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

TL;DR

This paper investigates how a single strategic agent can significantly improve the matching quality for others in a stable matching market, revealing the impact of strategic behavior on core stability.

Contribution

It demonstrates that a single woman's strategic preference report can dramatically enhance the match quality for women in the Gale-Shapley algorithm, with quantifiable improvements.

Findings

Women’s average rank improves to O(log^4 n) with strategic reporting.

Men’s average rank remains high at Ω(n / log^{2+ε} n) despite strategic behavior.

Results hold for any stable matching algorithm, indicating core convergence may stem from strategic actions.

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.