The Price of Fairness for Indivisible Goods

Abstract

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 envyfreeness up to one good (EF1), balancedness, maximum Nash welfare (MNW), and leximin. We mostly provide tight or asymptotically tight bounds on the worst-case efficiency loss for allocations satisfying these notions.

Optimal Bounds on the Price of Fairness for Indivisible Goods

“…First, we show that the price of EF1 is O( √ n). This matches the Ω( √ n) lower bound due to Bei et al [5]. The lower bound is for additive valuations, whereas our upper bound holds for the more general class of subadditive valuations.…”

“…For 1 /2-MMS our bound holds for additive valuations, whereas for EF1, it holds for the more general class of subadditive valuations. This resolves an open problem posed by Bei et al (2019).…”

“…Bei et al [5] instead focused on notions that can be guaranteed with indivisible goods and chores, such as envy-freeness up to one good (EF1). While they did not settle the price of EF1 (which is the focus of our work), they showed that the price of popular allocation rules such as the maximum Nash welfare rule [10], the egalitarian rule [24], and the leximin rule [7,19] is Θ(n).…”

“…For indivisible goods and additive valuations, Bei et al [5] studied the price of fairness for various notions of fairness. They showed that the price of envyfreeness up to one good (EF1) is between Ω( √ n) and O(n).…”

Truthful fair division without free disposal

“…The efficiency ratio is closely related to the price of fairness, which measures the worst-case welfare loss due to imposing fairness constraints(Bei et al 2019;Caragiannis et al 2011; Heydrich and van Stee 2015).3 Aziz and Ye (2014) call an allocation robust proportional if it satisfies the following property: even if an agent perturbs her value density function, as long as the ordinal information of the function is unchanged, then the allocation remains proportional. Robust proportionality is a stronger notion than standard proportionality.…”