Invariant Utility Functions and Certain Equivalent Transformations
Decision Analysis
The Role of Some Functional Equations in Decision Analysis
Decision Analysis
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Arrow and Pratt introduced a measure of risk aversion---the negative ratio of the second to the first derivative of the utility function. This measure has found widespread use in the valuation of uncertain lotteries and in the calculation of the risk premium of an investment. This paper introduces two new measures for characterizing changes in the valuation of uncertain lotteries when their outcomes are modified by a monotone transformation. The first is a characteristic transformation of a utility function, U, and a monotone transformation, g. The shape of the characteristic transformation determines an upper bound, lower bound, or equality on the magnitude of the certainty equivalent of the modified lottery. The second is a measure of change in certainty equivalent, ηg, whose sign also determines upper or lower bounds, and whose magnitude determines the change in value of a “small-risk” lottery when its outcomes are modified by a monotone transformation. For shift and scale transformations on the lottery outcomes, both the characteristic transformation and the measure of change, ηg, provide new characterizations for the notions of decreasing absolute and relative risk aversion with wealth.