Preferences, Risk Neutrality, and Utility Functions

A binary preference relation on a real vector space V with zero element 0 satisfying four axioms induces a utility function U = f ◦ u : V → R where u:V → R is linear as a map of vector spaces and f : R → R is weakly monotone. The fourth axiom is key: X − Y ( ) 0 ⇒ X ( ) Y (X, Y ∈ V ). The utility function is linear if and only if also satisfies the converse of the fourth axiom. A consequence is that previous axiomatizations of the discounted utility model in deterministic settings by T. C. Koopmans and others unwittingly imply risk neutrality because the axiom sets include the four and the converse.