 Note that ordinal utilities are not very mathematically “powerful”, so to speak. It does not make sense, for instance, to compare the probabilistic expectations of different sets of ordinal utilities. For example, consider the following two pairs of prospects: the elements of the first pair are assigned ordinal utilities of 2 and 4, while those in the second pair are assigned ordinal utilities of 0 and 5. Let us specify a “flat” probability distribution in each case, such that each element in the two pairs corresponds to a probability of 0.5. Relative to this probability assignment, the expectation of the first pair of ordinal utilities is 3, which is larger than 2.5, the expectation of the second pair. Yet when we transform the ordinal utilities in a permissible way—for instance by increasing the highest utility in the second pair from 5 to 10—the ordering of expectations reverses; now the comparison is between 3 and 5. The significance of this point will become clearer in what follows, when we turn to the comparative evaluation of lotteries and risky choices. An interval-valued or cardinal utility function is necessary for evaluating lotteries/risky prospects in a consistent way. By the same token, in order to construct or conceptualise a cardinal utility function, one typically appeals to preferences over lotteries. (Although see Alt 1936 for a “risk-free” construction of cardinal utility, that is, one that does not appeal to lotteries.)