Mixture-space theorem

In microeconomic theory and decision theory, the Mixture-space theorem is a utility-representation theorem for preferences defined over general mixture spaces.
The theorem generalizes the von Neumann–Morgenstern utility theorem and the usual utility-representation theorem for consumer preferences over. It was first proven by Israel Nathan Herstein and John Milnor in 1953, together with the introduction of the definition of a mixture space.

Mixture spaces

Definition

Mixture spaces, as introduced by Herstein and Milnor, are a generalization of convex sets from vector spaces. Formally:
Definition: A mixture space is a pair, where

is just any set, and
is a mixture function: it associates with each and each pair the -mixture of the two,, such that

Mixture spaces are essentially a special case of convex spaces, where the mixing operation is restricted to be over and not just an appropriately closed subset of a semiring.

Examples

Some examples and non-examples of mixture spaces are:

Vector spaces: any convex subset of a vector space over, with constitutes a mixture space.
Lotteries: given any finite set, the set of lotteries over constitutes a mixture space, with. Notice that this induces an "isomorphic" mixture space of CDFs over, with the naturally-induced mixture function.
Quantile functions: for any CDF, define as its quantile function. For any two CDFs and any, define the mixture operation as the CDF for the quantile function. This does not define a mixture over CDFs, but it does define a mixture over quantile functions.

Axioms and theorem

Axioms

Herstein and Milnor proposed the following axioms for preferences over when is a mixture space:Axiom 1 : is a weak order, in the sense that it is complete and transitive. Axiom 2 : For any,Axiom 3 : for any, the sets
are closed in with the usual topology.
The Mixture-Continuity Axiom is a way of introducing some form of continuity for the preferences without having to consider a topological structure over.

Theorem

Theorem : Given any mixture space and a preference relation over, the following are equivalent:

satisfies Axioms 1, 2, and 3.
There exists a mixture-preserving utility function that represents, where "mixture-preserving" represents a form of linearity: for any and any,