This paper investigates the consequences of the following modification of expected utility theory: Instead of requiring independence with respect to probability mixtures of risky prospects, require independence with respect to direct mixing of payments of risky prospects. A new theory of choice under risk--a so-called dual theory--is obtained. Within this new theory, the following questions are considered: (i) numerical representation of preferences; (ii) properties of the utility function; (iii) the possibility for resolving the "paradoxes" of expected utility theory; (iv) the characterization of risk aversion; (v) comparative statics. The paper ends with a discussion of other non-expected-utility theories proposed recently.