Mixture-space theorem

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 ${\displaystyle (X,h)}$, where

• ${\displaystyle X}$ is just any set, and

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

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

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

• Quantile functions: for any CDF ${\displaystyle F:\mathbb {R} \to [0,1]}$, define ${\displaystyle Q_{F}:[0,1]\to \mathbb {R} }$ as its quantile function. For any two CDFs ${\displaystyle F_{1},F_{2}}$ and any ${\displaystyle \alpha \in [0,1]}$, define the mixture operation ${\displaystyle \alpha F_{1}\boxplus (1-\alpha )F_{2}}$ as the CDF for the quantile function ${\displaystyle \alpha Q_{F_{1}}+(1-\alpha )Q_{F_{2}}}$. This does not define a mixture over CDFs, but it does define a mixture over quantile functions.

Axioms

Herstein and Milnor proposed the following axioms for preferences ${\displaystyle \succsim }$ over ${\displaystyle X}$ when ${\displaystyle (X,h)}$ is a mixture space:

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

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

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

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

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

are closed in ${\displaystyle [0,1]}$ 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 ${\displaystyle X}$.

Theorem

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

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

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

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