Matching145

# Matching145 - Matching Ichiro Obara UCLA January 27, 2009...

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Matching Ichiro Obara UCLA January 27, 2009 Obara (UCLA) Matching January 27, 2009 1 / 51

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Matching Problem Matching Problem Obara (UCLA) Matching January 27, 2009 2 / 51
Matching Problem Matching Problem Three boys m 1 , m 2 , m 3 and three girls w 1 , w 2 , w 3 . Assume that each boy is looking for a girlfriend and each girl is looking for a boyfriend (or looking for a prom partner). Each boy has a strict preference over girls (or staying alone) and each girl has a strict preference over boys (Ex. w 1 ± m 1 w 2 ± m 1 m 1 ± m 1 w 3 ) How to match these boys and girls? Obara (UCLA) Matching January 27, 2009 3 / 51

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Matching Problem Matching Problem Examples of matching ( m 1 , w 2 ) , ( m 2 , w 1 ) , ( m 3 , w 3 ) ( m 1 , w 2 ) , ( m 2 , w 3 ) , ( m 3 , w 1 ) ( m 1 , w 3 ) , ( m 2 , w 1 ) , ( m 3 ) , ( w 2 ) Obara (UCLA) Matching January 27, 2009 4 / 51
Matching Problem Formal Model Formal Model M = { m 1 , ..., m N } . W = { w 1 , ..., w M } . ± = ( ± m 1 , ..., ± m N , ± w 1 , ..., ± w M ). Obara (UCLA) Matching January 27, 2009 5 / 51

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Matching Problem Formal Model A matching μ describes which woman is matched with which man and which woman or man is unmatched. To be precise, μ is a function from M S W to itself that satisﬁes: I μ ( m ) = w for some w W or μ ( m ) = m (a man matches with a woman or remains unmatched). I μ ( w ) = m for some m M or μ ( w ) = w . I μ ( μ ( m )) = μ 2 ( m ) = m and μ 2 ( w ) = w (consistency). Obara (UCLA) Matching January 27, 2009 6 / 51
Stability Stability Obara (UCLA) Matching January 27, 2009 7 / 51

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Stability Individual Rationality A man or woman is acceptable if matching with him or her is better than staying unmatched. Formally, m is acceptable to w or w is accpetable to m when m ± w w w ± m m respectively. Obara (UCLA) Matching January 27, 2009 8 / 51
Stability Individual Rationality Deﬁne individual rationality . Individual Rationality A matching μ is individually rational if everyone with a partner is being matched with an acceptable partner. μ is blocked by some individual if it is not inidividually rational. Formally, μ is blocked by m ( w ) if m ± m μ ( m )( w ± w μ ( w )). Obara (UCLA) Matching January 27, 2009 9 / 51

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Stability Individual Rationality Remark We only consider individually rational matchings. So we can ignore the part of prefrences over unacceptable men or women. This allows us to represent the following preference w 2 ± m 1 w 3 ± m 1 m 1 ± m 1 w 1 ± m 1 w 4 simply by w 2 ± m 1 w 3 Obara (UCLA) Matching January 27, 2009 10 / 51
Stability Stability Supppose that there are two men and women, and w 2 ± m 1 w 1 w 2 ± m 2 w 1 m 2 ± w 1 m 1 m 2 ± w 2 m 1 . Consider a matching ( m 1 , w 2 ) , ( w 2 , m 1 ). Is this a good matching? w 2 and m 2 have an incentive to break up with their partners to match with each other. Obara (UCLA) Matching January 27, 2009 11 / 51

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Stability Stability Blocking A pair ( m * , w * ) blocks matching μ if w * ± m * μ ( m * ) and m * ± w * μ ( w * ). Obara (UCLA) Matching January 27, 2009 12 / 51
Stability Stability Stability Matching μ is stable if μ is not blocked by any individual or any pair of man and woman.

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## This note was uploaded on 03/06/2010 for the course ECON 102 taught by Professor Serra during the Spring '08 term at UCLA.

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Matching145 - Matching Ichiro Obara UCLA January 27, 2009...

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