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Show that in every stable matching, m₁, m₂ are matched to w₁, w₁, i.e., (m₁,w₁), (m₂, w₂) or (m₁,w₂), (m₂, w₁) must be part of any stable matching.

Show that in every stable matching, m₁, m₂ are matched to w₁, w₁, i.e., (m₁,w₁), (m₂, w₂) or (m₁,w₂), (m₂, w₁) must be part of any stable matching.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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Show that in every stable matching, m₁, m₂ are matched to w₁, W₂, i.e., (m₁,w₁), (m₂, W₂) or (m₁,w₂), (m₂,w₁) must be part of any stable matching.
Transcribed Image Text:Show that in every stable matching, m₁, m₂ are matched to w₁, W₂, i.e., (m₁,w₁), (m₂, W₂) or (m₁,w₂), (m₂,w₁) must be part of any stable matching.
Let m₁, m₂ be two of the men and w₁, W₂ be two of the women in an instance of the Stable Matching Problem with n men and n women. Assume m₁, M₂, W₁, W₂'s preference lists are: m₁'s preference: w₁ > W₂ >... m₂'s preference: W₂ > W₁ >... w₁'s preference: m₂ > m₁ >... w₂'s preference: m₁ > m₂ >... So, we only know the favorite and the second favorite of each of these four persons.
Transcribed Image Text:Let m₁, m₂ be two of the men and w₁, W₂ be two of the women in an instance of the Stable Matching Problem with n men and n women. Assume m₁, M₂, W₁, W₂'s preference lists are: m₁'s preference: w₁ > W₂ >... m₂'s preference: W₂ > W₁ >... w₁'s preference: m₂ > m₁ >... w₂'s preference: m₁ > m₂ >... So, we only know the favorite and the second favorite of each of these four persons.
Expert Solution
Step 1

Given

m1, m2 be two of the men, and w1, w2 be two of the women

The preference lists of m1, m2, w1, w2 are:

m₁'s preference: w1> w2 >...

m2's preference: w2> w1 >...

w1's preference: m2> m1 >...

w2's preference: m1> m2 >...

 

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