Fair and Almost Truthful Mechanisms for Additive Valuations and Beyond

TL;DR

This work investigates the incentive ratio as a relaxation of truthfulness in fair division of indivisible goods across several valuation classes. It establishes a constant-gap lower bound of $1.5$ for additive valuations under $(\tfrac{1}{2}+\varepsilon)$-EF1 and proves that in the cancelable setting, any $\varepsilon$-EF1 mechanism can have infinite incentive ratio, with a robust reduction even applying to multiplicative valuations. For subadditive cancelable valuations, Round-Robin achieves an incentive ratio of $2$, while a $(\varphi-1)$-EF1 guarantee yields a lower bound of $\varphi \approx 1.618$, and for submodular valuations, a generalized Round-Robin attains an incentive ratio of $n$, with matching upper and lower bounds. Collectively, the results delineate how stronger fairness constraints interact with strategic behavior across valuation classes and identify tight or near-tight boundaries, highlighting when traditional fair mechanisms retain meaningful incentive guarantees and when they do not. The findings advance understanding of fairness in strategic settings and point to future work on broader valuation classes and randomized mechanisms.

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 $ε> 0$, every $(\frac{1}{2} + ε)$-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 $ε> 0$, every $ε$-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 $n$.

Theorems & Definitions (26)

• Definition 2.1: $\alpha$-EF1
• Definition 2.2: Incentive Ratio
• Definition 2.3: $\alpha$-Control
• Lemma 2.4
• proof
• Theorem 3.1
• Theorem 4.1
• Lemma 4.2
• proof
• Lemma 4.3