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Lecture 12_College Admissions and the Stability of Marraige 2010

# Lecture 12_College Admissions and the Stability of Marraige 2010

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ECON 4109H LECTURE 12 The Stability of Marriage December 8, 2010 ECON 4109H LECTURE 12

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The Marriage Problem Suppose that there is a set M of men and a set W of women. For each m M there is a strict preference m over W { } , where corresponds to remaining alone. For each w W there is a strict preference w over M { } , where corresponds to remaining alone. ECON 4109H LECTURE 12
Matching Definition A matching is a function μ : M W M W { } such that for all m M and w W , μ ( m ) = w μ ( w ) = m ECON 4109H LECTURE 12

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Stability Definition A matching is stable if 1 For all i M W , 4 i μ ( i ) 2 For all i , j M W : μ ( i ) i j i j μ ( j ) ECON 4109H LECTURE 12
Example Suppose M = { α , β , γ } , W = { A , B , C } . Suppose that each agent ranks remaining single as the worst option. The following table gives the rankings of the matches, where a smaller number corresponds to a higher rank: A B C α 1, 3 2, 2 3, 1 β 3, 1 1, 3 2, 2 γ 2, 2 3, 1 1, 3 For example man α ranks woman A first, woman B second, and woman C third. A ranks β first, γ second, and α third. ECON 4109H LECTURE 12

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Example A B C α 1, 3 2, 2 3, 1 β 3, 1 1, 3 2, 2 γ 2, 2 3, 1 1, 3 6 possible sets of marriages 3 are stable: 1 Each man gets his first choice: μ ( α ) = A , μ ( β ) = B , μ ( γ ) = C . (Here each woman gets her last choice!) 2 Each woman gets her first choice. 3 Everyone gets their second choice. ECON 4109H LECTURE 12
Another example A B C D α 1, 3 2, 3 3, 2 4, 3 β 1, 4 4, 1 3, 3 2, 2 γ 2, 2 1, 4 3, 4 4, 1 δ 4, 1 2, 2 3, 1 1, 4 The underlined entries give the only stable matching.

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