Allocating Indivisible Goods to Strategic Agents: Pure Nash Equilibria and Fairness

TL;DR

This paper investigates the existence of mechanisms with pure Nash equilibria that guarantee fairness in the strategic allocation of indivisible goods, focusing on EF1 and EFX relaxations, and demonstrates positive results for specific algorithms.

Contribution

It proves that certain known algorithms produce pure Nash equilibria that ensure fairness, including EF1 and EFX, in strategic settings without monetary transfers.

Findings

Round-Robin equilibria are EF1 with true values.

Plaut and Roughgarden's algorithm yields EFX and maximin share fairness in equilibria.

Pure Nash equilibria always induce fair allocations for the studied algorithms.

Abstract

We consider the problem of fairly allocating a set of indivisible goods to a set of strategic agents with additive valuation functions. We assume no monetary transfers and, therefore, a mechanism in our setting is an algorithm that takes as input the reported -- rather than the true -- values of the agents. Our main goal is to explore whether there exist mechanisms that have pure Nash equilibria for every instance and, at the same time, provide fairness guarantees for the allocations that correspond to these equilibria. We focus on two relaxations of envy-freeness, namely envy-freeness up to one good (EF1), and envy-freeness up to any good (EFX), and we positively answer the above question. In particular, we study two algorithms that are known to produce such allocations in the non-strategic setting: Round-Robin (EF1 allocations for any number of agents) and a cut-and-choose algorithm of Plaut and Roughgarden [SIAM Journal of Discrete Mathematics, 2020] (EFX allocations for two agents). For Round-Robin we show that all of its pure Nash equilibria induce allocations that are EF1 with respect to the underlying true values, while for the algorithm of Plaut and Roughgarden we show that the corresponding allocations not only are EFX but also satisfy maximin share fairness, something that is not true for this algorithm in the non-strategic setting! Further, we show that a weaker version of the latter result holds for any mechanism for two agents that always has pure Nash equilibria which all induce EFX allocations.