Abstract
Prospect theory is the most popular theory for predicting decisions under risk. This paper investigates its predictive power for decisions under ambiguity, using its specification through the source method. We find that it outperforms its most popular alternatives, including subjective expected utility, Choquet expected utility, and three multiple priors theories: maxmin expected utility, maxmax expected utility, and a-maxmin expected utility.
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Notes
All extensions to more outcomes proposed in the literature have problems (Wakker 2010, Section 9.8).
Although prospect theory is often used to refer to OPT, the new version of 1992 has now replaced OPT and deserves this nontechnical and short name, rather than the technical and long “cumulative prospect theory” or CPT. Our terminology was Tversky’s preference.
HLM (p. 87 bottom) explain that the theories that they call prospect theory are not Tversky and Kahneman’s (1992) version or any other common version.
For descriptive violations of backward induction, see Barkan and Busemeyer (1999), Budescu and Fischer (2001), Carbone and Hey (2001), Cubitt et al. (1998), Dominiak et al. (2012), Hey and Knoll (2011), Hey and Lotito (2009), Hey and Panaccione (2011), Rachlin and Green (1972) and Yechiam et al. (2005).
See Andersen et al. (2012), Choi et al. (2007), Hsu et al. (2005), and Huettel et al. (2006). Only for rank dependence and PT have there been some quantitative studies considering more than two events:Abdellaoui et al. (2011), Abdellaoui et al. (2005), Baillon and Bleichrodt (2012) and Diecidue et al. (2007). We focus here on revealed-preference based studies. Hogarth and Einhorn (1990) present and cite influential work using introspective inputs.
Their first version was an extension of OPT to ambiguity. However, OPT has the aforementioned problems, making it ill suited for the three-outcome domain considered here. The second version was Schmeidler’s (1989) CEU with a level parameter of utility added to capture loss aversion. However, this second version did not incorporate the sign-dependent and reflected weighting of losses that is typical of prospect theory, and that is needed to make the added level parameter of utility identifiable.
Theoretical claims of uniqueness and identifiability are usually derived under the assumption of continuums of domains. For discrete observations as always obtained in practice, those claims provide a lower bound for uniqueness and an upper bound for identifiability.
Although the outcome 0 does not occur in our domain, it is essential for the decision weights that U(x) reflects the distance from x to 0 in utility units, for all x.
We use the terms maxmin EU and maxmax EU instead of HLM’s terms G&S maxmin and G&S maxmax because our terms are common in the literature.
It is even more so for generalizations of multiple priors models, including Chateauneuf and Faro (2009), Gajdos et al. (2008), Maccheroni et al. (2006), Nascimento and Riella (2010), Strzalecki (2011) andSiniscalchi (2009). Ghirardato et al. (2004) and Siniscalchi (2006) gave preference conditions, assuming an Anscombe-Aumann (1963) model, to determine whether or not a particular probability measure is contained in the set of priors. This requires a two-stage setup and then needs infinitely many observations (one for each possible prior) to determine the set of priors.
Although HLM do not state it explicitly (footnote 16 and p. 109), the boundaries ?? j are lower bounds and are not upper bounds.
HLM (footnote 18) point out that subject 35 was an outlier, performing very poorly for aMM and MnEU, and greatly influencing the average likelihoods of those models. Hence, HLM left this subject out. Although this means favoring the models affected, we follow HLM and also leave this subject out from Table 1. With subject 35 incorporated, SPT would still win, aMM and MnEU would be among the worst theories, and the other theories would not be affected seriously; see Table WA1 in the Web Appendix.
Now subject 35 is included, again following HLM.
SCEU can be seen to be the special case of SPT with dual, rather than identical, weighting for losses, which is why we incorporate it in this subsection. Because utility then becomes unique up to unit and level, the one parameter that we used for utility under SPT (through -£10) is equivalent to the one parameter used for utility in CEU.
References
Abdellaoui, M., Baillon, A., Placido, L., Wakker, P.P. (2011). The rich domain of uncertainty: source functions and their experimental implementation. American Economic Review, 101, 695–723.
Abdellaoui, M., Vossmann, F., Weber, M. (2005). Choice-based elicitation and decomposition of decision weights for gains and losses under uncertainty. Management Science, 51, 1384–1399.
Ahn, D.S., Choi, S., Gale, D., Kariv, S. (2013). Estimating ambiguity aversion in a portfolio choice experiment. Quantitative Economics, forthcoming.
Andersen, S., Fountain, J., Harrison, G.W., Hole, A.R., Rutström, E.E. (2012). Inferring beliefs as subjectively imprecise probabilities. Theory and Decision, 73, 161–184.
