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Unformatted text preview: ECON 4109H LECTURE 12 The Stability of Marriage December 8, 2010 ECON 4109H LECTURE 12 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 : M W M W { } such that for all m M and w W, ( m ) = w ( w ) = m ECON 4109H LECTURE 12 Stability Definition A matching is stable if 1 For all i M W , 4 i ( i ) 2 For all i , j M W: ( i ) i j i j ( j ) ECON 4109H LECTURE 12 Example Suppose M = { , , } , 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 1, 3 2, 2 3, 1 3, 1 1, 3 2, 2 2, 2 3, 1 1, 3 For example man ranks woman A first, woman B second, and woman C third. A ranks first, second, and third. ECON 4109H LECTURE 12 Example A B C 1, 3 2, 2 3, 1 3, 1 1, 3 2, 2 2, 2 3, 1 1, 3 6 possible sets of marriages 3 are stable: 1 Each man gets his first choice: ( ) = A , ( ) = B , ( ) = 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 1, 3 2, 3 3, 2 4, 3 1, 4 4, 1 3, 3 2, 2 2, 2 1, 4 3, 4 4, 1 4, 1 2, 2 3, 1 1, 4 The underlined entries give the only stable matching....
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This note was uploaded on 01/19/2012 for the course ECON 4109H taught by Professor Notsure during the Fall '09 term at Minnesota.
 Fall '09
 Notsure

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