Assignment
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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).
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Assignment
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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).
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Assignment – Preferences with Ties
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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 > w 2 = w3 = w 4 > w 5 = w6 > w 7
In the following question, all preferences can be
incomplete and have ties.
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Assignment – Preferences with Ties
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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).
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Question 3.1. Does a stable matching always exist?
If yes, give an algorithm to find one; otherwise, give a
counter example.
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Assignment
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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 Paretooptimal? If yes, prove it; otherwise, give a counter
example.
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Assignment
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Question 3.5. Given a matching f, we say f strongly stable
if there is no blocking pair (m,w) where either
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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.
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Reading Assignment
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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.
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