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 MatchingProblem 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.

Given m1, m2 be two of the men, and w1, w2 be two of the women The preference lists of m1, m2, w1,…

Answered: Show that in every stable matching, m₁,…

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.

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter15: Deterministic Eoq Inventory Models

Section15.2: The Basic Economic Order Quantity Model

Problem 15P

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Question

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.

Expert Solution

Step 1

Given

m_1,m₂ be two of the men, and w₁, w₂ be two of the women

The preference lists of m_1,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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