Abstract
The performance of rank dependent preference functionals under risk is comprehensively evaluated using Bayesian model averaging. Model comparisons are made at three levels of heterogeneity plus three ways of linking deterministic and stochastic models: differences in utilities, differences in certainty equivalents and contextual utility. Overall, the “best model”, which is conditional on the form of heterogeneity, is a form of Rank Dependent Utility or Prospect Theory that captures most behaviour at the representative agent and individual level. However, the curvature of the probability weighting function for many individuals is S-shaped, or ostensibly concave or convex rather than the inverse S-shape commonly employed. Also contextual utility is broadly supported across all levels of heterogeneity. Finally, the Priority Heuristic model is estimated within a stochastic framework, and allowing for endogenous thresholds does improve model performance although it does not compete well with the other specifications considered.
Similar content being viewed by others
Notes
We use PT to mean its cumulative variant which is sometimes termed Cumulative Prospect Theory.
See Pesaran and Weeks (2007) for an overview.
There are (n-1)n/2 combinations, which for the current paper means that the number of pairwise comparisons are of the order 10 9.
Note that in this paper we employ the term “Link” in a different manner than that used in Stott (2006) who refers to the “choice” function, which corresponds to what we call the outer link.
In the case of PT v and w take different forms in the gain and loss domains (and more generally may be asymmetric around a given reference point). In this study we only consider the gain domain.
For the LOG and EXPO-I functions if the parameters are to be equal ( α 2=α 3) and to achieve the same level of concavity (if \(\alpha _{2}<\frac {1}{x})\) α 3 needs to be higher. In effect, the prior for α 3 should be more diffuse with a higher mean unless the aim was to construct a prior supporting risk neutrality. However, we see that for values of α 3 equal to 100, we have a value at r that exceeds 99% of its possible value whereas at 10 it is at least equal to 90% of its possible value. We therefore placed 1% of the mass above 100 and 10% below 0.1, resulting in a relatively small shift in the mass above 10, at 13% rather than the 10% for the EXPO-I function.
Some expost sensitivity analysis was performed on these priors. For example, the two parameter probability weightings were re-estimated by doubling and halving the prior variances. These had no substantive impact on the results herein.
We note that while this makes complete sense, it is not a formal requirement that the two should equate. A particular aspect form could perform well when averaged across the other aspect forms, yet not actually be part of the model with the very highest LML.
Stott (2006) reports values of exactly 1 for both parameters of the PRELEC-II which is actually Linear, even though the PRELEC-I estimate is not linear. This seems unlikely, though is technically possible as the estimates are derived as medians of individuals, rather than using the representative agent model we are reporting here.
We note the observation of Wakker (2010) page 228 about the stability of probability weighting compared to utility curvature.
Although not explicitly reported the estimated parameter value is E(α 1)=0.197 with a standard deviation of 0.013. This result indicates a strongly concave form, which is consistent with Stott (2006) who reports 0.19.
Our search was not over the entire model space. We started by including all aspect forms for which elimination was not supported in Table 2. The search was then over all model spaces in which there was an elimination of one or more of these aspect forms.
References
Andersen, S., Harrison, G. W., Lau, M.I., & Rutström, E. (2008). Eliciting risk and time preferences. Econometrica, 76(3), 583–618.
Andersen, S., Harrison, G. W., Lau, M. I., & Rutström, E. (2010). Behavioral econometrics for psychologists. Journal of Economic Psychology, 31, 553–576.
Andrews, D.K. (1998). Hypothesis Testing with a restricted parameter space. Journal of Econometrics, 84, 155–199.
Andrieu, C., & Thoms, J. (2008). A tutorial on adaptive MCMC. Statistical Computation, 18, 343–373.
Barberis, N.C. (2013). Thirty years of prospect theory in economics: a review and assessment. Journal of Economic Perspectives, 27(1), 173–196.
Birnbaum, M. H. (2006). Evidence against prospect theories in gambles with positive, negative, and mixed consequences. Journal of Economic Psychology, 27, 737–761.
Birnbaum, M. H. (2008). Evaluation of the priority heuristic as a descriptive model of risky decision making: Comment on Brandstätter, Gigerenzer, Hertwig (2006). Psychological Review, 115(1), 253–260.
Birnbaum, M. H., & Chavez, A. (1997). Tests of theories of decision-making: Violations of branch independence and distribution independence. Organizational Behavior and Human Decision Processes, 71(2), 161–194.
Booij, A.S., van Praag, B.M.S., & van de Kuilen, G. (2010). A parametric analysis of prospect theory’s functionals for the general population. Theory and Decision, 68, 115–148.
Brandstätter, E., Gigerenzer, G., & Hertwig, R. (2006). The priority heuristic: Making choices without trade-offs. Psychological Review, 113(2), 409–432.
Brandstätter, E., Gigerenzer, G., & Hertwig, R. (2008). Risky choice with heuristics reply to Birnbaum (2008) Johnson Schulte-Mecklenbeck and Willemsen (2008) and Rieger and Wang (2008). Psychological Review, 115(1), 281–290.
Bruhin, A., Fehr-Duda, H., & Epper, T. (2010). Risk and rationality: uncovering heterogeneity in probability distortion. Econometrica, 78(4), 1375–1412.
Conte, A., Hey, J.D., & Moffat, J. (2011). Mixture models of choice under risk. Journal of Econometrics, 162, 79–88.
Cox, J.C., & Sadiraj, V. (2006). Small- and large-stakes risk aversion: implications of concavity calibration for decision theory. Games and Economic Behaviour, 56, 45–60.
