Marriage and Mathematics Lau Ting Sum Samson Suen Wai.

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Presentation transcript:

Marriage and Mathematics Lau Ting Sum Samson Suen Wai

What is the world record of the largest number of marriages a person have? A.6 B.7 C.8 D.9 E.29

Larry King (Talk show host) Married 8 Times!

Glynn Wolfe The World’s most married man Married 29 Times! Shortest marriage: 19 days Longest marriage: 7 years

Stable matching?! Does stable matching exist? If it exist, how do we find it?

Objectives Identify that some marriages are not stable Model marriages as mathematical games

Find stable matching using Deferred Acceptance Algorithm Use the Deferred Acceptance Algorithm to solve other daily- life problems Objectives

Marriage Game Players are of two types, Boys and Girls. Marrying players of the same gender is not allowed. Polygamy is not allowed.

“Side payments” are not allowed. Not getting married at all is last on everyone’s list.

Double List of Preference Orderings The essential data for a marriage game is a double list of preference orderings.

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.

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.

Example of the deferred acceptance algorithm: Boy Propose: Ans:

Girl Propose: Ans:

In-class Activity

Homework

Does it always work? YES!!! Theorem: The matchings obtained by the Deferred Acceptance Algorithm are stable.

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.

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.

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.

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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