Decision Making in Phantom Spaces

The phantom space, with its phantom probability measure, allows the framing of ambiguous probabilities of events, as well as their vague consequences. This paper introduces a decision-making model under uncertainty, established on a phantom generalization of the von Neumann-Morgenstern axiomatization. Beliefs in our model are subjective variations of objective probabilities, recorded in a framework comprising not only risk but also phantom effects. Uncertainty measures are carried over naturally into this setting such that many of the familiar attributes of objective probabilities are preserved. The degree of uncertainty, which is determined by the available information and subjective beliefs of the decision maker, is distinguished from the attitude toward uncertainty, which is drawn from her preferences. Decision making under ambiguity is a special case of our model in which probabilities are vague but outcomes of events are clearly forecasted. The Ellsberg paradox and an insurance dilemma are the main examples we present