  The goal of fair division is to distribute resources among competing players
in a "fair" way. Envy-freeness is the most extensively studied fairness notion
in fair division. Envy-free allocations do not always exist with indivisible
goods, motivating the study of relaxed versions of envy-freeness. We study the
envy-freeness up to any good (EFX) property, which states that no player
prefers the bundle of another player following the removal of any single good,
and prove the first general results about this property. We use the leximin
solution to show existence of EFX allocations in several contexts, sometimes in
conjunction with Pareto optimality. For two players with valuations obeying a
mild assumption, one of these results provides stronger guarantees than the
currently deployed algorithm on Spliddit, a popular fair division website.
Unfortunately, finding the leximin solution can require exponential time. We
show that this is necessary by proving an exponential lower bound on the number
of value queries needed to identify an EFX allocation, even for two players
with identical valuations. We consider both additive and more general
valuations, and our work suggests that there is a rich landscape of problems to
explore in the fair division of indivisible goods with different classes of
player valuations.
