Expected utility theory with non-commutative probability theory

In this paper, we extend von Neumann and Morgenstern's expected utility approach to a non-commutative probability theory. We introduce a new representation of the decision maker's set of events which extends the canonical representation. We reformulate von Neumann and Morgenstern's approach to modeling decision maker behavior by non-commutative probability theory. We introduce a set of preference axi- oms similar to von Neumann and Morgenstern's axioms, and show that they lead to a generalization of the expected utility theorem. Our generalization allows for decision makers to make an intuitive distinction between representations of a set of events. We find that this methodology enables several paradoxes and inconsistencies in traditional expected utility theory (e.g., Allais paradox, etc.) to be solved or better understood.