رفتار در دامنه افت: با استفاده از آزمایش شرایط سازگاری تجارت کردن احتمال

In gambles with two or more outcomes, the two versions of prospect theory, i.e., original prospect theory and cumulative prospect theory, make use of different composition rules and therefore yield different valuations of gambles. We test these composition rules in the loss domain using the probability trade-off consistency condition. The probability trade-off consistency condition offers a convenient and efficient way to compare gambles under risk and decision makers’ behavior. Experimental findings suggest that the rank dependent version of prospect theory, or cumulative prospect theory, cannot be rejected in the loss domain while original prospect theory is clearly rejected when a certainty effect is taken into account.

In their seminal paper in Econometrica, Kahneman and Tversky (1979) introduced prospect theory as a convincing decision theory to study and describe choices under risk. Kahneman and Tversky (1979) presented prospect theory as an alternative to expected utility theory based on three characteristics. First, carriers of value are gains and losses relative to a reference point rather than to final wealth. Moreover, the marginal impact of a gain or a loss decreases with the distance from the reference point. This psychological phenomena, known as diminishing sensitivity, contrasts with the usual assumption of diminishing marginal utility, which supposes an increasing impact of a loss with distance from the reference point. Second, losses loom larger than equivalent gains, a phenomenon known as loss aversion. Third, probabilities are weighted nonlinearly: typically, small probabilities are overweighted and large probabilities are underweighted. The so-called original prospect theory was originally applied to three-outcome gambles involving a zero outcome. Camerer and Ho, 1994 and Fennema and Wakker, 1997 formulated a more complete version of original prospect theory for n-outcome mixed prospects. The lack of generality of OriginalProspect Theory (OPT hereafter) and its failure to avoid stochastic dominance violations gave rise to a modified version of OPT, the Cumulative Prospect Theory (CPT hereafter). Compared to OPT, CPT offers three new key features. First, CPT extends to gambles involving any number of possible outcomes. Second, CPT is able to deal both with risk and uncertainty. Third, CPT includes the rank dependent framework introduced by Quiggin, 1982 and Schmeidler, 1989, which allows prospect theory to avoid stochastic dominance violations. Both versions of Prospect Theory involve a utility function and a probability weighting function but differ on the way those functions are related, which is often defined as composition rules. Even if CPT has become the most preeminent form of prospect theory, interest for OPT remains. Gonzalez and Wu (2003) showed that both OPT and CPT are able to explain some standard empirical patterns. As a result, both theories may be considered as equally valid from a descriptive point of view. Several studies investigated the descriptive power of both OPT and CPT in the gain domain. According to Camerer and Ho, 1994 and Wu and Gonzalez, 1996, OPT fits the experimental data better than CPT. In contrast, Fennema and Wakker (1997) reanalyzed the Lopes (1993) experimental results and found that CPT performed better than OPT. Such a result is in line with Schneider and Lopes (1986) evidence that people think of risk in terms of the probability of not achieving a target (i.e., in terms of decumulative probabilities). More recently, Wu, Zhang, and Abdellaoui (2005) showed that the design of gambles plays an important role in the composition rule used by the subjects. While gambles involving a certainty effect appear to be compatible with both CPT and OPT, more complex gambles are compatible with OPT only. Research in the gain domain has developed considerably, but only few results are available in the loss domain. Notably, experimental evidence is scarce concerning the descriptive power of CPT in the loss domain. However, such data would be useful in the field of insurance where CPT, or Rank Dependent Utility (the sign-independent version of CPT), has emerged as a viable alternative to Expected Utility (Doherty and Eeckhoudt, 1995, Schmidt and Zank, 2007 and Wang, 1995). Experimental findings recognized the specificity of losses since both utility and probability weighting could be sign-dependent. Measurements of the shape of utility for gains and losses have generally confirmed prospect theory’s assumption of concave utility for gains and convex utility for losses. Concerning the shape of the probability weighting function, recent studies showed a more pronounced overweighting of small probabilities in the loss domain than in the gain domain (under risk: Abdellaoui, 2000 and Lattimore et al., 1992, Wu, Zhang, & Gonzales (2003) on the basis of the 1992 Tversky and Kahneman data; under uncertainty: Abdellaoui, Vossman, and Weber (2005)). Etchart-Vincent (2004) found probability weighting to be insensitive to the magnitude of the consequences, a result compatible with prospect theory. As a result, the two elements of prospect theory’s composition rules (either for CPT or OPT), namely utility and probability weighting, are sign-dependent. None of these studies has directly addressed the question of the specific composition rule used by the subjects in the loss domain. The aim of this paper is to fill this gap. In order to study the composition rules used by individuals in the loss domain, we present an experiment using the probability trade-off consistency condition. Probability trade-off consistency offers a convenient and efficient way to compare gambles that involve risk. The probability trade-off consistency condition used in the axiomatic work of Abdellaoui (2002) requires the restrictions on probability transformations on one set of gambles to be consistent with restrictions on probability transformations on another set of gambles. In other words, this condition implies a consistent ordering of probability transformations, independently of the underlying outcomes. This approach is the dual of the trade-off consistency condition for outcomes developed by Wakker, 1994, Chateauneuf and Wakker, 1999, Köbberling and Wakker, 2003 and Schmidt and Zank, 2001. By focusing on the probability trade-off consistency, Wu et al. (2005) offer a simple way to test the composition rule used by individuals in the gain domain, the sets of related gambles implied by the condition being different in each specification. Based on the method introduced by Wu et al. (2005), our empirical investigation suggests that CPT cannot be rejected for any gambles, while OPT is clearly rejected when a certainty effect is at stake. The remainder of the paper is organized as follows: Section 1 describes the two versions of prospect theory and the corresponding probabilistic trade-off consistency conditions. Section 2 describes the design of the experiment. Section 3 reports the results and compares the composition rules. Section 4 concludes.

A large number of experimental studies have shown that non-linearity in probability is a crucial feature to obtain a good descriptive theory of decision under risk. In this paper, the probability trade-off consistency condition has been applied to study the composition rules used by individuals in the loss domain. When applied to gambles that involve more than two outcomes, both versions of the prospect theory, the original and cumulative ones, yield different trade-off consistency conditions. In particular, in three-outcome gambles these conditions can be considered to determine the fanning of indifference curves in the Marschak–Machina triangle. This property was used to evaluate the composition rules used in pure loss gambles. Our study showed that CPT could not be rejected in any simplexes we used, while OPT was clearly rejected in simplexes involving a certainty effect. Thus, even if CPT can be criticized for its complexity in real-world situations, our results suggest that it continues to be a useful way to describe choices under risk in the loss domain and remains, as Starmer (2000) notes, an elegant compromise between descriptive validity and mathematical rigor. One drawback of our method is that the trade-off valuation used in this paper is based on paired choices instead of indifferences (Köbberling & Wakker, 2004). The rationale for this option is twofold. First, paired choices provide a logical basis of comparison with the previous study by Wu et al. (2005). Second, eliciting indifferences significantly increases the number of tasks subjects have to perform. This could lead to an increase in task complexity and error in a setting involving comparisons between three-outcome gambles. The use of indifferences seems better suited to more general tests of composition rules that include both pure and mixed gambles.