Notes On The Stable Matching Problem

• 1. The stopping condition of the main loop is this:
  
  Every man is engaged, or has proposed to all the women (or both).
  
  
• 2. In each iteration of the loop, a man proposes to a woman, and there are only N² possible proposals. (N possible men proposing to N possible women)
  
  
• 3. The loop stops after no more than N² iterations. (Pf: 1+2 imply this.)
  
  (This establishes that the loop is not infinite, and moreover
  
  it tells us something about how efficient the algorithm may be. ( O(N²) )
  
  
• 4. Once a woman has become engaged, she remains engaged.
  
  
• 5. Once all women have received a proposal, all N women are engaged. (Pf: 4 implies this)
  
  
• 6. if all N women are engaged, then all N men are engaged. (Pf: The way the algorithm works, it is not possible for a woman to be engaged to more than one man at a time. So when all N women are engaged, they are engaged to N different men -- all the men.
  
  
• 7. if all N men are engaged, then all N women are engaged. (Pf: The algorithm works in a way that makes it impossible for a man to be engaged to two women at the same time, so the proof of #7 is like the proof of #6.
  
  
• 8. The loop stops, and when it does, all the men and women are engaged. (1+5+6+7 imply this.)
  
  So far we have done enough to verify that the algorithm performs what the book calls a "perfect matching" - it creates N pairs, and each of the N men and N women occur in exactly one of the N pairs.
  
  The other thing we want to know is whether this matching is "stable". Here is the idea: Suppose that
  
  
• 9. Alice is matched to Bill, and
  
  
• 10. Charlotte is matched to David.
  
  Is it possible that *both* the following things are true?
  
  
• 11. Does Alice like David better than Bill?
  
  
• 12. Does David like Alice better than Charlotte?
  
  If so then Alice and David both have an incentive to 'run off' with each other. This would be an instability in the matching.
  
  
• 13. If 12 is true, then David proposed to Alice before he proposed to Charlotte. (Pf: Men start at the top of their list and go down.)
  
  
• 14. If 12 is true, Alice rejected David ... either at the time David proposed, or later. (Pf: Otherwise Alice would still be paired to David.)
  
  
• 15. If 12 is true, after Alice rejected David, every man to whom she was engaged was someone she liked more than David. (Pf: Women reject a man only in favor of a man they like more.)
  
  
• 16. If 12 is true then Alice likes Bill better than David. (Pf: 15 + the fact that she was matched with Bill in the end.)
  
  
• We have shown that 9-12 can't all be true (We showed that if 9, 10, and 12 are true, then 11 must be false). This proves that the algorithm creates a matching with no instabilities.
  
  
• Our text goes on to define what it means for (m, w) to be valid partners. It means there exists a stable matching in which m and w are paired.
  
  
• Then the text proves that the Gale-Shapley algorithm always matches each man with his "best valid partner" - the valid partner who is highest on his list.
  
  This is a pretty amazing result. It means that the algorithm simultaneously gives all the men the "best" match they could receive in any stable matching. Also it means that the algorithm always results in the same matching - even though, each time through the main loop, there is freedom to choose which single man will make the next proposal.
  
  The central idea of the proof is to demonstrate that when the algorithm runs no man is rejected by a valid partner.
  
  
• The text also proves that the G-S algorithm pairs each woman with her worst valid partner.