AI 中文总结
该研究聚焦带可转移支付的多对一匹配市场，提出最大边轮算法等方法，推导了公平性与稳定性的权衡边界，还扩展了融资公式至更强公平性与容量受限场景。
AI 中文摘要
我们研究了带有可转移支付的多对一匹配市场中企业侧公平性与联盟稳定性之间的权衡。对于固定匹配X，我们通过瓶颈融资问题刻画了最大可支持的核心因子：α(X)=1/Φ(X)，其中Φ(X)=min_{z≥0}max_i R_i(X,z)。这为单个工人的重新分配提供了多项式时间线性规划和局部敏感度公式。随后，我们提出了最大边轮算法和更广泛的互斥顶层安全选择类。每一次安全执行均满足EF1公平性，且当t=δ(A)表示最小正边质量时，可保证α(X)≥max{t,1/[m-(m-1)t]}，同时社会福利SW(X)与最优值OPT的比值≥t+(1-t)/m。这些边界为有限数量企业的EF1-核心极小极大前沿提供了上下界，在两家企业时可得到精确结果，当δ≤1/2时三家企业也可得到精确结果；随着企业数量增加，紧的无标度稳定率为δ。我们还将该融资公式扩展至更强的EFX⁺公平性及容量受限市场。
英文摘要
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: $α(X)=1/Φ(X)$, where $Φ(X)=\min_{z\ge0}\max_i R_i(X,z)$. 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 $t=δ(A)$ denoting the minimum positive-edge quality, guarantees $α(X)\ge\max\{t,1/[m-(m-1)t]\}$ 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 $δ\le1/2$; as the number of firms grows, the tight scale-free stability rate is $δ$. We also extend the financing formulation to stronger $EFX^+$ fairness and capacity-constrained markets.
