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 GaleShapely 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 GaleShapely 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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 Fall '11
 RUDRA
 partner, Stable marriage problem, Stable roommates problem

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