Question

Prove in G-S all men and women get matched

a) Stable Matching Theorem
b) Gale-Shapley Algorithm
c) Maximal Matching Principle
d) Bipartite Matching Rule

Answer 1

The subject is asking to demonstrate that in the Gale-Shapley algorithm, all participants are matched in a stable manner. The algorithm proceeds with iterations of proposals and acceptances/rejections based on preferences until no unmatched individuals remain, ensuring everyone is paired. Mutual exclusivity is a separate concept related to probabilities, and without additional information, it cannot be numerically justified in this context.

Step-by-step explanation:

The question is asking to prove that in the Gale-Shapley algorithm, also known as the Deferred Acceptance algorithm, all men and women get matched, which is a concept within the field of Mathematics. While the provided probabilities P(G), P(G|E), P(G AND E), and P(G OR E) and the question regarding mutually exclusive events (G and E) are not immediately relevant to the proof, they are typical within the context of probability theory. The Gale-Shapley algorithm ensures a stable matching by iteratively matching men to women based on preferences until a stable outcome is reached where no pair would rather be with each other over their current partners.

To prove that all men and women get matched:

1. Initialize all men and women as unmatched.
2. While there exists an unmatched man:
3. The unmatched man proposes to the woman he prefers most who has not already rejected him.
4. If the woman is unmatched, she accepts the proposal. If she is matched but prefers the new proposal, she leaves her current match and accepts the new one. The rejected man becomes unmatched.
5. The process repeats until no unmatched men are left.

By the end of this process, every man has proposed to his list of women, and every woman has received proposals and chosen the best option according to her preference list. Hence, this leads to a situation where all men and women are matched, and the matching is stable since no one has an incentive to change their partner.

To address the concept of mutual exclusivity, two events G and E are mutually exclusive if they cannot both occur at the same time. In this content, without specific data, we cannot numerically justify mutual exclusivity. However, numerical justification would involve showing that P(G AND E) is equal to zero.