Round-Robin Beyond Additive Agents: Existence and Fairness of Approximate Equilibria

TL;DR

This paper extends the analysis of the Round-Robin mechanism's equilibria and fairness guarantees beyond additive agents, showing it works for cancelable and submodular valuation functions, including approximate fairness in complex settings.

Contribution

It proves the existence of equilibria and fairness guarantees for the Round-Robin mechanism with broader valuation classes, beyond additive functions.

Findings

Round-Robin has equilibria for cancelable valuations.

Approximate equilibria yield approximately EF1 allocations.

Fairness guarantees extend to submodular valuations, despite non-existence of exact equilibria.

Abstract

Fair allocation of indivisible goods has attracted extensive attention over the last two decades, yielding numerous elegant algorithmic results and producing challenging open questions. The problem becomes much harder in the presence of strategic agents. Ideally, one would want to design truthful mechanisms that produce allocations with fairness guarantees. However, in the standard setting without monetary transfers, it is generally impossible to have truthful mechanisms that provide non-trivial fairness guarantees. Recently, Amanatidis et al. [2021] suggested the study of mechanisms that produce fair allocations in their equilibria. Specifically, when the agents have additive valuation functions, the simple Round-Robin algorithm always has pure Nash equilibria and the corresponding allocations are envy-free up to one good (EF1) with respect to the agents' true valuation functions. Following this agenda, we show that this outstanding property of the Round-Robin mechanism extends much beyond the above default assumption of additivity. In particular, we prove that for agents with cancelable valuation functions (a natural class that contains, e.g., additive and budget-additive functions), this simple mechanism always has equilibria and even its approximate equilibria correspond to approximately EF1 allocations with respect to the agents' true valuation functions. Further, we show that the approximate EF1 fairness of approximate equilibria surprisingly holds for the important class of submodular valuation functions as well, even though exact equilibria fail to exist!