Certainty Equivalents for Three-Point Discrete-Distribution Approximations

Three-point discrete-distribution approximations are often used in decision and risk analyses to represent probability distributions for continuous random variables---e.g., as probability nodes in decision or probability trees. Performance evaluations of such approximations have generally been based on their accuracies in estimating moments for the underlying distributions. However, moment-based comparisons for recently proposed approximations have been limited. Moreover, very little research has addressed the accuracies of any such approximations in representing expected utilities or certainty equivalents of the underlying distributions. This is of potential concern since recent research shows three-point discrete-distribution approximations exist that match the first several moments of an underlying distribution exactly while, counterintuitively, approximating its certainty equivalents poorly. This paper compares the best two approximations for estimating means and variances identified in an earlier study with promising approximations proposed more recently. Specifically, it examines how accurately six simple general-purpose three-point approximations represent certainty equivalents for continuous random variables as the level of risk aversion is varied, as well as how accurately they estimate means and variances. The results show that several of these are quite accurate over a variety of test distributions when the level of risk aversion and the characteristics of the distributions are within reasonable bounds. Their robust performance is significant for decision analysis practice.