We study the efficiency of fair allocations using the well-studied price of fairness concept, which quantitatively measures the worst-case efficiency loss when imposing fairness constraints. Previous works provided partial results on the price of fairness with well-known fairness notions such as envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX). In this paper, we give a complete characterization for the price of envy-freeness in various settings. In particular, we first consider the two-agent case under the indivisible-goods setting and present tight ratios for the price of EF1 (for scaled utility) and EFX (for unscaled utility), which resolve questions left open in the literature. Next, we consider the mixed goods setting which concerns a mixture of both divisible and indivisible goods. We focus on envy-freeness for mixed goods (EFM), which generalizes both envy-freeness and EF1, as well as its strengthening called envy-freeness up to any good for mixed goods (EFXM), which generalizes envy-freeness and EFX. To this end, we settle the price of EFM and EFXM by providing a complete picture of tight bounds for two agents and asymptotically tight bounds for $n$ agents, for both scaled and unscaled utilities.

翻译：我们利用广受研究的公平价格概念来研究公平分配效率，该概念通过量化施加公平约束时的最坏情况效率损失进行评估。以往针对诸如"至多一种物品无嫉妒"(EF1)和"任意物品无嫉妒"(EFX)等经典公平性概念的研究，仅提供了公平价格的部分结果。本文首次完整刻画了多种情境下无嫉妒公平的价格特征：首先考虑不可分物品环境中的双智能体情形，针对EF1(规模化效用)与EFX(非规模化效用)给出其价格紧界，解决了文献中悬而未决的问题；其次研究包含可分与不可分物品的混合物品环境，聚焦于混合物品无嫉妒(EFM)及其强化版本——混合物品任意项无嫉妒(EFXM)——前者统一了无嫉妒与EF1概念，后者统一了无嫉妒与EFX概念。为此，我们给出双智能体环境下EFM与EFX价格的完整紧界刻画，并为n智能体场景（分别针对规模化与非规模化效用）提供渐近紧界，从而完整确立了这两类公平性概念的价格特征。