Gaussian fluctuations in mean field stable matchings

TL;DR

This paper analyzes the probabilistic properties of stable matchings in a bipartite setting with i.i.d. costs, revealing how the total matching cost fluctuates and converges under different pseudo-dimension regimes.

Contribution

It provides a detailed asymptotic analysis of stable matchings with costs following a specific density, including distributional limits, laws of large numbers, and central limit theorems.

Findings

Typical matching cost scales as n^{-1/d}

Total matching cost obeys a law of large numbers for d>1

Fluctuations of total cost follow a central limit theorem for d>2

Abstract

Two sets of objects of size $n$ are to be matched to each other based on i.i.d. costs associated to every pair of objects. Objects prefer to be matched as cheaply as possible, and a matching is said to be stable if there is no pair of objects that would prefer to match to each other rather than to their current partners. Properties of such matchings are analysed for cost distributions with a density $\rho$ satisfying $\rho(x)/(dx^{d-1})\to 1$ as $x\to 0^+$, where the number $d$ is known as the pseudo-dimension. For $d>0$, the typical matching cost is shown to be of order $n^{-1/d}$, with an explicit distributional limit. For $d>1$ the total matching cost is shown to be of order $n^{1-1/d}$, and to obey a law of large numbers. For $d>2$ the fluctuations of the total matching cost are shown to be of order $n^{1/2-1/d}$, and to obey a central limit theorem.