Sun, 17 May 2009
Bipartite matching and same-sex marriage
However, if same-sex marriages are permitted, there may not be a stable matching, so the character of the problem changes significantly.
A minimal counterexample is:
Suppose we match A–B, C–X. Then since B prefers C to A, and C prefers B to X, B and C divorce their mates and marry each other, yielding B–C, A–X.
But now C can improve her situation further by divorcing B in favor of A, who is only too glad to dump the miserable X. The marriages are now A–C, B–X.
B now realizes that his first divorce was a bad idea, since he thought he was trading up from A to C, but has gotten stuck with X instead. So he reconciles with A, who regards the fickle B as superior to her current mate C. The marriages are now A–B, C–X, and we are back where we started, having gone through every possible matching.
This should not be taken as an argument against same-sex marriage. The model fails to generate the following obvious real-world solution: A, B, and C should all move in together and live in joyous tripartite depravity, and X should jump off a bridge.