Comonotonicity, Rank-Dependent Utilities and a Search Problem

Abstract

Under the expected utility (EU) model, a decision maker (DM) is characterised by a utility function u, often assumed to be continuous and generally assumed to be non-decreasing. In this model, all the possible notions of risk aversion are merged and characterised by concavity of this utility function. Under the rank-dependent expected utility (RDEU) model (see [4, 12]), a DM is characterised by such a utility function (that plays the role of utility on certainty) in conjunction with a probability-perception function \(f: [0, 1] \to [0, 1]\) [0,1] that is non-decreasing and satisfies f(0) = 0, f(1) = 1. Such a DM prefers the random variable X to the random variable Y if and only if V(X) > V(Y), where the RDEU V (see [12, 16]) is given by $$V(Z) = V_u, f (Z) = -\int_{-\infty}^{\infty} u(x) df (P (Z > x)) = \int_{0}^{\infty} f(P (u(Z) > t))dt$$ , where the last equality holds for non-negative u(Z) but can be generalised in the usual way. It is easy to see that if the perception function f is the identity function \(f: (\upsilon ) \equiv \upsilon \), then \(V(Z) = V_u, I (Z)\) is simply the expected utility Eu(Z) of the random variable.