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

TL;DR

This paper investigates the minimal group of girls needed to manipulate the stable matching outcome from girl-pessimal to girl-optimal, characterizing this coalition and analyzing its expected size in random models.

Contribution

It characterizes the minimum winning coalition of girls in stable matchings and proves its expected size is about half the logarithm of the number of participants, resolving a conjecture.

Findings

The minimum coalition size can vary from 0 to floor(n/2).

Expected coalition size in random models is approximately (1/2) log n.

The work confirms a conjecture by Kupfer regarding coalition size.

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}.