I Firms’strategies
back