Non-Monotonicity of the Tversky-Kahneman Probability-Weighting Function: A Cautionary Note

Abstract

Cumulative Prospect Theory (CPT) has been used in a wide variety of economic models as an alternative to Expected Utility Theory (EUT). It is used to account for a number of anomalies in the observed behavior of economic agents. Like EUT, CPT uses a utility or value function to rate any given outcome. In CPT, however, the utility function is loss averse SS-shaped) rather than risk averse (concave). The other difference between the two theories is the use of decision weights rather than probabilities in computing the expectation.

This note provides a caution on the use of a popular form of decision weights - those proposed by Tversky and Kahneman (1992) in their introduction of the theory. In particular, the weights generated by their proposed probability weighting function can be negative for some parameter values. This could lead to evaluations that chose first-order stochastically dominated gambles rather than the dominating one.

The parameter values that can cause such problems are small relative to estimated values. Nevertheless, failure to recognize this problem may prevent the proofs of some theorems or allow the proof of some theorems which would be invalid with properly restricted probability weights.

Keywords: Tversky-Kahneman, Probability Weighting, Cumulative Prospect Theory, economic models, Expected Utility Theory