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Assignment

Explore the concept of stable matching with preferences and ties, including the existence of stable matchings, cardinality, optimality, and strong stability. Learn about algorithms and counterexamples in the context of preference-based assignments.

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Assignment

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  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

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