  We consider the problem of fairly allocating a set of indivisible goods to a
set of strategic agents with additive valuation functions. We assume no
monetary transfers and, therefore, a mechanism in our setting is an algorithm
that takes as input the reported -- rather than the true -- values of the
agents. Our main goal is to explore whether there exist mechanisms that have
pure Nash equilibria for every instance and, at the same time, provide fairness
guarantees for the allocations that correspond to these equilibria. We focus on
two relaxations of envy-freeness, namely envy-freeness up to one good (EF1),
and envy-freeness up to any good (EFX), and we positively answer the above
question. In particular, we study two algorithms that are known to produce such
allocations in the non-strategic setting: Round-Robin (EF1 allocations for any
number of agents) and a cut-and-choose algorithm of Plaut and Roughgarden [SIAM
Journal of Discrete Mathematics, 2020] (EFX allocations for two agents). For
Round-Robin we show that all of its pure Nash equilibria induce allocations
that are EF1 with respect to the underlying true values, while for the
algorithm of Plaut and Roughgarden we show that the corresponding allocations
not only are EFX but also satisfy maximin share fairness, something that is not
true for this algorithm in the non-strategic setting! Further, we show that a
weaker version of the latter result holds for any mechanism for two agents that
always has pure Nash equilibria which all induce EFX allocations.
