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

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.