Final answer:
The student's question deals with proving that the Gale-Shapley algorithm is woman-pessimal. Key points for such proofs are the Deferred Acceptance Lemma, Woman-Optimal Property, Pessimal Pair Lemma, and the Perfect Matching Theorem, all contributing to the understanding that every woman ends with the least preferred partner in any stable matching.
Step-by-step explanation:
The question asked refers to the Gale-Shapley algorithm and the issue of stability in matching problems, specifically within the context of the Stable Marriage Problem. To prove that Gale-Shapley (G-S) produces a woman-pessimal outcome, one can discuss several key lemmas and properties.
It states that in the G-S algorithm, once a woman accepts a proposal (which she does only if the new proposal is better than the current one), she will never receive a proposal from a less desirable man in subsequent rounds.
This property indicates that the matching obtained at the end of the G-S algorithm is the best possible outcome for all men, meaning that each woman ends up with the worst partner she could get in any stable matching.
The lemma implies that there isn't a stable matching where a woman can do better without making a man worse off than in their initial matching from the G-S algorithm. Hence, confirming it is indeed woman-pessimal.
This theorem ensures that the G-S algorithm will result in a perfect matching where all participants are matched if there are an equal number of men and women and preferences are complete and strict.