This chapter and Chapters 6 and 7 deviate considerably from what has been mainstream utility theoiy in that they introduce another primitive beyond gambles and preferences over them. As we shall see, doing so has a number of advantages. First of all, the primitive is extremely nat ural and from time to time has been used somewhat informally in the literature, e.g., we have already encountered informal uses by Kahneman and Tversky (1979) and Tversky (1967a). Second, it imposes a strong constraint on the possible utility functions. Third, it allows one to develop new approaches to the case of binaiy mixed gambles, which, in my opinion, have not been handled veiy effectively in the traditional literature.