Fair and Almost Truthful Mechanisms for Additive Valuations and Beyond

TL;DR

This paper investigates the limits of fair mechanisms in allocating indivisible goods among strategic agents, focusing on incentive ratios for various valuation classes and extending beyond additive valuations.

Contribution

It establishes new lower bounds on incentive ratios for different fairness notions and valuation classes, and analyzes the performance of Round-Robin in these contexts.

Findings

Round-Robin achieves an incentive ratio of 2 for subadditive cancelable valuations.

Any (rac{1}{2} + ext{epsilon})-EF1 mechanism for additive valuations has an incentive ratio of at least 1.5.

For submodular valuations, Round-Robin's incentive ratio is n.

Abstract

We study the problem of fairly allocating indivisible goods among $n$ strategic agents. It is well-known that truthfulness is incompatible with any meaningful fairness notions. We bypass the strong negative result by considering the concept of incentive ratio, a relaxation of truthfulness quantifying agents' incentive to misreport. Previous studies show that Round-Robin, which satisfies envy-freeness up to one good (EF1), achieves an incentive ratio of $2$ for additive valuations. In this paper, we explore the incentive ratio achievable by fair mechanisms for various classes of valuations besides additive ones. We first show that, for arbitrary $\epsilon > 0$, every $(\frac{1}{2} + \epsilon)$-EF1 mechanism for additive valuations admits an incentive ratio of at least $1.5$. Then, using the above lower bound for additive valuations in a black-box manner, we show that for arbitrary $\epsilon > 0$, every $\epsilon$-EF1 mechanism for cancelable valuations admits an infinite incentive ratio. Moreover, for subadditive cancelable valuations, we show that Round-Robin, which satisfies EF1, achieves an incentive ratio of $2$, and every $(\varphi - 1)$-EF1 mechanism admits an incentive ratio of at least $\varphi$ with $\varphi = (1 + \sqrt{5}) / 2 \approx 1.618$. Finally, for submodular valuations, we show that Round-Robin, which satisfies $\frac{1}{2}$-EF1, admits an incentive ratio of $n$.