# CSE331 HW1Answer - Lecture 5 CSE 331 Sep 9 2011 Homework 1...

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Sarah Jordan September 16, 2011 HW1 CSE331 1. A) Decide whether the following statement is true or false: In every Stable Marriage problem instance, there always exists a stable matching where for every matched pair (m,w), it is true that both m and w do not have the other as their least preferred partner. If you state true then you will have to formally argue why the statement is correct. If you state false, then you have to give a counter example. Proof idea: false; (m,w) may have the other as their least preferred partner and this can be proved by contradiction. Because of the Gale-Shapely theorem, a stable matching occurs when w women continue finding their preferred m men until a stable matching occurs. If the w woman is finding her most preferred partner, there will eventually be a m man that will be with his least preferred partner OR if the womam’s preferred partner(s) are already taken, she may end up with her least preferred partner. Proof: For example, 3 cases – w proposes to m, but he’s take; m’ is also taken and rejects her. W has to end up with m” because he is the only one left and therefore, she is with her least preferred of the three matchings. The matching would end up as the following: (m, w”), (m’,w’), (m”,w) The preferred choice is proven by Lemma2, which shows w’>w and m>m’ and that by contradicting Gale-Shapely theorem, not everyone ends up in their most preferred matching. the GS theorem is meant to prove that there will always be a stable matching, but women will continuously be choosing better and better men unless they are unavailable. B) Decide whether you think the following statement is true or false. If it is true, give a short

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CSE331 HW1Answer - Lecture 5 CSE 331 Sep 9 2011 Homework 1...

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