  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.
