lecture13 - Instructor: Professor Amin Saberi...

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Unformatted text preview: Instructor: Professor Amin Saberi (saberi@stanford.edu) MS&E 111: Introduction to Optimization - Lecture 13 Marriage, Honesty and Stability in 1963, Gale and Shapley decided to study whether there was a stable marriage in a set of n men and n women where each men (and women respectively) would rank every other womam (man)...and that each man/woman was only married to one person! Definition 1 Stable matching Assume we have n men and n women. Each man (woman) has a ranking associated to each woman (man). We call M a matching a one to one correspondence between men and women. Notation: m, P M ( m ) is the woman corresponding to m in M ( P M ( w ) is defined the same way) We say that m and w are a blocking pair if P M ( m ) 6 = w and m prefers w to P M ( m ) and w prefers m to P M ( w ). a matching M is stable if there are no blocking pairs Naturally we ask ourselves if there is a stable matching, and if one can find it should it exists. 2 MS&E 111: Introduction to Optimization - Lecture 13 Algorithm 1 Gale and Shapley (men propose, women dispose) Set all men and women free while there is still a free man m let w be the next woman in m s list if w is free, set P M ( m ) = w if w is not free and w prefers m to m = P M ( w ), free m and set P M ( w ) = m else w rejects m and he stays free. He updates his list to the next woman after w Theorem 1 Gale and Shapleys algorithm always finds a stable matching...
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lecture13 - Instructor: Professor Amin Saberi...

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