Curvature of the Probability Weighting Function

When individuals choose among risky alternatives, the psychological weight attached to an outcome may not correspond to the probability of that outcome. In rank-dependent utility theories, including prospect theory, the probability weighting function permits probabilities to be weighted nonlinearly. Previous empirical studies of the weighting function have suggested an inverse S-shaped function, first concave and then convex. However, these studies suffer from a methodological shortcoming: estimation procedures have required assumptions about the functional form of the value and/or weighting functions. We propose two preference conditions that are necessary and sufficient for concavity and convexity of the weighting function. Empirical tests of these conditions are independent of the form of the value function. We test these conditions using preference "ladders" (a series of questions that differ only by a common consequence). The concavity-convexity ladders validate previous findings of an S-shaped weighting function, concave up to p < 0.40, and convex beyond that probability. The tests also show significant nonlinearity away from the boundaries, 0 and 1. Finally, we fit the ladder data with weighting functions proposed by Tversky and Kahneman (Tversky, Amos, Daniel Kahneman. 1992. Advances in prospect theory: Cumulative representation of uncertainty. J. Risk and Uncertainty 5 297--323.) and Prelec (Prelec, Dra\v{z}en. 1995. The probability weighting function. Unpublished paper.).