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 ${\displaystyle \mathbb {R} ^{n}}${\displaystyle \mathbb {R} ^{n}}. It was first proven by Israel Nathan Herstein and John Milnor in 1953,^[1] together with the introduction of the definition of a mixture space.

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

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

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

1. ${\displaystyle h_{1}(x,y)=x}${\displaystyle h_{1}(x,y)=x}.
2. ${\displaystyle h_{\alpha }(x,y)=h_{1-\alpha }(y,x)}${\displaystyle h_{\alpha }(x,y)=h_{1-\alpha }(y,x)}.
3. ${\displaystyle h_{\alpha }(h_{\beta }(x,y),y)=h_{\alpha \beta }(x,y)}${\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),^[2] where the mixing operation is restricted to be over ${\displaystyle [0,1]}${\displaystyle [0,1]} and not just an appropriately closed subset of a semiring.

Some examples and non-examples of mixture spaces are:

• Vector spaces: any convex subset ${\displaystyle X}${\displaystyle X} of a vector space ${\displaystyle (V,+,\cdot )}${\displaystyle (V,+,\cdot )} over ${\displaystyle \mathbb {R} }${\displaystyle \mathbb {R} }, with ${\displaystyle h_{\alpha }(x,y)=\alpha x+(1-\alpha )y}${\displaystyle h_{\alpha }(x,y)=\alpha x+(1-\alpha )y} constitutes a mixture space ${\displaystyle (X,h)}${\displaystyle (X,h)}.
• Lotteries: given any finite set ${\displaystyle X}${\displaystyle X}, the set ${\displaystyle {\mathcal {L}}(x)=\left\{p:X\to [0,1]:\sum _{x}p(x)=1\right\}}${\displaystyle {\mathcal {L}}(x)=\left\{p:X\to [0,1]:\sum _{x}p(x)=1\right\}} of lotteries over ${\displaystyle X}${\displaystyle X} constitutes a mixture space, with ${\displaystyle h_{\alpha }(p,q)(x):=\alpha p(x)+(1-\alpha )q(x)}${\displaystyle h_{\alpha }(p,q)(x):=\alpha p(x)+(1-\alpha )q(x)}. Notice that this induces an "isomorphic" mixture space of CDFs over ${\displaystyle X}${\displaystyle X}, with the naturally-induced mixture function.

• Quantile functions: for any CDF ${\displaystyle F:\mathbb {R} \to [0,1]}${\displaystyle F:\mathbb {R} \to [0,1]}, define ${\displaystyle Q_{F}:[0,1]\to \mathbb {R} }${\displaystyle Q_{F}:[0,1]\to \mathbb {R} } as its quantile function. For any two CDFs ${\displaystyle F_{1},F_{2}}${\displaystyle F_{1},F_{2}} and any ${\displaystyle \alpha \in [0,1]}${\displaystyle \alpha \in [0,1]}, define the mixture operation ${\displaystyle \alpha F_{1}\boxplus (1-\alpha )F_{2}}${\displaystyle \alpha F_{1}\boxplus (1-\alpha )F_{2}} as the CDF for the quantile function ${\displaystyle \alpha Q_{F_{1}}+(1-\alpha )Q_{F_{2}}}${\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.^[3]

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

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

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

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

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

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

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

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

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

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

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