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

Truthful and Fair Mechanisms for Matroid-Rank Valuations