On the Existence of Pareto Efficient and Envy Free Allocations

TL;DR

This paper proves the existence of ex ante mixed allocations that are both envy-free and Pareto efficient for any number of items and players under certain conditions, advancing welfare economics understanding.

Contribution

It introduces a new existence result for envy-free and Pareto efficient allocations in the ex ante setting with mixed allocations, under swap-closure conditions.

Findings

Existence of envy-free and Pareto efficient ex ante allocations proven for any number of items and players.

The problem remains open for divisible items.

Analyzes communication complexity for finding such allocations.

Abstract

Envy-freeness and Pareto Efficiency are two major goals in welfare economics. The existence of an allocation that satisfies both conditions has been studied for a long time. Whether items are indivisible or divisible, it is impossible to achieve envy-freeness and Pareto Efficiency ex post even in the case of two people and two items. In contrast, in this work, we prove that, for any cardinal utility functions (including complementary utilities for example) and for any number of items and players, there always exists an ex ante mixed allocation which is envy-free and Pareto Efficient, assuming the allowable assignments are closed under swaps, i.e. if given a legal assignment, swapping any two players' allocations produces another legal assignment. The problem remains open in the divisible case. We also investigate the communication complexity for finding a Pareto Efficient and envy-free allocation.