Anscombe, F.J., & Aumann, R.J. (1963). A definition of subjective probability. Annals of Mathematical Statistics, 34, 199–205.
Baillon, A., & Bleichrodt, H. (2012). Testing ambiguity models through the measurement of probabilities for gains and losses. The Netherlands: Erasmus School of Economics, Erasmus University, Rotterdam.
Barberis, N., Huang, M., Santos, T. (2001). Prospect theory and asset prices. Quarterly Journal of Economics, 66, 1–53.
Barkan, R., & Busemeyer, J.R. (1999). Changing plans: dynamic inconsistency and the effect of experience on the reference point. Psychonomic Bulletin and Review, 6, 547–554.
Bernasconi, M. (1994). Nonlinear preference and two-stage lotteries: theories and evidence. Economic Journal, 104, 54–70.
Budescu, D.V., & Fischer, I. (2001). The same but different: an empirical investigation of the reducibility principle. Journal of Behavioral Decision Making, 14, 187–206.
Camerer, C.F. (1989). An experimental test of several generalized utility theories. Journal of Risk and Uncertainty, 2, 61–104.
Carbone, E., & Hey, J.D. (2001). A test of the principle of optimality. Theory and Decision, 50, 263–281.
Chateauneuf, A. (1991). On the use of capacities in modeling uncertainty aversion and risk aversion. Journal of Mathematical Economics, 20, 343–369.
Chateauneuf, A., Eichberger, J., Grant, S. (2007). Choice under uncertainty with the best and worst in mind: NEO-additive capacities. Journal of Economic Theory, 137, 538–567.
Chateauneuf, A., & Faro, J.H. (2009). Ambiguity through confidence functions. Journal of Mathematical Economics, 45, 535–558.
Chesson, H.W., & Viscusi, W.K. (2003). Commonalities in time and ambiguity aversion for long-term risks. Theory and Decision, 54, 57–71.
Choi, S., Fishman, R., Gale, D., Kariv, S. (2007). Consistency and heterogeneity of individual behavior under uncertainty. American Economic Review, 97, 1921–1938.
Cozman, F.G. (2012). Sets of probability distributions, independence, and convexity. Synthese, 186, 577–600.
Cubitt, P.R., Starmer, C., Sugden, R. (1998). Dynamic choice and the common ratio effect: an experimental investigation. Economic Journal, 108, 1362–1380.
Diecidue, E., Wakker, P.P., Zeelenberg, M. (2007). Eliciting decision weights by adapting de Finetti’s betting-odds method to prospect theory. Journal of Risk and Uncertainty, 34, 179–199.
Dominiak, A., Duersch, P., Lefort, J.-P. (2012). A dynamic Ellsberg urn experiment. Games and Economic Behavior, 75, 625–638.
Drèze, J.H. (1961). Les fondements logiques de l’utilité cardinale et de la probabilité subjective. La Décision (pp. 73–83). Paris: CNRS.
Drèze, J.H. (1987). Essays on economic decision under uncertainty. London: Cambridge University Press.
Ergin, H., & Gul, F. (2009). A theory of subjective compound lotteries. Journal of Economic Theory, 144, 899–929.
Eron, S.A., & Schmeidler, D. (2012). Purely subjective maxmin expected utility. Mimeo.
Etner, J., Jeleva, M., Tallon, J.M. (2012). Decision theory under ambiguity. Journal of Economic Surveys, 26, 234–270.
Fehr, E., & Schmidt, K. (1999). A theory of fairness, competition and cooperation. Quarterly Journal of Economics, 114, 817–868.
Gächter, S., Herrmann, A., Johnson, E.J. (2007). Individual-level loss aversion in riskless and risky choice. Working Paper, University of Nottingham.
Gajdos, T., Hayashi, T., Tallon, J.M., Vergnaud, J.C. (2008). Attitude towards imprecise information. Journal of Economic Theory, 140, 27–65.
Ghirardato, P., Maccheroni, F., Marinacci, M. (2004). Differentiating ambiguity and ambiguity attitude. Journal of Economic Theory, 118, 133–173.
Gilboa, I. (1987). Expected utility with purely subjective non-additive probabilities. Journal of Mathematical Economics, 16, 65–88.
Gilboa, I., & Schmeidler, D. (1989). Maxmin expected utility with a non-unique prior. Journal of Mathematical Economics, 18, 141–153.
Gilboa, I., Postlewaite, A., Schmeidler, D. (2012). Rationality of belief or: Why Savage’s axioms are neither necessary nor sufficient for rationality. Synthese, 187, 11–31.
