Truthful and Fair Mechanisms for Matroid-Rank Valuations

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