Gaussian Cumulative Prospect Theory
Mederic Motte
TL;DR
Addresses decision-making under risk using Cumulative Prospect Theory for Gaussian rewards. Proposes parametric families for rewards, values, and weights with validity and richness, plus a closed-form gamble valuation for $R$ distributed as $N(mu,sigma)$. Demonstrates analytic gradients with respect to model parameters, enabling efficient gradient-based optimization in large-scale settings. Illustrates two population-oriented applications—design optimization and equilibrium computation—that benefit from fast, scalable CPT valuations.
Abstract
We propose a novel parametrization of Cumulative Prospect Theory (CPT), as developed by Daniel Kahneman and Amos Tversky, that yields an explicit gamble valuation formula for Gaussian reward distributions. Specifically, we define parametric functions $ v_θ $, $ w^{-}_θ $, and $ w^{+}_θ $ satisfying three key properties. The first, \emph{validity}, ensures that for any parameter $θ$, the functions conform to the qualitative principles of CPT: $ v_θ $ is concave over gains and convex over losses with a steeper slope for losses; $ w^{-}_θ $ and $ w^{+}_θ $ are increasing, exhibit inverse S-shaped curves, and map 0 to 0 and 1 to 1. The second, \emph{richness}, guarantees that the parametrization is expressive enough to capture a wide range of behaviors: $ v_θ $ can exhibit arbitrary asymptotic behavior and convergence rates, while $ w^{-}_θ $ and $ w^{+}_θ $ can achieve any specified crossover points and slopes. The third, \emph{explicit valuation}, ensures that for any $θ$, the CPT valuation of a Gaussian-distributed gamble (with arbitrary mean and variance) can be computed in closed form -- enabling efficient approximations for bell-shaped reward distributions. This framework is designed for scalable and rapid computation, making it particularly suited for applications involving large populations. We demonstrate its practicality through two illustrative examples in population-level CPT modeling.