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Marriage and Mathematics Lau Ting Sum Samson Suen Wai
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What is the world record of the largest number of marriages a person have? A.6 B.7 C.8 D.9 E.29
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Larry King (Talk show host) Married 8 Times!
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Glynn Wolfe The World’s most married man Married 29 Times! Shortest marriage: 19 days Longest marriage: 7 years
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Stable matching?! Does stable matching exist? If it exist, how do we find it?
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Objectives Identify that some marriages are not stable Model marriages as mathematical games
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Find stable matching using Deferred Acceptance Algorithm Use the Deferred Acceptance Algorithm to solve other daily- life problems Objectives
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Marriage Game Players are of two types, Boys and Girls. Marrying players of the same gender is not allowed. Polygamy is not allowed.
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“Side payments” are not allowed. Not getting married at all is last on everyone’s list.
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Double List of Preference Orderings The essential data for a marriage game is a double list of preference orderings.
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What is Stable? Definition: A matching W is said to be stable if no boy and girl not matched in W prefer each other to their W-mates.
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Consider the matching : Aa, Bc, Cd, Db. In A’s list, b is higher than a. In b’s list, A is higher than D. A will dump a and b will dump D to form a pairing of Ab. Therefore, this matching is not stable.
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Example of the deferred acceptance algorithm: Boy Propose: Ans:
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Girl Propose: Ans:
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In-class Activity
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Homework
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Does it always work? YES!!! Theorem: The matchings obtained by the Deferred Acceptance Algorithm are stable.
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Proof Suppose Aa, Bb are matchings in the boy- propose algorithm and that A prefers b over a. Then, A will appear at b’s list before A shows up at a’s list.
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Since Aa is the result, b rejects A in favor of B at certain stage. This means that b prefers B over A. Therefore, b will not dump B to match with A. Thus, matchings in the boy-propose algorithm are stable. Similar argument goes for girl-propose algorithm.
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Definition: Call a boy (girl) feasible for a girl (boy) if there exists a stable matching in which they are matched. Theorem: In a girl-propose algorithm, no girl is ever rejected by a boy feasible for her. Proof: Exercise.
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What Does that mean? In the girl-propose algorithm, the girls are matched to optimal feasible boys. Therefore, girl-propose algorithm is girl-optimal.
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JUPAS JUPAS = Joint University Programmes Admissions System
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Exchange Programme
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And more… Share your thoughts at LMES
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