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 $\mathbb {R} ^{n}$. 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 $(X,h)$, where

• $X$ is just any set, and

• $h:[0,1]\times X\times X\to \mathbb {R}$ is a mixture function: it associates with each $\alpha \in [0,1]$ and each pair $x,y\in X\times X$ the $\alpha$-mixture of the two, $h_{\alpha }(x,y)\equiv h(\alpha ,x,y)$, such that

1. $h_{1}(x,y)=x$.
2. $h_{\alpha }(x,y)=h_{1-\alpha }(y,x)$.
3. $h_{\alpha }(h_{\beta }(x,y),y)=h_{\alpha \beta }(x,y)$.

Mixture spaces are essentially a special case of convex spaces (also called barycentric algebras), where the mixing operation is restricted to be over $[0,1]$ 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 $X$ of a vector space $(V,+,\cdot )$ over $\mathbb {R}$, with $h_{\alpha }(x,y)=\alpha x+(1-\alpha )y$ constitutes a mixture space $(X,h)$.
• Lotteries: given any finite set $X$, the set ${\mathcal {L}}(x)=\left\{p:X\to [0,1]:\sum _{x}p(x)=1\right\}$ of lotteries over $X$ constitutes a mixture space, with $h_{\alpha }(p,q)(x):=\alpha p(x)+(1-\alpha )q(x)$. Notice that this induces an "isomorphic" mixture space of CDFs over $X$, with the naturally-induced mixture function.

• Quantile functions: for any CDF $F:\mathbb {R} \to [0,1]$, define $Q_{F}:[0,1]\to \mathbb {R}$ as its quantile function. For any two CDFs $F_{1},F_{2}$ and any $\alpha \in [0,1]$, define the mixture operation $\alpha F_{1}\boxplus (1-\alpha )F_{2}$ as the CDF for the quantile function $\alpha Q_{F_{1}}+(1-\alpha )Q_{F_{2}}$. 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 $\succsim$ over $X$ when $(X,h)$ is a mixture space:

• Axiom 1 (Preference Relation): $\succsim$ is a weak order, in the sense that it is complete (for all $x,y\in X$, it's true that $x\succsim y$ or $y\succsim x$) and transitive.
• Axiom 2 (Independence): For any $x,y,z\in X$,

$x\sim y\implies h_{1/2}(x,z)\sim h_{1/2}(y,z).$

• Axiom 3 (Mixture Continuity): for any $x,y,z\in X$, the sets

$\{\alpha \in [0,1]:h_{\alpha }(x,y)\succsim z\},$

$\{\alpha \in [0,1]:h_{\alpha }(x,y)\precsim z\}$

are closed in $[0,1]$ 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 $X$.

Theorem

Theorem (Herstein & Milnor 1953): Given any mixture space $(X,h)$ and a preference relation $\succsim$ over $X$, the following are equivalent:

• $\succsim$ satisfies Axioms 1, 2, and 3.

• There exists a mixture-preserving utility function $U:X\to \mathbb {R}$ that represents $\succsim$, where "mixture-preserving" represents a form of linearity: for any $x,y\in X$ and any $\alpha \in [0,1]$,

$U(h_{\alpha }(x,y))=\alpha U(x)+(1-\alpha )U(y)$.