Expected utility as an expression of linear preference intensity

Following Gilboa and Schmeidler (Games Econ Behav 44:184–194, 2003) we consider a scenario where the decision maker holds, for every possible probabilistic belief about the states, a preference relation over his choices. For this setting, Gilboa and Schmeidler have offered conditions that allow for an expected utility representation. Their central condition is the diversity axiom which states that for every strict ordering of at most four choices there should be a belief at which it obtains. It turns out that this axiom excludes many natural cases, even when there are no weakly dominated choices. We replace the diversity axiom by two new axioms—three choice and four choice linear preference intensity, which reflect the assumption that the preference intensity between two choices varies linearly with the belief. It is shown that in the absence of weakly dominated choices, the resulting set of axioms characterizes precisely those scenarios that admit an expected utility representation. In particular, our set of axioms covers a significantly broader class of scenarios than the Gilboa-Schmeidler axioms.