## Abstract

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 axioms 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.

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### Cite this article

Borie, D. Expected utility theory with non-commutative probability theory.
*J Econ Interact Coord* **8, **295–315 (2013). https://doi.org/10.1007/s11403-012-0098-1

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### Keywords

- Expected utility
- Decision theory
- Non-expected utility
- Quantum decision theory

### JEL Classification

- C60
- D03
- D81