Expected utility of a one-dimensional classical risky generalized lottery of I and III type With linearly interpolated CDF

The paper deals with ranking risky alternatives whose consequences form a one-dimensional piece-wise continuous set where the uncertainty is described by a classical one-dimensional CDF, with possible discontinuities. The alternatives can be modeled as one-dimensional classical risky generalized lotteries of either type I or III. The expected utility of such lotteries is represented as a Riemann-Stieltjes integral, whose form in the special cases of continuous and of discrete CDF is known. This paper gives a formula of the expected utility integral for the case when CDF is of mixed character, and is linearly interpolated with discontinuities, whereas the utility function is continuous. A special case for linearly interpolated utility function at arbitrary preferences is developed. The results are useful when the uncertainty in each alternative is described by p-boxes, which depending on the selected Q criterion under strict uncertainty are approximated by classical risky generalized lotteries of type I or III, called Q-lotteries. Often, the latter have extreme expected utilities from the set of all classical risky lotteries with classical CDF described by the p-box. Therefore the Q-lotteries often coincide with the type of lotteries analyzed here. Therefore, the results of the paper are important when making choices with interval probabilities.