Stable_Matching

Stable_Matching - S t a b le M a t c h i n g P r o b le m...

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5 Stable Matching Problem Perfect matching: everyone is matched monogamously. ! Each man gets exactly one woman. ! Each woman gets exactly one man. Stability: no incentive for some pair of participants to undermine assignment by joint action. ! In matching M, an unmatched pair m-w is unstable if man m and woman w prefer each other to current partners. ! Unstable pair m-w could each improve by eloping. Stable matching: perfect matching with no unstable pairs. Stable matching problem. Given the preference lists of n men and n women, find a stable matching if one exists. 6 Stable Matching Problem Q. Is assignment X-C, Y-B, Z-A stable? Zeus Amy Clare Bertha Yancey Bertha Clare Amy Xavier Amy Clare Bertha 1 st 2 nd 3 rd Men’s Preference Profile Clare Xavier Zeus Yancey Bertha Xavier Zeus Yancey Amy Yancey Zeus Xavier 1 st 2 nd 3 rd Women’s Preference Profile favorite least favorite favorite least favorite 7 Stable Matching Problem Q. Is assignment X-C, Y-B, Z-A stable? A. No. Bertha and Xavier will hook up. Zeus Amy Clare Bertha Yancey Bertha Clare Amy Xavier Amy Clare Bertha Clare Xavier Zeus Yancey Bertha Xavier Zeus Yancey Amy Yancey Zeus Xavier 1 st 2 nd 3 rd 1 st 2 nd 3 rd favorite least favorite favorite least favorite 8 Stable Matching Problem Q. Is assignment X-A, Y-B, Z-C stable? A. Yes. Zeus Amy Clare Bertha Yancey Bertha Clare Amy Xavier Amy Clare Bertha Clare Xavier Zeus Yancey Bertha Xavier Zeus Yancey Amy Yancey Zeus Xavier 1 st 2 nd 3 rd 1 st 2 nd 3 rd favorite least favorite favorite least favorite
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9 Stable Roommate Problem Q. Do stable matchings always exist? A. Not obvious a priori. Stable roommate problem. ! 2n people; each person ranks others from 1 to 2n-1. ! Assign roommate pairs so that no unstable pairs. Observation. Stable matchings do not always exist for stable roommate problem. B Bob Chris Adam C A B D D Doofus A B C D C A 1 st 2 nd 3 rd A-B, C-D ! B-C unstable A-C, B-D ! A-B unstable A-D, B-C ! A-C unstable is core of market nonempty? 10 Propose-And-Reject Algorithm Propose-and-reject algorithm. [Gale-Shapley 1962] Intuitive method that guarantees to find a stable matching. Initialize each person to be free. while (some man is free and hasn't proposed to every woman) { Choose such a man m w = 1 st woman on m's list to whom m has not yet proposed if (w is free) assign m and w to be engaged else if (w prefers m to her fiancé m') assign m and w to be engaged, and m' to be free else w rejects m } 11 Proof of Correctness: Termination
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This note was uploaded on 01/24/2011 for the course CS 578 taught by Professor Staff during the Fall '08 term at USC.

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Stable_Matching - S t a b le M a t c h i n g P r o b le m...

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