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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 pdecision making, expected utility, nonexpected utility theory, prospect theory, risk, risk aversion

Expected utility theory, prospect theory, and most other models of risky choice are based on the fundamental premise that individuals choose among risky prospects by balancing the value of the possible consequences. These models, therefore, require that the value of a risky prospect lie between the value of that prospect's highest and lowest outcome. Although this requirement seems essential for any theory of risky decision-making, we document a violation of this condition in which individuals value a risky prospect less than its worst possible realization. This demonstration, which we term the uncertainty effect, draws from more than 1000 experimental participants, and includes hypothetical and real pricing and choice tasks, as well as field experiments in real markets with financial incentives. Our results suggest that there are choice situations in which decision-makers discount lotteries for uncertainty in a manner that cannot be accommodated by standard models of risky choice. From the time of Bernoulli on, it has been common to argue that (a) individuals tend to display aversion to the taking of risks, and (b) that risk aversion in turn is an explanation for many observed phenomena in the economic world [Arrow 1971, p. 90].

and Rongchen Zhu provided invaluable research assistance. We also thank Dave McGillivray and Marc Davis of the Boston Athletic Association for providing us with historical data on Boston Marathon qualifying times. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peerreviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications.