The Need for Discontinuous Probability Weighting Functions: How Cumulative Prospect Theory is torn between the Allais Paradox and the St. Petersburg Paradox

Abstract

Cumulative Prospect Theory (CPT) must embrace probability weighting functions with a discontinuity at probability zero to pass the two most prominent litmus tests for descriptive decision theories under risk: the Allais paradox and the St. Petersburg paradox. We prove in a nonparametric framework that, with continuous preference functions, CPT cannot explain both paradoxes simultaneously. Thus, Kahneman and Tversky’s (1979) originally proposed discontinuous probability weighting function has - when applied in a rank-dependent framework, of course - much more predictive power compared to all other popular, but continuous weighting functions, including e.g. Tversky and Kahneman's (1992) proposal. Neo-additive weighting functions constitute another parsimonious, yet promising class of discontinuous weighting functions. In other words, if we rashly restricted CPT to continuous preference functions we might erroneously jump to the conclusion that risk preferences are not stable over similar tasks or even reject CPT.