Problem Set 2 (Two-Sided Matching)
Marek Pycia
March 15, 2018
This problem set is optional and ungraded. Its goal is to help you learn the material not
to test your mastery of it. In particular, some questions should be easy, while other might
well be challenging at this time. The solutions will be discussed in tutorial and posted on
OLAT.
Consider the economy from Problem Set 1. There are 4 men and 4 women. The preferences of the men are:
m1
w1
w3
w2
w4
m2
w2
w1
w4
w3
m3 m4
w4 w2
w3 w3
w2 ∅
w1
w2
m1
m3
m2
m4
w3 w4
m2 m4
m1 m1
m3 ∅
m4
The preferences of the women are:
w1
m4
m3
m2
m1
Questions
1. What is the outcome of the DA (deferred acceptance) with men proposing?
2. What is the outcome of the DA with women proposing?
Definition: a matching is core stable if no subset of agents can form new matches
(and drop old ones) in such a way that all of them are strictly better off.
3. Is the outcome of the DA with men proposing core stable?
4. (*) Prove the following Theorem: A matching is stable if and only if it is core stable.
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In the next two questions, agents report preference rankings and then the
men-proposing DA is run to determine who matches with whom.
5. Assume that all agents report their true preferences except possibly man m1 . Can man
m1 improve his matching outcome by reporting a preference ranking different from his
true ranking? If yes, please provide (i) a preference ranking such that if m1 reports
it then he gets a better match than he would by reporting his true ranking, and (ii)
please provide the resulting matching. If no, please give an argument why not.
6. Assume that all agents report their true preferences except possibly woman w1 . Can
woman w1 improve her matching outcome by reporting a preference ranking different
from her true ranking? If yes, please provide (i) a preference ranking such that if w1
reports it then she gets a better match than she would by reporting her true ranking,
and (ii) please provide the resulting matching. If no, please give an argument why not.
7. Are there stable matchings different from the outcomes of DA with men proposing and
DA with women proposing?
8. Find a Nash equilibrium of the game of reporting preferences to men-proposing DA.
9. Find a different Nash equilibrium of the game of reporting preferences to men-proposing
DA.
10. Can an unstable matching be an outcome of a Nash equilibrium of the game of reporting
preferences to men-proposing DA?
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