Comparing Probability Measures on a Set with an Intransitive Preference Relation

Recently Fishburn (Fishburn, P. C. 1978. Stochastic dominance without transitive preferences. Management Sci. 24 1268-1277.) proposed a definition of stochastic dominance for probability measures defined on a set with an intransitive preference-or-indifference relation. A significant feature of Fishburn's definition is that the stochastic dominance relation inherits the intransitivity. We propose several alternative definitions of stochastic dominance that are transitive even though the preference relation on the underlying sample space is not. The basic idea is to use ordinary first-order stochastic dominance after appropriately transforming the intransitive relation into a transitive relation. As extreme cases or bounds, strong (weak) stochastic dominance is defined to be ordinary stochastic dominance for probability measures defined on the same set after the intransitive relation is replaced by the transitive relation obtained by regarding any two alternatives that were connected by an intransitive cycle as noncomparable (equivalent). Fishburn's definition is shown to be included within these bounds.