Decide whether the following statement is true or false: For any instance of the

Question

Decide whether the following statement is true or false:

For any instance of the Stable Matching Problem, if there exists a perfect matching that is not stable due to m and w wanting to be together, then (m, w) will be in every stable matching of that instance. In other words, if (m, w') and (m', w) are both in a perfect matching and m prefers w over w' and w prefers m over m', then (m, w) will be in every stable matching.

please provide formal arguments if the answer is true, but if answer is false then provide the counter arguments

Explanation / Answer

Since m prefers w to w', the pair(m,w') is not stable. Similarly, since w prefers m over m', the pair (m',w) will also not be stable. Since m prefers w to w' and w prefers m to m', therefore (m,w) will always form a stable matching. Therefore (m,w) will appear in all stable matches of the instance. The following statement is true.

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