One Dollar Each Eliminates Envy

TL;DR

This paper proves that a small, fixed subsidy per agent guarantees envy-free allocations in fair division problems with indivisible goods, improving previous bounds and extending results to more general valuation functions.

Contribution

It confirms that a subsidy of at most one dollar per agent suffices for additive valuations and establishes a bound of 2(n-1) dollars for general monotonic valuations, advancing fair division theory.

Findings

A subsidy of at most one dollar per agent guarantees envy-freeness for additive valuations.

For general monotonic valuations, a subsidy of at most 2(n-1) dollars per agent suffices.

The total subsidy needed for monotonic valuations does not depend on the number of items.

Abstract

We study the fair division of a collection of $m$ indivisible goods amongst a set of $n$ agents. Whilst envy-free allocations typically do not exist in the indivisible goods setting, envy-freeness can be achieved if some amount of a divisible good (money) is introduced. Specifically, Halpern and Shah (SAGT 2019, pp.374-389) showed that, given additive valuation functions where the marginal value of each item is at most one dollar for each agent, there always exists an envy-free allocation requiring a subsidy of at most $(n-1)\cdot m$ dollars. The authors also conjectured that a subsidy of $n-1$ dollars is sufficient for additive valuations. We prove this conjecture. In fact, a subsidy of at most one dollar per agent is sufficient to guarantee the existence of an envy-free allocation. Further, we prove that for general monotonic valuation functions an envy-free allocation always exists with a subsidy of at most $2(n-1)$ dollars per agent. In particular, the total subsidy required for monotonic valuations is independent of the number of items.