Assume that the ranking lists of all women by the men are the same, and analogou

Question

Assume that the ranking lists of all women by the men are the same, and analogously, the ranking of all men by the women are the same. In other words, there is a consensus between the women who is the most favorite man, the second favorite man and so on.

Prove that then there is only one stable matching. What is it? (note that in general, the TMA finds one of possible multiple stable matchings). Don’t forget to prove its stability.

How many rejections are occuring during the exection of the TMA, if this is indeed the input?

Explanation / Answer

According to the given similar problem:

Consider 'n' people of each gender and now number the men and women 1 through n in order of their rank by the opposite gender. From the assumption the one and only stable matching has women i match with man i for all i = 1...n.

To prove this we need to show two things.

          1) No other similar is stable

          2) Whether this similar is stable

In order to prove first (1):

In order to prove second (2):

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