Risk Neutrality and Ordered Vector Spaces

The following result clarifies when preferences over time and under risk correspond to discounting and are not risk neutral. If a binary relation on a real vector space V satisfies four axioms, then there is a utility function U=fu from V to R where u from V to R is linear as a map of vector spaces and f from R to R is continuous and weakly monotone. Three axioms are familiar: weak ordering, continuity, and non-triviality. The fourth axiom is X-Y is weakly preferred to 0 (where 0 is the zero element in V) implies X is weakly preferred to Y (X and Y in V). The function f from R to R is linear if and only if the binary relation also satisfies the converse of the fourth axiom. When V is a real vector space of stochastic processes and 0 is the zero process, it is known that the four axioms imply the existence of discount factors and the linearity of an intra-period utility function. So preferences correspond to discounting and are not risk neutral only if the converse of the fourth axiom is not satisfied.