Choices, Values, and Frames

INTRODUCTION

In their article on Prospect Theory, Kahneman and Tversky (1979) introduced a nonlinear transformation of probabilities, p → w(p) (or π(p) in the 1979 notation), which is also called a probability weighting function. The purpose of the transformation was to explain several key expected utility violations, including the classical paradoxes of Allais (1953). Taking each violation as an independent constraint on w(p), they composed a conjecture about its shape - a conjecture that has been refined but not substantially altered by later work (Camerer and Ho 1994, Wu and Gonzalez 1996a).

Although its empirical picture has come into focus, the weighting function has remained a somewhat tricky object to analyze - at least in comparison with utility functions. A glance at Figure 4.1, displaying some recent estimates, reveals the nature of the problem. In the figure, the x-axis represents probability of an outcome, and the y-axis the weight associated with that probability. Unlike utility functions, in which the deviation from linearity has an essentially one-dimensional character (i.e., concavity), here we see both concavity and convexity. Curiously, the function is asymmetrical, with the convex region being about twice as large as the concave region. Overall, it does not look like a shape that one would draw unless compelled by strong empirical evidence.