One of the frequently applied normative and descriptive model of decision making under risk is the Cumulative Prospect Theory developed by Tversky and Kahneman (1992). Under this model decision maker preferences over a given family of lotteries Δ is represented by the functional V:Δ→R of the form V(X)=E_{gh}u(X), where E_{gh} is the generalized Choquet integral with respect to the probability distortion functions g and h for gains and losses, respectively, and u:R→R is a value function, that is an increasing and continuous function such that u(0)=0. A certainty equivalent is a functional C:Δ→R defined as follows C(X)=u^{-1}(E_{gh}u(X)). The certainty equivalent is said to be positively homogeneous provided C(λ X)=λ C(X). Tversky and Kahneman (1992) stated, without proof, that if Δ consists of all two-payoff lotteries then the certainty equivalent is positively homogeneous if and only if the value function is of the form u(x) = x^{α} for x≥0 and u(x)=-λ(-x)^{β} for x < 0 with some α, β, λ>0. A formal proof of this fact is presented by al-Nowaihi, Bradley and Dhami (2007). In the talk we generalize this result in various directions. |