Vector Representation of Preferences on $\sigma$-Algebras and Fair Division in Saturated Measure Spaces

TL;DR

This paper develops a vector measure-based utility framework for preferences on sigma-algebras in saturated measure spaces and applies it to establish fair division solutions like Pareto optimal and envy-free partitions.

Contribution

It axiomatizes preferences via vector measures and proves the existence of fair division solutions in saturated measure spaces with nonadditive preferences.

Findings

Existence of Pareto optimal partitions

Existence of Walrasian equilibria

Existence of envy-free partitions

Abstract

The purpose of this paper is twofold. First, we axiomatize preference relations on a $\sigma$-algebra of a saturated measure space represented by a vector measure and furnish a utility representation in terms of a nonadditive measure satisfying the appropriate requirement of continuity and convexity. Second, we investigate the fair division problems in which each individual has nonadditive preferences on a $\sigma$-algebra invoking our utility representation result. We show the existence of individually rational Pareto optimal partitions, Walrasian equilibria, core partitions, and Pareto optimal envy-free partitions.