# How are relative rankings of previously numbered men and women matter?

Consider a marriage market (MWP) where M is the set of men and W is the set of women. Suppose tha Show more Consider a marriage market (MWP) where M is the set of men and W is the set of women. Suppose that there are equal number of men and women and preferences are strict. Consider the following condition on preferences of men and women: The set of men and the set of women are ordered M={m1 m2 mn} and W={w1 w2 wn} such that I. w_iWm_i _(w_i ) m_j for all j>i II. m_iMw_i _(m_i ) w_j for all j>i. 1) Prove that if M and W can be ordered such that the preference profile satisfies the above conditions (I and II) then ^M (m_i )=w_i for all i=12n when ^M is the men-optimal stable matching. 2) Prove that if M and W can be ordered such that the preference profile satisfies the above conditions (I and II) then ^M=^w where ^M and ^W are the men-optimal and women-optimal matchings respectively. Hint: 1) Remember what we said in the classroom about a potential candidate for m1 and w1. They must be ranked top in each others preference list. You can start with this idea. Since there is a couple like that you know immediately that they must be matched in any stable matching not only in the men-optimal one. Once you got (m1 and w1) then you will look for m2 and w2. What do the conditions say about them? How are relative rankings of previously numbered men and women matter? You can continue with this idea. There are other ways to prove it. This is just a suggestion. You can start with the assumption and then say Suppose there exists a man mi and women wi such that ^M (m_i )w_i.Suppose ^M (m_i )=w_j with ij and then find a contradiction. When you are looking for a contradiction think about whether mi can be matched with a woman wj such that j < i. That is Can m2 be matched with w1? or Can m5be matched with w1 or w2 or w3 or w4?. Now one you show that we cannot have j < i you must show that we cannot have j > i either. Show less

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