 The basic idea of interval preference measurement is to assume that acts have uncertain consequences, and that each act is equivalent to a lottery between these outcomes. An agent who expresses a preference for an act over others by choosing it thus expresses a preference for the equivalent lottery over the lotteries equivalent to other acts. The utilities of these acts are then determined as the expected utilities of the equivalent lotteries, calculated as the probability-weighted average of the lottery’s consequences. This approach was pioneered by Ramsey (1928) and refined by von Neumann and Morgenstern (1944); other approaches have been presented by Savage (1954/72) and Jeffrey (1965/90). There are substantial differences between these approaches and their respective assumptions. For more detail, see  decision theory.