Two Birds With One Stone: Fairness and Welfare via Transfers

TL;DR

This paper explores how transfer payments can simultaneously achieve fairness and high welfare in dividing indivisible goods, demonstrating that envy-freeness and near-optimal Nash social welfare are compatible with minimal transfers.

Contribution

It proves the existence of envy-free allocations with high Nash social welfare using transfers for general valuations and provides polynomial-time algorithms for additive valuations.

Findings

Envy-free allocations with high Nash welfare exist with minimal transfers for general valuations.

For additive valuations, such allocations can be found efficiently with negligible transfers.

Non-negligible transfers are necessary for achieving high utilitarian social welfare in envy-free allocations.

Abstract

We study the question of dividing a collection of indivisible goods amongst a set of agents. The main objective of research in the area is to achieve one of two goals: fairness or efficiency. On the fairness side, envy-freeness is the central fairness criterion in economics, but envy-free allocations typically do not exist when the goods are indivisible. A recent line of research shows that envy-freeness can be achieved if a small quantity of a homogeneous divisible good (money) is introduced into the system, or equivalently, if transfer payments are allowed between the agents. A natural question to explore, then, is whether transfer payments can be used to provide high welfare in addition to envy-freeness, and if so, how much money is needed to be transferred. We show that for general monotone valuations, there always exists an allocation with transfers that is envy-free and whose Nash social welfare (NSW) is at least an $e^{-1/e}$-fraction of the optimal Nash social welfare. Additionally, when the agents have additive valuations, an envy-free allocation with negligible transfers and whose NSW is within a constant factor of optimal can be found in polynomial time. Consequently, we demonstrate that the seemingly incompatible objectives of fairness and high welfare can be achieved simultaneously via transfer payments, even for general valuations, when the welfare objective is NSW. On the other hand, we show that a similar result is impossible for utilitarian social welfare: any envy-freeable allocation that achieves a constant fraction of the optimal welfare requires non-negligible transfers. To complement this result we present algorithms that compute an envy-free allocation with a given target welfare and with bounded transfers.