Towards Optimal Subsidy Bounds for Envy-Freeable Allocations

We study the fair division of indivisible items with subsidies among n agents, where the absolute marginal valuation of each item is at most one. Under monotone valuations (where each item is a good), Brustle et al. [9] demonstrated that a maximum subsidy of 2(n -1) and a total subsidy of 2(n -1) 2 are sufficient to guarantee the existence of an envy-freeable allocation. In this paper, we improve upon these bounds, even in a wider model. Namely, we show that, given an EF1 allocation, we can compute in polynomial time an envy-free allocation with a subsidy of at most n -1 per agent and a total subsidy of at most n(n -1)/2. Moreover, we present further improved bounds for monotone valuations.