In fair division problems, the notion of price of fairness measures the loss in welfare due to a fairness constraint. Prior work on the price of fairness has focused primarily on envy-freeness up to one good (EF1) as the fairness constraint, and on the utilitarian and egalitarian welfare measures. Our work instead focuses on the price of equitability up to one good (EQ1) (which we term price of equity) and considers the broad class of generalized $p$-mean welfare measures (which includes utilitarian, egalitarian, and Nash welfare as special cases). We derive fine-grained bounds on the price of equity in terms of the number of agent types (i.e., the maximum number of agents with distinct valuations), which allows us to identify scenarios where the existing bounds in terms of the number of agents are overly pessimistic. Our work focuses on the setting with binary additive valuations, and obtains upper and lower bounds on the price of equity for $p$-mean welfare for all $p \leqslant 1$. For any fixed $p$, our bounds are tight up to constant factors. A useful insight of our work is to identify the structure of allocations that underlie the upper (respectively, the lower) bounds simultaneously for all $p$-mean welfare measures, thus providing a unified structural understanding of price of fairness in this setting. This structural understanding, in fact, extends to the more general class of binary submodular (or matroid rank) valuations. We also show that, unlike binary additive valuations, for binary submodular valuations the number of agent types does not provide bounds on the price of equity.

翻译：在公平分配问题中，公平性代价的概念衡量了因公平约束所导致的福利损失。以往关于公平性代价的研究主要关注至多一个物品的无嫉妒性（EF1）作为公平约束，并侧重于功利主义和平等主义福利度量。而我们的工作则聚焦于至多一个物品的公平性（EQ1）的代价（我们称之为公平性代价），并考虑广义的$p$-均值福利度量（其中包括功利主义、平等主义和纳什福利作为特例）。我们根据代理类型数量（即具有不同估值的代理的最大数量）推导出公平性代价的精细边界，这使我们能够识别出以代理数量表示的现有边界过于悲观的情形。本文研究了具有二元可加估值的设置，并为所有$p \leqslant 1$的$p$-均值福利得到了公平性代价的上界和下界。对于任意固定的$p$，我们的边界在常数因子内是紧的。一个重要的研究见解是，我们识别出同时支撑所有$p$-均值福利度量上界（分别对应下界）的分配结构，从而为此设置下的公平性代价提供了统一的结构性理解。这种结构性理解实际上可推广至更一般的二元子模（或拟阵秩）估值。我们还表明，与二元可加估值不同，对于二元子模估值，代理类型数量并不提供公平性代价的边界。