A Generalization Of Pratt-Arrow Measure To Non-Expected-Utility Preferences And Inseparable Probability And Utility

By Robert F. Nau

Abstract:  A measure of local risk aversion is developed for choices under uncertainty with non-expected-utility preferences, inseparable subjective probabilities and state-dependent utilities, and unobserved stochastic prior wealth.  It is assumed only that the decision maker can assign arbitrage-free marginal prices to risky assets in a manner that varies smoothly in response to changes in wealth.  The risk aversion measure generalizes the Pratt-Arrow measure and depends only on the risk neutral probabilities supporting the arbitrage-free prices and on the matrix of their derivatives with respect to wealth.    For decision makers with non-expected-utility preferences, the measure also incorporates aversion to uncertainty, i.e., ambiguity or lack of information about probabilities.  As illustrations, the Ellsberg and Allais paradoxes are explained using a model of “partitionable” smooth preferences that exhibits local uncertainty aversion at all wealth distributions, unlike Choquet expected utility. Under this model, the decision maker satisfies the independence axiom selectively within partitions of the state space whose elements have similar degrees of uncertainty.  As such, she may behave like an expected-utility maximizer with respect to assets in the same uncertainty class, while exhibiting higher degrees of risk aversion toward assets that are more uncertain.”