GAME THEORY 2017-2018 5 cfu

7 Matching problem

Exercise 89. Given the matching problem M = {a, b, c, d}, W = {A, B, C, D}, with preferences

D a C a B a A b A a A c A d

B b A b C b D c B a B d B b

B c D c C c A d C a C c C b

A d D d C d B d D c D b D a

Find a stable set and a non stable set. Verify that M = W.

Solution A stable set is given by {(a, C), (b, A), (c, B), (d, D)}. A non stable set is given by {(a, A), (b, B), (c, C), (d, D)}, since a C c and C a A. M = W = {(a, C), (b, A), (c, B), (d, D)}.

Exercise 90. Consider the matching problem M = {a, b, c} and W = {A, B, C}, with the following system of preferences:

A a B a C b A c A a

B b C b A c B a B b

C c A c B a C b C c.

- Find the stable set M provided by the men’s courtship algorithm.

- Find the stable set W provided by the women’s courtship algorithm.

- Is there any other stable set?

Solution M = {(a, A), (b, B), (c, C)} is the stable solution obtained when we suppose that men go to visit women, W = {(a, C), (b, A), (c, B)} is the stable solution obtained when we suppose that women go to visit men. A third stable solution is J = {(a, B), (b, C), (c, A)}.

1 Exercise 91. Consider the following matching problem with W = {Catwoman, W onderwoman} and M = {Superman, Batman, F lash}. Suppose that Catwoman prefers Batman to Superman and Superman to Flash, while Wonderwoman prefers Superman to Flash and Flash to Batman. 1. If the women are visiting men, what is the stable set?

2. Find a preference profile for the men such that there is another stable set.

3. If all the men prefer to be paired than to be alone, is there a man that will be alone in all the stable sets? (independently from the men’s preferences) Solution 1. The stable set is (Batman, Catwoman), (Superman, Wonderwoman), Flash alone.

2. For instance if the preferences are

Batman : W onderwoman ∅ Catwoman Superman : Catwoman W onderwoman ∅ F lash : Catwoman W onderwoman ∅

(where ∅ stands for being alone), then the stable set is (Superman, Catwoman), (Flash, Wonder- woman), Batman alone.

3. If all men prefer to be paired than to be alone, then in all the stable sets Flash is alone. Flash can not be paired with Catwoman, since he is her last preference and if one of the other two is alone Catwoman will prefer him to Flash (and vice-versa any of the two will prefer Catwoman than being alone). On the other hand, suppose Flash is paired with Wonderwoman, this couple is stable if Superman is not available, that is if he is paired with Catwoman. But if Batman is alone and he prefers to be paired, the couple (Catwoman, Superman) is not stable. Thus, the only stable solution is (Batman, Catwoman), (Superman, Wonderwoman), Flash alone. Exercise 92. Find an example of a matching problem with 4 women and 4 men such that there is a unique stable set. Is it convenient for somebody to lie about his/her preferences? Exercise 93. Four boys are to be divided into two rooms with two beds each. Is this the same type of problem as the marriage problem? If not, does a stable set always exist? Solution Let A, B, C and D be the four guys, we consider the following preferences

B A C A D,

C B A B D,

A C B C D.

2 Preferences of D are irrelevant. Nobody wants to stay with him and every matching is unstable since the person matched to him can make a proposal that will be accepted. For example if the match is given by (A, B), (C,D), C may make a proposal to B, who will accept since he prefer C instead of A. This happens with every possible match we choose.

Exercise 94. Consider the following situations and prove that in each of them there is at least one couple that is fixed in every stable matching. a) Alan is at the top of Kelly’s preference list, and Kelly is at the top of Alan’s preference list; b) Janet is preferred by every man to all the other women; c) all the men have the same preferences.

Solution a) Alan and Kelly are matched in every possible stable matching; b) Janet is always matched with her favourite man; c) the stable set is unique because the men visiting algorithm and the women visiting algorithm will give the same result.

Exercise 95. Paul is at the bottom of Anne’s preference list, and Anne is at the bottom of Paul’s preference list. Is it possible that there is a stable matching that matches Paul and Anne? Justify your answer.

3