The Influence of One Strategic Agent on the Core of Stable Matchings

Abstract: In this work, we analyze the influence of a single strategic agent on the quality of the other agents' matchings in a matching market. We consider a stable matching problem with $n$ men and $n$ women when preferences are drawn uniformly from the possible $(n!)^{2n}$ full ranking options. We focus on the effect of a single woman who reports a modified preferences list in a way that is optimal from her perspective. We show that in this case, the quality of the matching dramatically improves from the other women's perspective. When running the Gale--Shapley men-proposing algorithm, the expected women-rank is $O(\log^4 n)$ and almost surely the average women-rank is $O(\log^{2+\epsilon}n)$, rather than a rank of $O(\frac{n}{\log n})$ in both cases under a truthful regime. On the other hand, almost surely, the average men's rank is no better than $\Omega\left(\frac{n}{\log^{2+\epsilon}n}\right)$, compared to a rank of $O({\log n})$ under a truthful regime. All of the results hold for any matching algorithm that guarantees a stable matching, which suggests that the core convergence observed in real markets may be caused by the strategic behavior of the participants.