Egalitarian Price of Fairness for Indivisible Goods

TL;DR

This paper extends the price-of-fairness framework to egalitarian welfare with indivisible goods, analyzing fairness constraints (EF1, balanced, and round-robin) and welfare maximizers (MUW and MNW). It derives precise asymptotic bounds: the price of fairness for EF1, balanced, and RR is $\Theta(n)$, while MUW and MNW exhibit unbounded price of fairness except MNW with $n=2$, where the bound lies between approximately $1.754$ and $2$. The analysis shows RR allocations inherently satisfy EF1 and Ba, and provides both lower and upper bounds that reveal how fairness constraints impact the best possible egalitarian welfare. The results highlight a notable gap between egalitarian and utilitarian perspectives and point to future work on additional properties and alternative welfare notions within fair division of indivisible goods.

Abstract

In the context of fair division, the concept of price of fairness has been introduced to quantify the loss of welfare when we have to satisfy some fairness condition. In other words, it is the price we have to pay to guarantee fairness. Various settings of fair division have been considered previously; we extend to the setting of indivisible goods by using egalitarian welfare as the welfare measure, instead of the commonly used utilitarian welfare. We provide lower and upper bounds for various fairness and efficiency conditions such as envy-freeness up to one good (EF1) and maximum Nash welfare (MNW).

Theorems & Definitions (9)

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• proof : Theorem \ref{['thm:RRupper']}
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