Risk-Aversion, Risk-Seeking, and Subjective Probability Weighting:

A Response to Wakker’s Critique


Moshe Levy and Haim Levy

Hebrew University

February 2003


Prospect Theory (PT) has two main components relevant to the Levy and Levy (2002a) study: subjective probability distortion, and the S-shape value function.  In our paper we focus on testing the S-shape value function hypothesis, ignoring probability distortion. Wakker’s note is technically correct: if one incorporates the CPT subjective probability distortion then the S-shape value function hypothesis cannot be rejected. However, we would like to present two arguments for why our results do in fact reject the S-shape value function: 1) As other studies show, probability distortion is irrelevant for the case of equally likely outcomes (uniform probability distributions) with moderate outcomes (like the ¼,¼,¼,¼ case in our study), hence Wakker’s calculation is irrelevant here; 2) Even if CPT probability distortion is taken into account, we have shown experimentally elsewhere that the S-shape value function is rejected. Let us elaborate on these two arguments.


1) The idea that in the case of uniform probabilities no subjective distortion takes place has been previously presented in the literature. Quiggin (1982), Officer and Halter (1968) and Anderson, Dillon and Hardaker (1977) argue that in uniform 50-50 bets no probability distortion would take place. Quiggin writes:


The claim that the probabilities of 50-50 bets will not be subjectively distorted seems reasonable, and, as stated above, has proved a satisfactory basis for practical work” (Quiggin, 1982, p.328).


Though Quiggin does employ probability distortion for uniform bets other than 50-50 bets, in Viscusi’s (1989) Prospective Reference Theory there is no probability distortion in the general symmetric uniform case. Also, in the original PT framework (Kahneman and Tversky 1979), in which the probabilities are transformed directly, the choice among prospects is unaffected by subjective probability distortion in the general uniform case. Thus, as our study was conducted with uniform probabilities and moderate outcomes, we consciously ignore the effects of subjective probability distortion in this case.

2) We have shown elsewhere (Levy and Levy 2002b) that even when probability distortion is taken into account exactly as suggested by CPT (and according to the formula employed by Wakker), the S-shape value function is rejected. For example, in Table 1 prospect G dominates F by Prospect Stochastic Dominance (PSD) with both the objective probabilities and with the CPT decision weights, yet 50% of the subjects preferred prospect F. This implies that the preferences of at least 50% of the subjects do not conform with an S-shape value function.


Table 1:  G dominates F by PSD Even with CPT Decision Weights

(Task II of Experiment 3 in Levy and Levy 2002b)











Gain or Loss



CPT decision



Gain or Loss


CPT decision

















The entanglement of probability distortion with the shape of preferences is not a simple issue. Indeed, even Kahneman and Tversky’s (1979) original claim of an S-shape value function is made ignoring probability distortion. It can be easily shown that with CPT probability distortion their findings are consistent with linear preferences, and even with a reversed S-shape value function. Matters are further complicated by studies showing that probability distortion is generally a function of the magnitude of the outcomes, and studies that find probability distortion which is exactly opposite to the probability distortion suggested by CPT (see, for example, Birnbaum and McIntosh,  1996, and Birnbaum 1999). Although our study is novel in employing the PSD and Markowitz Stochastic Dominance (MSD) criteria for testing the S-shape value function hypothesis, contrary to Wakker’s claim, there are many other studies in the literature rejecting this hypothesis. Thus, the issue of the shape of preferences is far from being settled. We hope that this debate will stimulate more research towards disentangling probability distortion and the shape of preferences for uniform and non-uniform probability distributions.


Anderson, J., Dillon, J., and Hardaker, B. 1977.  Agricultural Decision Analysis. Iowa State University Press, Ames, IA.


Birnbaum, M.H. 1999. “The Paradoxes of Allais, Stochastic Dominance, and Decision Weights”, in J. Shanteau, B.A. Mellers, and D.A. Schum (Eds.) Decision Science and Technology: Reflections on the Contributions of Ward Edwards,  Kluwer Academic Publishers, Boston.


Birnbaum, M.H., and McIntosh, W.R. 1996. “Violations of Branch Independence in Choices Between Gambles”, Organizational Behavior and Human Decision Processes 67, 91-110.


Kahneman, D., and  Tversky, A. 1979. “Prospect Theory of Decisions under Risk,” Econometrica, 47(2), 263-291.


Levy, M., and Levy, H. 2002a. “Prospect Theory: Much Ado About Nothing?”, Management Science, 48, 1334-1349.


Levy, H. and  Levy, M. 2002b. ”Experimental Test of the Prospect Theory Value Function”, Organizational Behavior and Human Decision Processes 89, 1058-1081.


Officer, R. and Halter A. 1968. “Utility Analysis in a Practical Setting,” American Journal of Agricultural Economics, 50, 257-277.


Quiggin, J. 1982. “A Theory of Anticipated Utility,” Journal of Economic Behavior and Organization, 3, 323-343.


Wakker, P.P. 2003. “The Data of Levy and Levy (2002). “Prospect Theory: Much Ado About Nothing?” Actually Support Prospect Theory”, Management Science.