Stable matchings with switching costs

TL;DR

This paper introduces a relaxed notion of stability in matching problems, showing that small changes in the stability threshold cause a sudden increase in the expected number of stable matchings, revealing an explosion phenomenon.

Contribution

It models preferences as utilities and analyzes how slight relaxations in stability criteria lead to a dramatic rise in the number of stable matchings.

Findings

Expected number of ε-stable matchings is polynomial for ε ~ n^{-1} log n.

A small increase in ε causes the expected number to become super-polynomial.

The explosion phenomenon persists regardless of imbalance in the matching sizes.

Abstract

In a stable matching problem there are two groups of agents, with agents on one side having their individual preferences for agents on another side as a potential match. It is assumed silently that agents can freely and costlessly ``switch" partners. A matching is called stable if no two unmatched agents prefer each other to their matches. Half a century ago, for equinumerous sides, Knuth demonstrated existence of preferences for which there are exponentially many stable matchings, and he posed a problem of evaluating an expected number of stable matchings when the preferences are uniformly random and independent. It was shown later by Pittel that this expectation is quite moderate, asymptotic to $e^{-1}n\log n$, $n$ being the number of agents on each side. The proof used Knuth's integral formula for the expectation based on a classic inclusion-exclusion counting. In later papers by Pittel this and other integral formulas were obtained by generating preference lists via a pair of random matrices, with independently random, $[0,1]$-uniform entries, rather than from a progressively problematic counting approach. The novelty of this paper is that we view the matrix entries as a basis for cardinal utilities that each agent ascribes to the agents on the other side. Relaxing the notion of stability, we declare matching stable if no unmatched pair of agents {\it strongly\/} prefer each other to their partners, with ``strength'' measured by a parameter $\eps>0$, $\eps=0$ corresponding to classic stability. We show that for $\eps$ of order $n^{-1}\log n$ the expected number of $\eps$-stable matchings is polynomially large, but for $\eps\gg n^{-1}\log n$ {\it however slightly\/} the expectation is suddenly super-polynomially large. This ``explosion'' phenomenon for a very small strength parameter $\eps$ continues to hold for imbalanced matchings, regardless of the imbalance size.