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Mixture-space theorem

Utility-representation theorem in Decision Theory From Wikipedia, the free encyclopedia

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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,[1] together with the introduction of the definition of a mixture space.

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Mixture spaces

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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
  1. .
  2. .
  3. .

Mixture spaces are essentially a special case of convex spaces (also called barycentric algebras),[2] 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.[3]
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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 ,
[nb 1]
  • 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 .[1]

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 ,
.
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Notes

  1. This version of the Indepence Axiom is equivalent to the more usual one of von Neumann-Morgenstern which requires a general instead of just .[1]

References

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