Descending the Stable Matching Lattice: How many Strategic Agents are required to turn Pessimality to Optimality?

Abstract

The set of stable matchings induces a distributive lattice. The supremum of the stable matching lattice is the boy-optimal (girl-pessimal) stable matching and the infimum is the girl-optimal (boy-pessimal) stable matching. The classical boy-proposal deferred-acceptance algorithm returns the supremum of the lattice, that is, the boy-optimal stable matching. In this paper, we study the smallest group of girls, called the {\em minimum winning coalition of girls}, that can act strategically, but independently, to force the boy-proposal deferred-acceptance algorithm to output the girl-optimal stable matching. We characterize the minimum winning coalition in terms of stable matching rotations and show that its cardinality can take on any value between $0$ and $\left\lfloor \frac{n}{2}\right\rfloor$, for instances with $n$ boys and $n$ girls. Our main result is that, for the random matching model, the expected cardinality of the minimum winning coalition is $(\frac{1}{2}+o(1))\log{n}$. This resolves a conjecture of Kupfer \cite{Kup18}.