is preferred over x1; this set of outcomes, , can be assumed to be a calculable money-prize in a controlled game of chance, unique up to one positive proportionality factor depending on the currency unit. Let and be two lotteries with probabilities p1, p2, and p3 of x1, x2, and x''3 respectively being Assume that someone has the following preference structure under risk: meaning that is preferred over . By modifying the values of and in , eventually there will be some appropriate values () for which she is found to be indifferent between it and —for example Expected utility theory tells us that and so In this example from Majumdar fixing the zero value of the utility index such that the utility of is 0, and by choosing the scale so that the utility of equals 1, gives Intertemporal utility Models of utility with several periods, in which people discount future values of utility, need to employ cardinalism in order to have well-behaved utility functions. According to Paul Samuelson the maximization of the discounted sum of future utilities implies that a person can rank utility differences. Controversies Some authors have commented on the misleading nature of the terms "cardinal utility" and "ordinal utility", as used in economic jargon: There remain economists who believe that utility, if it cannot be measured, at least can be approximated somewhat to provide some form of measurement, similar to how prices, which have no uniform unit to provide an actual price level, could still be indexed to provide an "inflation rate" (which is actually a level of change in the prices of weighted indexed products). These measures are not perfect but can act as a proxy for the utility. Lancaster's characteristics approach to consumer demand illustrates this point. See also Expected utility theory Level of measurement Marginal utility Multi-attribute utility Utility References External links Ordinal utility vs. Cardinal utility Murray N. Rothbard, "Towards a Reconstruction of Utility and Welfare Economics" Utility