General Changes in Uncertainty

Abstract

In decision problems involving uncertainty, one is often interested in analyzing the effect on the optimal value of a decision variable resulting from a change in the nature of the uncertainty. One approach to this problem that has frequently been used in the literature is to subject the random variable in question to some deterministic transformation, usually of the linear type. Most researchers have chosen those transformations which bring about a mean-preserving increase in the riskiness of the random variable. Some examples in which such transformations are used are the works of Arrow [1, 104-106], Sandmo [12, 67-69], Batra and Ullah [2, 542-45], Ishii [8, 768-69], and Meyer and Ormiston [9, 181-85]. The advantage of using the transformation method for the analysis of changes in uncertainty is its simplicity, and the fact that determinate results can be obtained by imposing an additional assumption such as decreasing absolute (or increasing relative) risk aversion. However, the transformation method is not without shortcomings. The most obvious of these is the restriction imposed on the relationship between the distribution functions of the two random variables. Specifically, if the random variable X has the distribution F(x), and the transformed variable Y, where Y = t(X), has the distribution G(y), then the distribution function of Y must satisfy G(y) = F[t -(y)], assuming t is monotone. Thus, if the transformation t is linear, the two distribution functions can intersect at most once. While this feature facilitates the analysis, it clearly restricts the admissible types of changes in uncertainty, and this, in turn, may preclude certain outcomes which more general types of shifts in the distributions could bring about. Another aspect of the transformation method that could prove to be somewhat problematic in some contexts is the fact that some transformations, such as linear ones, cause a change in the effective range of the random variable. In some cases, however, the random variable has a natural bounded range (such as the unit interval, or the non-negative half-line), so that the transformed variable may have to take values outside the meaningful range. The most important aspect of the restrictiveness of the transformation method is its lack of robustness; that is, when more general changes in uncertainty are allowed, the formal propositions obtained under the transformation method may no longer be valid. The limitations of the transformation-of-variable approach to changes in uncertainty can be avoided altogether by allowing the distribution function to shift in an arbitrary fashion, at least