We investigate the efficiency of fair allocations of indivisible goods using the well-studied price of fairness concept. Previous work has focused on classical fairness notions such as envy-freeness, proportionality, and equitability. However, these notions cannot always be satisfied for indivisible goods, leading to certain instances being ignored in the analysis. In this paper, we focus instead on notions with guaranteed existence, including envy-freeness up to one good (EF1), balancedness, maximum Nash welfare (MNW), and leximin. We also introduce the concept of strong price of fairness, which captures the efficiency loss in the worst fair allocation as opposed to that in the best fair allocation as in the price of fairness. We mostly provide tight or asymptotically tight bounds on the worst-case efficiency loss for allocations satisfying these notions, for both the price of fairness and the strong price of fairness.

我们使用经过精心研究的公平概念价格研究不可分割商品的公平分配效率。先前的工作集中在经典的公平概念上，例如嫉妒自由，相称性和公平性。但是，对于不可分割的商品，这些概念可能无法始终满足，从而导致某些实例在分析中被忽略。在本文中，我们将重点放在具有保证存在的概念上，包括至多一种产品的嫉妒自由（EF1），平衡性，最大纳什福利（MNW）和Leximin。我们还介绍了公平的价格坚挺的概念，它记录了最差公平分配中的效率损失，而不是公平价格中最佳公平分配中的效率损失。对于公平性的价格和公平性的强价格，对于满足这些概念的分配，我们大多为最坏情况的效率损失提供了严格的或渐近的严格的界限。