In this problem, we will consider what happens in stable matching when we are guaranteed addi-...

 

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In this problem, we will consider what happens in stable matching when we are guaranteed addi- tional structure on the preferences. (a) [5 points]. Suppose that among the n ≥ 2 men, there are two men m, m' such that every woman prefers m to m'. Prove that for every stable matching, m prefers his assigned partner to the assigned partner of m'. (b) [10 points]. Suppose that all women have the exact same preference ranking Prove that there is now a unique stable matching, and describe this matching in words. Hint: There is a simpler algorithm than Gale-Shapley for computing the unique stable matching in this special case. Thinking about what that algorithm might be should guide you to the proof. In this problem, we will consider what happens in stable matching when we are guaranteed addi- tional structure on the preferences. (a) [5 points]. Suppose that among the n ≥ 2 men, there are two men m, m' such that every woman prefers m to m'. Prove that for every stable matching, m prefers his assigned partner to the assigned partner of m'. (b) [10 points]. Suppose that all women have the exact same preference ranking Prove that there is now a unique stable matching, and describe this matching in words. Hint: There is a simpler algorithm than Gale-Shapley for computing the unique stable matching in this special case. Thinking about what that algorithm might be should guide you to the proof.

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