Fair Division with Subsidy.

When allocating a set of goods to a set of agents, a classic fairness notion called envy-freeness requires that no agent prefer the allocation of another agent to her own. When the goods are indivisible, this notion is impossible to guarantee, and prior work has focused on its relaxations. However, envy-freeness can be achieved if a third party is willing to subsidize by providing a small amount of money (divisible good), which can be allocated along with the indivisible goods. In this paper, we study the amount of subsidy needed to achieve envy-freeness for agents with additive valuations, both for a given allocation of indivisible goods and when we can choose the allocation. In the former case, we provide a strongly polynomial time algorithm to minimize subsidy. In the latter case, we provide optimal constructive results for the special cases of binary and identical valuations, and make a conjecture in the general case. Our experiments using real data show that a small amount of subsidy is sufficient in practice.