Cox, J.C., Sadiraj, V., Vogt, B., & Dasgupta, U. (2013). Is there a plausible theory for decision under risk? A Dual Calibration Critique. Economic Theory, 53 (2), 305–333.
Fiedler, S., & Glöckner, A. (2012). The dynamics of decision making in risky choice: an eye-tracking analysis. Frontiers in Psychology, 3, Article 335.
Fehr-Duda, H., & Epper, T. (2012). Probability and risk: foundations and economic implications of probability-dependent risk preferences. Annual Review of Economics, 4(1), 567–593.
Fernandez, C., Ley, E., & Steel, M.F. (2001). Benchmark priors for Bayesian model averaging. Journal of Econometrics, 100, 381–427.
Findley, D.F. (1990). Making difficult model comparisons, Bureau of the Census Statistical Research Division Report Series, SRD Research Report Number: CENSUS/SRD/RR-90/11.
Gelfand, A., & Dey, D. (1994). Bayesian model choice: asymptotics and exact calculations. Journal of the Royal Statistical Society (Series B), 56, 501–504.
Green, P.J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82(4), 711–32.
Goldstein, W. M., & Einhom, H. J. (1987). Expression theory and the preference reversal phenomena. Psychological Review, 94(2), 236–254.
Harless, D.W., & Camerer, C.F. (1994). The predictive utility of generalized expected utility theories. Econometrica, 62(6), 1251–1289.
Harrison, G.W., Humphrey, S.J., & Verschoor, A. (2010). Choice under uncertainty: evidence from Ethiopia, India and Uganda. Economic Journal, 120, 80–104.
Harrison, G.W., & Rutström, E. (2009). Expected utility and prospect theory: one wedding and a decent funeral. Experimental Economics, 12(2), 133–158.
Hey, J. D., & Orme, C. (1994). Investigating generalizations of expected utility theory using experimental data. Econometrica, 62(6), 1291–1326.
Ingersoll, J. (2008). Non-monotonicity of the Tversky-Kahneman probability-weighting function: a cautionary note. European Financial Management, 14(3), 385–390.
Kahneman, D., & Tversky, A. (1979). Prospect theory: an analysis of decision under risk. Econometrica, 47, 263–291.
Koop, G. (2003). Bayesian econometrics. Chichester: Wiley.
Nilsson, H., Rieskampa, J., & Wagenmakers, E.J (2011). Hierarchical Bayesian parameter estimation for cumulative prospect theory. Journal of Mathematical Psychology, 55, 84–93.
Pesaran, M. H., & Weeks, M. (2007). Nonnested hypothesis testing: an overview. In Baltagi, B.H. (Ed.) A Companion to Theoretical Econometrics. Malden, MA, USA: Blackwell Publishing Ltd.
Rabin, M. (2000). Risk aversion and expected-utility theory: a calibration theorem. Econometrica, 68, 1281–1292.
Rabin, M., & Thaler, R. H. (2001). Anomalies risk aversion. Journal of Economic Perspectives, 15(1), 219–232.
Sala-i-Martin, X., Doppelhofer, G., & Miller, R. (2004). Determinants of long-term growth: a Bayesian averaging of classical estimates (BACE) approach. American Economic Review, 94, 813–835.
Shleifer, A. (2012). Psychologists at the gate: a review of Daniel Kahneman’s Thinking, Fast and Slow. Journal of Economic Literature, 50(4), 1–12.
Stott, H.P. (2006). Cumulative prospect theory’s functional menagerie. Journal of Risk and Uncertainty, 32, 101–130.
Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: cumulative representations of uncertainty. Journal of Risk and Uncertainty, 5, 297–323.
Vuong, Q.H. (1989). Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 57(2), 307–333.
Wakker, P.P. (2010). Prospect theory: for risk and ambiguity. Cambridge: Cambridge University Press.
Wilcox, N.T. (2011). Stochastically more risk averse: A contextual theory of stochastic discrete choice under risk. Journal of Econometrics, 162, 89–104.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Appendices
Appendix A1: Transformations
The parameters of interest in the models take 𝜃 in only one of two forms. That is, we parameterise our model by using \(\theta =t_{1}( \vartheta ;\delta _{l},\delta _{u}) =\delta _{l}+ ( \delta _{u}-\delta _{l}) \frac {e^{\vartheta }}{1+e^{\vartheta }}\) or 𝜃=t 2(𝜗)= exp(𝜗) where 𝜗∈R. In the case of t 1(𝜗;δ l ,δ u ) the transformed parameter lies within the interval (δ l ,δ u ). We set the values for {δ i }a priori in accordance with the inequality constraints. The priors for parameters of the form t 1(𝜗;δ l ,δ u ) are ( 𝜗∼N(0,ζ)) where they are assigned a variance ζ equal to \(\frac {9}{4}\) , yielding an approximately uniform prior within the specified interval, although there is less mass at the very extremes. Thus, in a sense we are being ‘non-informative’ about the values except that we have specified the interval over which the parameters lie. For parameters of the form t 2(𝜗) we assume that 𝜗 is normally distributed so that the implied prior distribution for the transformed parameter is log-normal.
Appendix A2: Pratt coefficients
The v-forms in the text are as follows:
Rights and permissions
About this article
Cite this article
Balcombe, K., Fraser, I. Parametric preference functionals under risk in the gain domain: A Bayesian analysis. J Risk Uncertain 50, 161–187 (2015). https://doi.org/10.1007/s11166-015-9213-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11166-015-9213-8