BVP_pf.txt

(Latest Revision: Wed Feb 5 23:00:42 PST 2014 ) BVP_pf.txt
BVP_pf.txt




Claim: The GS algorithm matches each man to his best valid partner.

Proof:

1) The GS Algorithm always finds a stable matching for its input (N men and N
   women, plus their preference lists).  We already verified that.

2) A matching (X, A) is said to be valid if there is a stable matching that
   pairs X to A.

3) Each man's preference list begins with 0 or more non-valid partners.  Next
   comes the best valid partner (BVP), followed by 0 or more additional names,
   which may or may not be valid partners.

4) 1+2+3 imply that, in any particular execution of the GS algorithm, each man
   eventually proposes to his BVP.

5) Is it possible for a guy's BVP to reject him? 

   Suppose it is possible and consider the first time this happens during some
   execution of the GS algorithm - say it is the rejection of Pete by Gladys.

   This could happen right when Pete proposes to Gladys, or at some later time
   when she sets him free so she can accept another.  In either case, right
   after Gladys rejects Pete, she is engaged to someone, call him Mark.

   Since Mark and Gladys are now engaged, in this execution of GS, Mark has
   already proposed to all the women he likes better than Gladys, and they
   have all rejected him.  Since the rejection of Pete by Gladys was the FIRST
   rejection of a guy by a (B)VP, we can conclude that all the women that Mark
   prefers to Gladys are NOT valid partners of Mark.

   Now consider a stable matching S in which Pete and Gladys are matched.
   Under S, Mark has to be matched to someone - call her Violet.  Violet and
   Mark are valid partners.  So Mark likes Gladys MORE than Violet. However we
   also know that Gladys likes Mark more than Pete.

   That is a contradiction, because S is a stable matching.

   This contradiction shows that the answer to question 5 is "No"

   So we can conclude that each man's BVP does NOT reject him.  In other
   words, when GS executes, each guy eventually proposes to his BVP, and
   remains engaged to her.

   So we see that GS matches each guy to his BVP.

   
   Claim: The GS algorithm matches each woman to her worst valid partner

   Proof: 

   Suppose that the GS algorithm pairs Pete with Gladys, and suppose Pete is
   not the worst valid partner (WVP) of Gladys.

   Then there is a stable matching M in which Gladys is paired with Yuck-o,
   whom she likes less than Pete.

   Suppose Pete is paired under M with Nancy.

   Since Gladys is Pete's BVP, Pete likes Nancy less than Gladys.

   Therefore Pete & Gladys is an instability of M.

   This contradiction shows that the supposition that Pete is not the (WVP) of
   Gladys is false.

   Therefore the claim is established: The GS algorithm matches each woman to
   her worst valid partner.