Symmetric von Neumann-Morgenstern stable sets in pure exchange economies

We show that in a (finite) neoclassical pure exchange economy (with a weaker monotonicity condition than the usual one): if the number of traders of each type is the same and every type has a corner on one of the commodities, then the set of all symmetric Pareto-optimal allocations in the economy (i.e., Pareto-optimal allocations which assign indifferent commodity bundles to traders of the same type) is a von Neumann-Morgenstern stable set. Moreover, this is the unique stable set of symmetric allocations. A similar result holds in atomless economies with a finite number of types without the convexity assumption on the preference relations. It is also not necessary to assume (in the atomless case) that the members of the types' partition have the same measure.