Flexibility allocation in random bipartite matching markets: exact matching rates and dominance regimes

TL;DR

This paper develops an exact formula for optimal flexibility allocation in bipartite matching markets, revealing when one-sided or two-sided flexibility strategies are superior, based on a probabilistic model.

Contribution

It extends the local weak convergence framework to multi-type models, providing an explicit optimization formula for asymptotic matching rates.

Findings

One-sided flexibility can outperform two-sided in certain regimes.

Exact variational formula derived for asymptotic matching rate.

Analytical comparison extends previous approximate bounds.

Abstract

This paper studies how a fixed flexibility budget should be allocated across the two sides of a balanced bipartite matching market. We model compatibilities via a sparse bipartite stochastic block model in which flexible agents are more likely to connect with agents on the opposite side, and derive an exact variational formula for the asymptotic matching rate under any flexibility allocation. The derivation extends the local weak convergence framework of [BLS11] from single-type to multi-type unimodular Galton-Watson trees, reducing the matching rate to an explicit low-dimensional optimization problem. Using this formula, we analytically investigate when the one-sided allocation, which concentrates all flexibility on one side, dominates the two-sided allocation and vice versa, sharpening and extending the comparisons of [FMZ26] which relied on approximate algorithmic bounds rather than an exact characterization of the matching rate.