Stable Sets in Multi-Good Pillage Games are Small

Abstract

It is known that, in one – good pillage games, stable sets are finite. For m goods, it has been conjectured that the stable sets have measure zero. We introduce a class of sets, termed pseudo-indiff erence sets, which includes level sets of utility functions, quasi-indiff erence classes associated with a preference relation not given by a utility function, production possibility frontiers, and Pareto efficient sets. We establish the truth of the conjecture by proving that pseudo-indifference sets in R^p have p-dimensional measure zero. This implies that stable sets in n-agent, m-good pillage games have m(n-1) – dimensional measure zero.We then prove that each pseudo-indi fference set in R^p has Hausdor ff dimension at most p-1, a much stronger result than measure zero. Finally, we establish a stronger version of the conjecture: stable sets in n-agent, m-good pillage games have dimension at most m(n-1)-1.