# Competitive Equilibrium For Almost All Incomes: Existence and Fairness

## TL;DR

This paper completes the characterization of the existence of competitive equilibrium for almost all income vectors in resource allocation problems, proving non-existence in certain cases and existence in others, with implications for fairness.

## Contribution

It provides a complete characterization of CE existence for almost all incomes in various preference domains, including new non-existence and existence results, and introduces a novel tool using subgame-perfect equilibrium.

## Key findings

- CE does not exist for four goods and three or four agents with increasing preferences.
- CE exists for almost all incomes with four goods and three agents with additive preferences.
- CE allocations satisfy strong fairness properties.

## Abstract

Competitive equilibrium (CE) is a fundamental concept in market economics. Its efficiency and fairness properties make it particularly appealing as a rule for fair allocation of resources among agents with possibly different entitlements. However, when the resources are indivisible, a CE might not exist even when there is one resource and two agents with equal incomes. Recently, Babaioff and Nisan and Talgam-Cohen (2017) have suggested to consider the entire space of possible incomes, and check whether there exists a competitive equilibrium for almost all income-vectors --- all income-space except a subset of measure zero. They proved various existence and non-existence results, but left open the cases of four goods and three or four agents with monotonically-increasing preferences. This paper proves non-existence in both these cases, thus completing the characterization of CE existence for almost all incomes in the domain of monotonically increasing preferences. Additionally, the paper provides a complete characterization of CE existence in the domain of monotonically decreasing preferences, corresponding to allocation of chores. On the positive side, the paper proves that CE exists for almost all incomes when there are four goods and three agents with additive preferences. The proof uses a new tool for describing a CE, as a subgame-perfect equilibrium of a specific sequential game. The same tool also enables substantially simpler proofs to the cases already proved by Babaioff et al. Additionally, this paper proves several strong fairness properties that are satisfied by any CE allocation, illustrating its usefulness for fair allocation among agents with different entitlements.