Optimal Bounds on the Price of Fairness for Indivisible Goods

TL;DR

This paper establishes tight bounds on the social welfare loss caused by fairness constraints in indivisible goods allocation, specifically for EF1 and 1/2-MMS fairness notions, using efficient algorithms.

Contribution

It provides the first tight bounds on the price of fairness for EF1 and 1/2-MMS in indivisible goods allocation, resolving an open problem.

Findings

Price of fairness is O(√n) for EF1 and 1/2-MMS.

Bounds are tight and achieved via efficient algorithms.

Results hold for subadditive valuations in EF1 and additive valuations in MMS.

Abstract

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).