 In order to get a cardinal (interval-valued) utility representation of a preference ordering—i.e., a measure that represents not only how an agent orders the options but also says something about the desirabilistic “distance” between options—we need a richer setting; the option set and the corresponding preference ordering will need to have more structure than for an ordinal utility measure. One such account, owing to John von Neumann and Oskar Morgenstern (1944), will be cashed out in detail below. For now, it is useful to focus on the kind of option that is key to understanding and constructing a cardinal utility function:  lotteries.[1]