Question: In this problem, we will consider what happens in stable matching when we are guaranteed addi- tional structure on the preferences. (a) [5 points].

In this problem, we will consider what happens in stable matching when

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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