Efficient reallocation of indivisible resources: Pair-efficiency versus Pareto-efficiency

TL;DR

This paper proves that in resource reallocation problems with single-peaked or single-dipped preferences, pair-efficiency and Pareto-efficiency are equivalent, highlighting the significance of preference structure in efficiency outcomes.

Contribution

It establishes the equivalence between pair-efficiency and Pareto-efficiency for single-peaked and single-dipped preferences, and identifies these domains as maximal for this property.

Findings

Pair-efficiency and Pareto-efficiency are equivalent under single-peaked preferences.

The same equivalence holds for single-dipped preferences.

These domains are maximal where the equivalence is valid.

Abstract

In the object reallocation problem, achieving Pareto-efficiency is desirable, but may be too demanding for implementation purposes. In contrast, pair-efficiency, which is the minimal efficiency requirement, is more suitable. Despite being a significant relaxation, however, pair-efficiency ensures Pareto-efficiency for any strategy-proof and individually rational rule when agents' preferences are unrestricted. What if agents' preferences have specific restricted structures, such as single-peakedness or single-dippedness? We often encounter such situations in real-world scenarios. This study aims to investigate whether pair-efficiency is sufficient to ensure Pareto-efficiency in such cases. Our main contribution in this paper is establishing the equivalence between pair-efficiency and Pareto-efficiency when dealing with single-peaked or single-dipped preference profiles. This equivalence holds without needing to assume any other properties of the rule. We further show that both the single-peaked domain and the single-dipped domain are the "maximal" domains where this equivalence holds.