The Landscape of Almost Equitable Allocations

Equitability is a fundamental notion in fair division which requires that all agents derive equal value from their allocated bundles. We study, for general (possibly non-monotone) valuations, a popular relaxation of equitability known as equitability up to one item (EQ1). An EQ1 allocation may fail to exist even with additive non-monotone valuations; for instance, when there are two agents, one valuing every item positively and the other negatively. This motivates a mild and natural assumption: all agents agree on the sign of their value for the grand bundle. Under this assumption, we prove the existence and provide an efficient algorithm for computing EQ1 allocations for two agents with general valuations. When there are more than two agents, we show the existence and polynomial-time computability of EQ1 allocations for valuation classes beyond additivity and monotonicity, in particular for (1) doubly monotone valuations and (2) submodular (resp. supermodular ) valuations where the value for the grand bundle is non-negative (resp. non-positive ) for all agents. Furthermore, we settle an open question of Bilò et al. by showing that an EQ1 allocation always exists for non-negative (resp. non-positive) valuations, i.e., when every agent values each subset of items non-negatively (resp. non-positively). Finally, we complete the picture by showing that for general valuations with more than two agents, EQ1 allocations may not exist even when agents agree on the sign of the grand bundle, and that deciding the existence of an EQ1 allocation is computationally intractable.