Gale-Shapley Algorithm Observations

• The proofs show that the performance is at worst $O(n^2)$.
  • This is an example where we do not "count" the number of loops
• The algorithm where men choose can lead to women receiving a poorer match then if they proposed
  • m: w, w'
  • m': w', w
  • w : m', m
  • w': m, m'
  • S = { (m, w), (m', w')}, both women get their "second" choice.
• There can be multiple solutions to the problem.
  • In the above example, S'={ (m, w'), (m', w)} is also a solution.
  • This would have been the result if women proposed.
• They prove that all executions of Gale-Shapley yield the same result.
  • I don't want to do this proof, but you should read through it.
  • They characterize the "best possible partner"
  • By the algorithm, it is fairly easy to see that the men will automatically get their best possible partner.
    • The proposals are made in best possible order
    • The proposals and are eliminated in order.
    • Thus they keep the first non-eliminated which is their best possible.
• So for men, this is the best of times.
• They claim that women are matched with their "worst possible" valid partner.