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 (also called barycentric algebras), 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 (Preference Relation): is a weak order, in the sense that it is complete (for all , it's true that or ) and transitive. Axiom 2 (Independence): For any ,

Axiom 3 (Mixture Continuity): 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 (Herstein & Milnor 1953): 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 ,

.

Notes

References