A Probability Weighting Function for Cumulative Prospect Theory and Mean-Gini Approach to Optimal Portfolio Investment

Abstract

This paper presents a new two-parameter probability weighting function for Tversky and Kahneman (1992) cumulative prospect theory as well as its special cases — Quiggin (1981) rank-dependent utility and Yaari (1987) dual model. The proposed probability weighting function can be inverse S-shaped (concave near probability zero and convex near probability one), S-shaped, globally convex and globally concave. Utility function of Yaari (1987) dual model with the proposed probability weighting function is a linear tradeoff between the lottery’s expected value (i.e. the first L-moment), Gini (1912) mean difference statistic (or the second L-moment, known as L-scale) and the third L-moment (measuring the lottery’s skewness). Two parameters of the proposed probability weighting function can be interpreted as a decision maker’s sensitivity to the dispersion and skewness of lottery’s outcomes. A decision maker who prefers positively skewed distributions (e.g., a small chance to win a highly desirable outcome) and dislikes negatively skewed distributions generally has an inverse S-shaped probability weighting function. This function crosses the 45° line at a probability smaller (greater) than 0.5 if a decision maker is also averse (attracted) to the dispersion of outcomes. Cumulative prospect theory with our proposed probability weighting function can rationalize the common ratio effect (i.e. a systematic fanning-out of indifference curves) in one type of common ratio problems as well as the reverse common ratio effect (i.e. a systematic fanning-in) — in another type of common ratio problems, in accordance with the recent experimental evidence.