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