Optimal rules for ordering uncertain prospects

In this paper, we obtain the optimal selection rule for ordering uncertain prospects for all individuals with decreasing absolute risk averse utility functions. The optimal selection rule minimizes the admissible set of alternatives by discarding, from among a given set of alternatives, those that are inferior (for each utility function in the restricted class) to a member of the given set. We show that the Third Order Stochastic Dominance (TSD) rule is the optimal rule when comparing uncertain prospects with equal means. We also show that in the general case of unequal means, no known selection rule uses both necessary and sufficient conditions for dominance, and the TSD rule may be used to obtain a reasonable approximation to the smallest admissible set. The TSD rule is complex and we provide an efficient algorithm to obtain the TSD admissible set. For certain restrictive classes of the probability distributions (of returns on uncertain prospects) which cover most commonly used distributions in finance and economics, we obtain the optimal rule and show that it reduces to a simple form. We also study the relationship of the optimal selection rule to others previously advocated in the literature, including the more popular mean-variance rule as well as the semi-variance rule.