On expected utility theorems on mixture sets

Abstract This note discusses expected utility theorems on mixture sets by Herstein and Milnor (1953) and Fishburn (1970). We show that the Archimedean axiom in Fishburn’s expected utility theorem can be weakened to a one-sided Archimedean axiom. We also show that the mixture continuity and transitivity axioms in Herstein–Milnor expected utility theorem can be further weakened under the strong independence axiom.