Measuring utility without mixing apples and oranges and eliciting beliefs about stock prices

In their classic paper, Herstein and Milnor axiomatically generalise convex sets to mixture sets. They then provide necessary and sufficient conditions for a mixture preserving utility representation that is cardinal. In this paper we derive the same utility representation for partial mixture sets: where the mixture operation is only partially defined. The resulting model has an interesting application to stochastic processes. In particular, it allows us to use paths instead of events to elicit utility and beliefs. This feature is promising for finance settings, where the dimension of the state space is infinite.