Multivariate risk aversion with applications

Abstract

Risk aversion has played a fundamental part in applied decision models under uncertainty. The measure which has found wide applications is the Arrow-Pratt (r A) measure of absolute risk aversion [1,10]: $$ \mathop r\nolimits_A = - \mathop \partial \nolimits^2 u(z)/\partial u(z)$$ ((1)) defined on the space of real-valued utility functionsu(z). Herez may be a scalar i.e. wealth or income, or it may be a vector of goods over which the scalar utility function is defined. If\(z = z(\tilde c,x)\) represents the consequences of a lottery generated by the random variables\({\tilde c}\) with a probability distribution\(\begin{gathered}F(\tilde c\left| {\theta )} \right. \hfill \\{\tilde z} \hfill \\\end{gathered}\) indexed by its parametersθ (e.g. mean, variance), then the decision-maker (DM) has a problem of optimal decision-making under the uncertain environment. Two types of uses are then usually made. One is the notion of certainty equivalence of a lottery which has random outcomes denoted by\({\tilde z}\). For all monotonic utility functions\(u(\tilde z)\) defined on the space of\({\tilde z}\), a DM is said to be risk averse, if he prefers\(u(E(\tilde z))\) over\(E(u(\tilde z))\) whereE is the expectation operator over the nondegenerate distribution of the random variable\({\tilde z}\) or, of\({\tilde c}\) givenθ. If these expectations are finite, then the certainty equivalent (CE) of the lottery is defined by an amount\({\hat z}\) such that $$u(\hat z) = E[u(\tilde z)]$$ ((2)) i.e. the DM is in different between thelottery and the amount\({\hat z}\) for certain.KeywordsUtility FunctionRisk AversionGeodesic DistanceCertainty EquivalentMinimax StrategyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.