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Unformatted text preview: CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi ([email protected]) January 5, 2010 Lecture 1: Marriage, Stability and Honesty 1 Stable Matching This lecture is on stable matching problem. Consider a community with a set of n men, M , and a set of n women, W . Each man, m , has a ranking of women representing his preferences i.e., if in m ’s list, woman w comes before woman w , it means that m prefers to marry w rather than w . Similarly each woman w has a ranking of her preferred men. The stable marriage problem asks to pair (match) the men and women in such a way that no two persons prefer each other over their matched partners. More formally: Definition: A matching , P M , is a onetoone mapping from M to W (or equivalently we can define a matching as a onetoone mapping from W to M ). Definition: A pair ( m,w ) is a rogue pair iff 1. m prefers w to his matched partner P M ( m ). 2. w prefers m to to her matched partner m ( w = P M ( m )), A matching is stable if it doesn’t have a rogue pair. Example: Assume that we have M = { A,B,C } , and W = { 1 , 2 , 3 } with preferences (rankings) given by, A : 123 1 : BAC B : 213 2 : ABC C : 321 3 : CBA Let matching P M be as follows: P M ( C ) = 1 P M ( B ) = 2 P M ( A ) = 3 It is easy to see that the matching above is not stable since ( A, 1) is a rogue pair. But the following matching has no rogue pairs (1 , 3 ,A,C get their first choices), and so is stable. P M ( A ) = 1 P M ( B ) = 2 P M ( C ) = 3 Question: How can we find a stable matching in general? Gale and Shapley, in 1962, proposed the Deferred Acceptance Algorithm (1). Here, we assume that each m (w) ranks all the possible women (men), i.e. m’s (w’s) list is complete....
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