Rental harmony with roommates

TL;DR

This paper proves the existence of envy-free allocations in markets with indivisible goods and money, considering agents' preferences dependent on entire price vectors, and applies the results to cake-cutting and exchange economies.

Contribution

It introduces a novel existence proof for envy-free allocations allowing preferences to depend on full price vectors, extending previous models.

Findings

Envy-free allocations exist under the new model.

The theorem applies to cake-cutting problems.

The theorem guarantees equilibrium in certain exchange economies.

Abstract

We prove existence of envy-free allocations in markets with heterogenous indivisible goods and money, when a given quantity is supplied from each of the goods and agents have unit demands. We depart from most of the previous literature by allowing agents' preferences over the goods to depend on the entire vector of prices. Our proof uses Shapley's K-K-M-S theorem and Hall's marriage lemma. We then show how our theorem may be applied in two related problems: Existence of envy-free allocations in a version of the cake-cutting problem, and existence of equilibrium in an exchange economy with indivisible goods and money.