Strategy-proof and Envy-free Mechanisms for House Allocation

TL;DR

This paper investigates the trade-offs between efficiency, fairness, and incentives in house allocation, showing inherent limitations of strategy-proof and envy-free mechanisms and introducing a new class called Pairwise Exchange mechanisms.

Contribution

It introduces a weaker efficiency notion, proves the incompatibility with strategy-proofness and envy-freeness, and characterizes strategy-proof mechanisms within the new Pairwise Exchange class.

Findings

Strategy-proof and envy-free mechanisms cannot be fully efficient.

Within Pairwise Exchange mechanisms, strategy-proofness and envy-freeness are equivalent.

All such mechanisms have a simple linear representation.

Abstract

We consider the problem of allocating indivisible objects to agents when agents have strict preferences over objects. There are inherent trade-offs between competing notions of efficiency, fairness and incentives in assignment mechanisms. It is, therefore, natural to consider mechanisms that satisfy two of these three properties in their strongest notions, while trying to improve on the third dimension. In this paper, we are motivated by the following question: Is there a strategy-proof and envy-free random assignment mechanism more efficient than equal division? Our contributions in this paper are twofold. First, we further explore the incompatibility between efficiency and envy-freeness in the class of strategy-proof mechanisms. We define a new notion of efficiency that is weaker than ex-post efficiency and prove that any strategy-proof and envy-free mechanism must sacrifice efficiency even in this very weak sense. Next, we introduce a new family of mechanisms called Pairwise Exchange mechanisms and make the surprising observation that strategy-proofness is equivalent to envy-freeness within this class. We characterize the set of all neutral and strategy-proof (and hence, also envy-free) mechanisms in this family and show that they admit a very simple linear representation.