Vector Representation of Preferences on σ-Algebras and Fair Division in Saturated Measure Spaces

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.