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IN THIS ESSAY, a new theory of choice under risk is being proposed. It is a theory which, in a sense that will become clear, is to expected utility theory, hence the title dual Risky prospects are evaluated in this theory by a cardinal numerical scale which resembles an expected utility, except that the roles of payments and probabilities are reversed. This theme-the reversal of the roles of probabilities and payments-will recur throughout the paper. I should emphasize that playing games, with probabilities masquerading as payments and payments masquerading as probabilities, is not my object. Rather, I hope to convince the reader that the theory has intrinsic economic significance and that, in some areas, its predictions are superior to those of expected utility theory (while in other areas the reverse will be the case). Two reasons have prompted me to look for an alternative to expected utility theory. The first reason is methodological: In expected utility theory, the agent's attitude towards risk and the agent's attitude towards wealth are forever bonded together. At the level of fundamental principles, risk aversion and diminishing marginal utility of wealth, which are synonymous under expected utility theory, are horses of different colors. The former expresses an attitute towards risk (increased uncertainty hurts) while the latter expresses an attitude towards wealth (the loss of a sheep hurts more when the agent is poor than when the agent is rich). A question arises, therefore, as to whether these two notions can be kept separate from each other in a full-fledged theory of cardinal utility. The theory will have this property. The second reason that leads me to look for an alternative to expected utility theory is empirical: Behavior patterns which are systematic, yet inconsistent with expected utility theory, have often been observed. (Two prominent references, among many others, are Allais (1953) and Kahneman-Tversky (1979).) So deeply