Preferences, risk neutrality and risk-sensitive MDPs

A binary preference relation on a real vector space satisfying four (natural) axioms is shown to induce a utility function composed of a linear function to the reals and a weakly monotonic function. The key axiom is decomposition, and the utility function can be taken to be linear if and only if this axiom's converse is also satisfied. Important consequences follow for risk-sensitive discounted Markov decision processes, decision trees, and the discounted utility model in economics. Since the four axioms imply that preferences correspond to discounting, the four axioms without the converse imply that preferences are consistent with discounting without risk neutrality.