Expected utility theory on mixture spaces without the completeness axiom

A mixture preorder is a on space (such as convex set) that compatible with the mixing operation. In decision theoretic terms, it satisfies central expected utility axiom of strong independence. We consider when has multi-representation consists real-valued, mixture-preserving functions. If does, must satisfy continuity Herstein and Milnor (1953). Mixture sufficient for dimension countable, but not uncountable. Our strongest positive result in conjunction novel we call countable domination, which constrains order complexity terms its Archimedean structure. also what happens given natural weak topology. Continuity (having closed upper lower sets) closedness graph) are stronger than continuity. show necessary to have multi-representation. Closedness necessary; leave an open question whether sufficient. end results concerning existence multi-representations consist entirely strictly increasing functions, uniqueness result.