Assignment

1. Assignment Question 1 (blocking lemma). For a given instance with set of men M and set of women W, let f be the men-optimal stable matching. Let f’ be any unstable matching and S be the subset of men who prefer f’ to f. If S ≠ Ø , then there is a blocking pair (m,w) for f’ such that m∈M-S and w∈f’(S). 1

2. Assignment Question 2. [Dubins and Freedman] Assume that f is the men-optimal stable matching when all men and women state their true preferences. Let f’ be the men-optimal stable matching when a coalition of men S ⊆ M lie on their preferences. Then there is m∈S such that f(m) ≥m f’(m). 2

3. Assignment – Preferences with Ties In Gale-Shapley model, preferences are strict. We generalize this condition to ties. That is, a man (or woman) can be indifferent between a few women (or men) on his (or her) preference list. For example, a preference of a man m can be w1 > w2 = w3 = w4 > w5 = w6 > w7 In the following question, all preferences can be incomplete and have ties. 3

4. Assignment – Preferences with Ties When there are ties, we generalize the notion of stable matching as follows: Given a matching f, we say f(weakly) stable if there is no blocking pair (m,w) where both of them strictly prefer each other. That is, w >m f(m) and m >w f(w). Question 3.1. Does a stable matching always exist? If yes, give an algorithm to find one; otherwise, give a counter example. 4

5. Assignment Assume a stable matching exists, answer the following questions: Question 3.2.Are all stable matchings have the same cardinality? If yes, prove it; otherwise, give a counter example. Question 3.3.Are men/women optimal stable matching always exist? If yes, prove it; otherwise, give a counter example. Question 3.4.Are stable matchings always Pareto-optimal? If yes, prove it; otherwise, give a counter example. 5

6. Assignment Question 3.5.Given a matching f, we say fstrongly stable if there is no blocking pair (m,w) where either w >m f(m) and m >w f(w), or w >m f(m) and m =w f(w), or w =m f(m) and m >w f(w) Does a strongly stable matching always exist? If yes, give an algorithm to find one; otherwise, give a counter example. 6

7. Reading Assignment R. W. Irving, Stable Marriage and Indifference, Discrete Applied Mathematics, V.48, 261-272, 1994. Gabrielle Demange, David Gale, Marilda Sotomayor, Multi-Item Auctions. Journal of Political Economy, V.94, 863-872,1986. 7