Achieving Envy-Freeness with Limited Subsidies under Dichotomous Valuations

Abstract

We study the problem of allocating indivisible goods among agents in a fair manner. While envy-free allocations of indivisible goods are not guaranteed to exist, envy-freeness can be achieved by additionally providing some subsidy to the agents. These subsidies can be alternatively viewed as a divisible good (money) that is fractionally assigned among the agents to realize an envy-free outcome. In this setup, we bound the subsidy required to attain envy-freeness among agents with dichotomous valuations, i.e., among agents whose marginal value for any good is either zero or one. We prove that, under dichotomous valuations, there exists an allocation that achieves envy-freeness with a per-agent subsidy of either 0 or 1. Furthermore, such an envy-free solution can be computed efficiently in the standard value-oracle model. Notably, our results hold for general dichotomous valuations and, in particular, do not require the (dichotomous) valuations to be additive, submodular, or even subadditive. Also, our subsidy bounds are tight and provide a linear (in the number of agents) factor improvement over the bounds known for general monotone valuations.

Mixed Fair Division: A Survey

“…Goko et al (2022) showed that when the utilities are submodular functions with binary marginals, a total subsidy payment of n − 1 suffices. Subsequently, Barman et al (2022) showed that for general set valuations with binary marginals total subsidy payment of n − 1 suffices.…”

The budgeted maximin share allocation problem

“…The budgeted maximin share allocation problem is also closely related to relaxed envy free allocation with budget constraints [8,9,19,21,30] . Other related results can be found in the recent surveys [4,23].…”

“…Since RG 0 ⊇ RG 1 ⊇ • • • ⊇ RG, by (9), there must exist an iteration τ , such that v ′ i (A(RG t , i)) ≥ β for each t ∈ {0, 1, . .…”