 Modern Economy, 2011, 2, 691-700 doi:10.4236/me.2011.24077 Published Online September 2011 (http://www.SciRP.org/journal/me) Copyright © 2011 SciRes. ME Too Risk-Averse for Prospect T h e o ry? Marc Oliver Rieger1, Thuy Bui2 1University of Trier, Trier, Germany 2Project Finance Department, Vingroup, Ha Noi, Vietnam E-mail: mrieger@uni-trier.de, buithuy84@gmail.com Received May 4, 2011; revised June 25, 2011; accepted July 12, 2011 Abstract We observe that the standard variant of Prospect Theory cannot describe very risk-averse choices in simple lotteries. This makes it difficult to accommodate it with experimental data. Using an exponential value func-tion can solve this problem and allows to cover the whole spectrum of risk-averse behavior. Further evidence in favor of the exponential value function comes from the evaluation of data from a large scale survey on preferences over lotteries where the exponential value function produces the best fits. The results enhance the understanding on what types of lotteries pose potential problems for the classical value function. Keywords: Cumulative Prospect Theory, Decisions under risk, Risk-Aversion, Probability Weighting, Value Function 1. Introduction Imagine you are faced with the following gamble: with a probability of 90% you win 100 Euro, otherwise you win only 10 Euro. Which safe amount of money would be equally as good for you as participating in the gamble? Obviously, depending on your risk attitudes you could choose any amount between 10 and 100 Euro. If you are risk-averse, you will choose an amount between 10 and 91 Euro (the latter being the expected value of the lottery). Let us say, a person states 25 Euro as the according amount. We want to be able to model the preferences of this person in the framework of Prospect Theory (PT), the most commonly used descriptive model for choices under risk. Can we do this? It would be natural to answer yes: we just have to adapt the risk-aversion parameter in the model appro- priately. In this article, however, we will show that the answer is no! We cannot model the preference in the standard framework of PT. The person is too risk-averse to be described by this theory. We will generalize this surprising result and prove it in Section 3.1. Moreover, we will demonstrate that this effect also causes problems when measuring PT-para- meters in experiments. In Section 3.2 we see how the problem can be solved by using an exponential value function, and in Section 3.3 we study quadratic value functions. In section 4 we discuss empirical evidence which confirms the advantages of exponential value functions. Before that, we start with a short review of PT (Section 2). 2. Prospect Theory Prospect Theory (PT) has been introduced by  as a descriptive model for decision making under risk, adding certain behavioral effects to the classical Expected Uti- lity Theory:  Decisions are framed as gains and losses. The utility function is replaced by a value function v which has two parts, a concave part in the gain domain and a convex part in the loss domain, capturing risk-averse behavior in gains and risk-seeking behavior in losses.  Probabilities are weighted by an S-shaped probability weighting function w, overweighting small and un- derweighting large probabilities. In this article we will—for simplicity only—consider two- outcome lotteries in gains, a case where the version of PT we use in this article coincides with PT’s modern variant Cumulative Prospect Theory . The value of a lottery with outcomes A and B (A. Such lotteries can already be found in the article of . It is, of course, possible to circumvent the problem by shifting the reference point to the lower outcome, thus effectively only considering gambles with A = 0 (with respect to the reference point). With this trick any amount of risk-averseness can be explained within the framework of standard PT for two-outcome lotteries. There are, however, two major concerns about this ad hoc method: First, one might complain that this would not be problem solving, but rather “sweeping the pro- blem under the carpet” by arbitrarily changing the re- ference point. Second, the question of what to do if there is a third outcome arises, e.g. 0, with a very low pro- bability. Then the trick is not applicable and we are left in the same situation as before. In short, it is difficult or impossible to define a consistent rule to choose the re- ference point that circumvents the problem we have encountered. Therefore, we decided to refrain from changing the reference point. But we still haven’t seen whether the theoretical bound of risk-averseness is a practical problem. In other words: are there people who are so risk-averse that their behavior cannot be explained within the standard for- mulation of PT? To check this, we had a look at the lotteries with 0A> in the data of  and computed the lower bound for the CE in their model with their parameters (in- cluding their probability weighting parameter) for all lotteries with outcomes . Then we compared the results with the median answer of their respondents. Moreover, we computed the percentage of respondents who gave a CE below the theoretical threshold. In other words: the percentage of participants for each question who could not be described by the standard form of PT. (The detailed results are given in the appendix, see Table 1). 02², for 2xx xvxxx x (5) Such functions allow to model mean-variance pre- ferences as a special case of PT (for and ) . Of course, the parameters have to be chosen α=1λ= M. O. RIEGER ET AL.695 so that the highest and lowest outcome of all lotteries are still in the area where is non-decreasing.4 vOne can now ask whether with such value functions, arbitrarily strong risk-aversion can be modeled. The an- swer is again negative. More precisely we have the following result: Proposition 4. Using the quadratic value function (5), the CE of a two-outcome lottery with positive outcomes A and (where BAAαα 1ln 1ee1 lne1e1 lneln1e1 =ln1e1.αAαBαBAαAαBAαAαBACEz+ zαz+zα+z+zαA+zα   Since and 0α>A1ln 1ln 1e10.αBAz< +z< Inserting these inequalities into (6), we obtain bounds for the CE, namely: 1ln 1.A