Fair Division through Information Withholding

Envy-freeness up to one good (EF1) is a well-studied fairness notion for indivisible goods that addresses pairwise envy by the removal of at most one good. In the worst case, each pair of agents might require the (hypothetical) removal of a different good, resulting in a weak aggregate guarantee. We study allocations that are nearly envy-free in aggregate, and define a novel fairness notion based on information withholding. Under our notion, an agent can withhold (or hide) some of the goods in its bundle and reveal the remaining goods to the other agents. We observe that in practice, envy-freeness can be achieved by withholding only a small number of goods overall. We show that finding allocations that withhold an optimal number of goods is computationally hard even for highly restricted classes of valuations. On our way, we show that for binary valuations, finding an envy-free allocation is NP-complete---somewhat surprisingly, this fundamental question was unresolved prior to our work. In contrast to the worst-case results, our experiments on synthetic and real-world preference data show that existing algorithms for finding EF1 allocations withhold close-to-optimal amount of information.