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), it is known 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.