Expected utility theory on mixture spaces without the completeness axiom

TL;DR

This paper explores conditions under which a mixture preorder on a mixture space can be represented by real-valued, mixture-preserving functions, especially focusing on the roles of mixture continuity, topology, and the new axiom of countable domination.

Contribution

It introduces the countable domination axiom and analyzes its role in ensuring mixture-preserving multi-representations in uncountable spaces.

Findings

Mixture continuity, combined with countable domination, guarantees multi-representations in uncountable spaces.

Continuity and closedness are necessary but not sufficient for mixture-preserving representations.

The paper discusses conditions for the existence and uniqueness of strictly increasing multi-representations.

Abstract

A mixture preorder is a preorder on a mixture space (such as a convex set) that is compatible with the mixing operation. In decision theoretic terms, it satisfies the central expected utility axiom of strong independence. We consider when a mixture preorder has a multi-representation that consists of real-valued, mixture-preserving functions. If it does, it must satisfy the mixture continuity axiom of Herstein and Milnor (1953). Mixture continuity is sufficient for a mixture-preserving multi-representation when the dimension of the mixture space is countable, but not when it is uncountable. Our strongest positive result is that mixture continuity is sufficient in conjunction with a novel axiom we call countable domination, which constrains the order complexity of the mixture preorder in terms of its Archimedean structure. We also consider what happens when the mixture space is given its natural weak topology. Continuity (having closed upper and lower sets) and closedness (having a closed graph) are stronger than mixture continuity. We show that continuity is necessary but not sufficient for a mixture preorder to have a mixture-preserving multi-representation. Closedness is also necessary; we leave it as an open question whether it is sufficient. We end with results concerning the existence of mixture-preserving multi-representations that consist entirely of strictly increasing functions, and a uniqueness result.