Truthful and Fair Mechanisms for Matroid-Rank Valuations

TL;DR

This paper investigates fair division mechanisms for indivisible goods with strategic agents having matroid-rank valuations, establishing positive results for envy-freeness up to one good and negative results for maximin share fairness.

Contribution

It proves group strategy-proofness of a Pareto-efficient mechanism for EF1 and shows the non-existence of truthful, Pareto-efficient, MMS mechanisms for matroid-rank valuations.

Findings

Group strategy-proofness for EF1 allocations under matroid-rank valuations.

Impossibility of truthful, Pareto-efficient, MMS mechanisms for matroid-rank valuations.

Characterization of truthful mechanisms for matroid-rank and binary XOS valuations.

Abstract

We study the problem of allocating indivisible goods among strategic agents. We focus on settings wherein monetary transfers are not available and each agent's private valuation is a submodular function with binary marginals, i.e., the agents' valuations are matroid-rank functions. In this setup, we establish a notable dichotomy between two of the most well-studied fairness notions in discrete fair division; specifically, between envy-freeness up to one good (EF1) and maximin shares (MMS). First, we show that a Pareto-efficient mechanism of Babaioff et al. (2021) is group strategy-proof for finding EF1 allocations, under matroid-rank valuations. The group strategy-proofness guarantee strengthens the result of Babaioff et al. (2021), that establishes truthfulness (individually for each agent) in the same context. Our result also generalizes a work of Halpern et al. (2020), from binary additive valuations to the matroid-rank case. Next, we establish that an analogous positive result cannot be achieved for MMS, even when considering truthfulness on an individual level. Specifically, we prove that, for matroid-rank valuations, there does not exist a truthful mechanism that is index oblivious, Pareto efficient, and maximin fair. For establishing our results, we develop a characterization of truthful mechanisms for matroid-rank functions. This characterization in fact holds for a broader class of valuations (specifically, holds for binary XOS functions) and might be of independent interest.