This paper introduces a new model of decision making under uncertainty. Aiming to provide a more realistic depiction of decision making, it generalizes the von Neumann–Morgenstern theory by including additional tiers of uncertainty. In this model, beliefs about the probabilities of events are ambiguous and their consequential utilities are vague; both are naturally formulated in the phantom space using phantom numbers. The degree of uncertainty, determined by the decision maker’s beliefs, is distinguished from the attitude toward uncertainty, which is drawn from her preferences. Decision making under ambiguity is a particular case of our model in which probabilities are ambiguous, but resulting utilities of events are knowable.
Bibliographical noteThe authors thank two anonymous referees, Adam Brandenburger, Xavier Gabaix, Sergiu Hart, Ruth Kaufman, Efe Ok, Benjamin Polak, Jacob Sagi, Larry Samuelson, and especially Itzhak Gilboa and Thomas Sargent for valuable discussions and suggestions. We would also like to thank the seminar participants at Lund University, New York University, Technion, Tel Aviv University, The Hebrew University of Jerusalem, and Yale University. The first author acknowledges the support of the AXA research fund. The second author acknowledges the support of the Chateaubriand scientific fellowship, Ministry of Science France.
- Phantom probability
- Decision making under uncertainty
- Expected utility
- Imprecise risk
- Ellsberg paradox