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arXiv · 2608.17295

Fairness--Stability Trade-offs in Many-to-One Matching

Genjie Qin
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Abstract

We study the trade-off between firm-side fairness and coalition stability in many-to-one matching markets with transferable payments. For a fixed matching $X$, we characterize the largest supportable core factor by a bottleneck financing problem: $\alpha(X)=1/\Phi(X)$, where $\Phi(X)=\min_{z\ge0}\max_i R_i(X,z)$. This yields a polynomial-time linear program and local sensitivity formulas for one-worker reallocations. We then develop a maximum-edge round algorithm and a broader class of mutual-top safe choices. Every safe execution is EF1 and, with $t=\delta(A)$ denoting the minimum positive-edge quality, guarantees $\alpha(X)\ge\max\{t,1/[m-(m-1)t]\}$ and $SW(X)/OPT\geq t+(1-t)/m$. These bounds give finite-firm lower and upper bounds for the EF1--core minimax frontier, with exact results for two firms and for three firms when $\delta\le1/2$; as the number of firms grows, the tight scale-free stability rate is $\delta$. We also extend the financing formulation to stronger $EFX^+$ fairness and capacity-constrained markets.

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Genjie Qin. 2026-08-18. Fairness--Stability Trade-offs in Many-to-One Matching. https://arxiv.org/abs/2608.17295

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