Online bipartite matching, where agents are known in advance but items arrive sequentially and must be irrevocably assigned, is fundamental to problems ranging from ride-sharing to online advertising. When agents belong to classes such as demographic groups or geographic regions, fairness demands equitable treatment across these groups. Recent work introduced class envy-freeness (CEF), a natural extension of the classical fair division notion: an algorithm is $α$-CEF if each class receives value at least an $α$ fraction of what it could extract from any other class's bundle. However, all known algorithms achieving constant-factor CEF guarantees attain utilitarian social welfare (total matching value) of at most $\frac{1}{2}$ times the optimum, far below the $1-\frac{1}{e} \approx 0.632$ achievable without fairness constraints. We resolve the open question of whether fairness necessitates this efficiency loss, by introducing threshold-based algorithms parameterized by $γ\in [0,1]$ that equalize allocations across classes until threshold $γ$, then maximize efficiency. For divisible matching, this yields simultaneous $(1-e^{-γ})$-CEF and $(1 - \frac{e^{γ-1}}{γ+1})$-USW guarantees; for indivisible matching, $\fracγ{2}$-CEF with the same USW. Setting $γ> 0$ produces the first algorithms beating $\frac{1}{2}$-USW while maintaining constant CEF. We complement this with a novel upper bound construction, proving no non-wasteful $α$-CEF algorithm can exceed $\frac{1 +α- e^{α-1}}{1+α}$-USW and correcting prior bounds that were vacuous for $α< 0.58$. Our upper bound nearly matches our algorithms' performance, giving the first substantive characterization of the price of fairness in online class matching.

翻译：在线二分匹配中，代理事先已知，但物品顺序到达且必须不可撤销地分配，这是从拼车到在线广告等问题的核心所在。当代理属于人口统计群体或地理区域等类别时，公平性要求在这些群体间进行平等对待。近期工作引入了类别无嫉妒性（CEF），这是经典公平分配概念的自然扩展：如果一个算法是$α$-CEF的，则每个类别获得的价值至少为它从其他类别组合中可能提取价值的$α$倍。然而，所有已知实现常数因子CEF保证的算法，其实现的社会总福利（总匹配价值）最多为最优值的$\frac{1}{2}$，远低于无公平约束时可达的$1-\frac{1}{e} \approx 0.632$。我们通过引入由$γ\in [0,1]$参数化的基于阈值的算法，解决了公平性是否必然导致效率损失的开放问题；该算法在阈值$γ$之前均衡分配，然后最大化效率。对于可分匹配，这同时保证了$(1-e^{-γ})$-CEF和$(1 - \frac{e^{γ-1}}{γ+1})$-USW；对于不可分匹配，则保证$\fracγ{2}$-CEF和相同的USW。设定$γ> 0$产生了首批在保持常数CEF的同时超越$\frac{1}{2}$-USW的算法。我们通过一个新的上界构造来补充这一结果，证明不存在非浪费的$α$-CEF算法能超过$(1 +α- e^{α-1})/(1+α)$-USW，并修正了此前对$α<0.58$无效的界限。我们的上界几乎匹配算法的性能，首次给出了在线类别匹配中公平性代价的实质性刻画。