Decisions under risk and one objective

Summary For decisions under risk the theory of expected utility establishes the basis of rational action. A preference regarding risky alternatives can be represented by a utility function, if the preference fulfills the axioms of “complete ordering”, “continuity” and “independence”. There is a measure by Arrow and Pratt for a decision maker’s risk attitude concerning lotteries. The measure is greater (smaller) than zero, if the utility function is concave (convex). A concave utility function is equivalent to risk aversion, a convex utility function is equivalent to risk proclivity. Using the terms risk aversion or risk proclivity, it must be pointed out that aspects of risk and aspects of value are inseparably inter-connected. Only in a few cases is the mean variance approach compatible with the theory of expected utility; it should hence only be applied with great caution. You will learn five methods of determining a utility function: the bisection version of the variable certainty equivalent method, the quantile version of the variable certainty equivalent method, the variable probability method, the method of equal utility difference, and the trade-off method. All methods are based on the repeated evaluation of simple lotteries. Consistency checks are an essential component of the methods of determining a utility function. The expected utility can easily be calculated in the single-stage model; the optimal alternative is the one with the highest expected utility. In the multistage model, the optimal alternative is computed with the aid of the rollback method.