Optimal Subsidy Bounds for Goods and Chores: One Dollar Each Suffices

We study the fair allocation of $m$ indivisible items to $n$ agents with additive utilities. In our setting, each indivisible item may be a good, yielding non-negative utility to some agents, or a chore, yielding negative utility to others. Whilst envy-free allocations may not exist in the indivisible-items setting, envy-freeness can be achieved if some amount of divisible good (i.e., \emph{money}) is introduced. When each item's utility or disutility is bounded by one, we show that a subsidy of at most one dollar per agent suffices to guarantee the existence of an envy-free allocation, and that this bound is tight. Moreover, such an allocation can be computed in polynomial time. Since at least one agent need not receive any subsidy, our results imply that a total subsidy of at most $n-1$ dollars suffices to ensure envy-freeness.