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Optimal Bounds on the Price of Fairness for Indivisible Goods
arXiv - CS - Computer Science and Game Theory Pub Date : 2020-07-13 , DOI: arxiv-2007.06242
Siddharth Barman, Umang Bhaskar, Nisarg Shah

In the allocation of resources to a set of agents, how do fairness guarantees impact the social welfare? A quantitative measure of this impact is the price of fairness, which measures the worst-case loss of social welfare due to fairness constraints. While initially studied for divisible goods, recent work on the price of fairness also studies the setting of indivisible goods. In this paper, we resolve the price of two well-studied fairness notions for the allocation of indivisible goods: envy-freeness up to one good (EF1), and approximate maximin share (MMS). For both EF1 and 1/2-MMS guarantees, we show, via different techniques, that the price of fairness is $O(\sqrt{n})$, where $n$ is the number of agents. From previous work, it follows that our bounds are tight. Our bounds are obtained via efficient algorithms. For 1/2-MMS, our bound holds for additive valuations, whereas for EF1, our bound holds for the more general class of subadditive valuations. This resolves an open problem posed by Bei et al. (2019).

中文翻译:

不可分割商品公平价格的最优边界

在向一组代理分配资源时,公平保证如何影响社会福利?这种影响的量化衡量标准是公平的代价,它衡量了由于公平约束而造成的最坏情况下的社会福利损失。虽然最初研究的是可分割的商品,但最近关于公平价格的工作也研究了不可分割的商品的设置。在本文中,我们解决了用于分配不可分割商品的两个经过充分研究的公平概念的价格:最多一种商品的嫉妒(EF1)和近似最大份额(MMS)。对于 EF1 和 1/2-MMS 保证,我们通过不同的技术表明公平的代价是 $O(\sqrt{n})$,其中 $n$ 是代理的数量。从以前的工作来看,我们的界限很紧。我们的边界是通过有效的算法获得的。对于 1/2-彩信,我们的界限适用于加法估值,而对于 EF1,我们的界限适用于更一般的次加法估值。这解决了贝等人提出的一个开放问题。(2019)。
更新日期:2020-11-03
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