Glöckner, A., & Pachur, T. (2012). Cognitive models of risky choice: parameter stability and predictive accuracy of prospect theory. Cognition, 123, 21–32.
Goldstein, W.M., & Einhorn, H.J. (1987). Expression theory and the preference reversal phenomena. Psychological Review, 94, 236–254.
Grabisch, M., & Labreuche, C. (2008). A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. 4OR, 6, 1–44.
Harless, D.W., & Camerer, C.F. (1994). The predictive utility of generalized expected utility theories. Econometrica, 62, 1251–1289.
Hayashi, T., & Wada, R. (2010). Choice with imprecise information: an experimental approach. Theory and Decision, 69, 355–373.
Hey, J.D., & Di Cagno, D. (1990). Circles and triangles: an experimental estimation of indifference lines in the Marschak-Machina triangle. Journal of Behavioral Decision Making, 3, 279–306.
Hey, J.D., & Knoll, J.A. (2011). Strategies in dynamic decision making – an experimental investigation of the rationality of decision behaviour. Journal of Economic Psychology, 32, 399–409.
Hey, J.D., & Lotito, G. (2009). Naïve, resolute or sophisticated? A study of dynamic decision making. Journal of Risk and Uncertainty, 38, 1–25.
Hey, J.D., Lotito, G., Maffioletti, A. (2010). The descriptive and predictive adequacy of theories of decision making under uncertainty/ambiguity. Journal of Risk and Uncertainty, 41, 81–111.
Hey, J.D., Lotito, G., Maffioletti, A. (2011). The descriptive and predictive adequacy of theories of decision making under uncertainty/ambiguity. Corrigendum, Mimeo.
Hey, J.D., & Orme, C. (1994). Investigating generalizations of expected utility theory using experimental data. Econometrica, 62, 1291–1326.
Hey, J.D., & Panaccione, L. (2011). Dynamic decision making: what do people do?Journal of Risk and Uncertainty, 42, 85–123.
Hogarth, R.M., & Einhorn, H.J. (1990). Venture theory: a model of decision weights. Management Science, 36, 780–803.
Hsu, M., Bhatt, M., Adolphs, R., Tranel, D., Camerer, C.F. (2005). Neural systems responding to degrees of uncertainty in human decision making. Science, 310, 1680–1683.
Huettel, S.A., Stowe, C.J., Gordon, E.M., Warner, B.T., Platt, M.L. (2006). Neural signatures of economic preferences for risk and ambiguity. Neuron, 49, 765–775.
Ivanov, A. (2011). Attitudes to ambiguity in one-shot normal-form games: an experimental study. Games and Economic Behavior, 71, 366–394.
Jaffray, J.-Y. (1994). Dynamic decision making with belief functions. In R.R. Yager, M. Fedrizzi, J. Kacprzyk (Eds.), Advances in the Dempster-Shafer theory of evidence (pp. 331–352). New York: Wiley.
Kahneman, D., & Tversky, A. (1979). Prospect theory: an analysis of decision under risk. Econometrica, 47, 263–291.
Klibanoff, P., Marinacci, M., Mukerji, S. (2005). A smooth model of decision making under ambiguity. Econometrica, 73, 1849–1892.
Köbberling, V., & Wakker, P.P. (2003). Preference foundations for nonexpected utility: a generalized and simplified technique. Mathematics of Operations Research, 28, 395–423.
Luce, R.D., & Raiffa, H. (1957). Games and decisions. New York: Wiley.
Maafi, H. (2011). Preference reversals under ambiguity. Management Science, 57, 2054–2066.
Maccheroni, F., Marinacci, M., Rustichini, A. (2006). Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica, 74, 1447–1498.
Machina, M.J., & Schmeidler, D. (1992). A more robust definition of subjective probability. Econometrica, 60, 745–780.
Narukawa, Y., & Torra, V. (2011). On distorted probabilities and m-separable fuzzy measures. International Journal of Approximate Reasoning, 52, 1325–1336.
Nascimento, L., & Riella, G. (2010). On the uses of the monotonicity and independence axioms in models of ambiguity aversion. Mathematical Social Sciences, 59, 326–329.
Nau, R.F. (2006). Uncertainty aversion with second-order utilities and probabilities. Management Science, 52, 136–145.
Neilson, W.S. (2010). A simplified axiomatic approach to ambiguity aversion. Journal of Risk and Uncertainty, 41, 113–124.
Prelec, D. (1998). The probability weighting function. Econometrica, 66, 497–527.
Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behaviour and Organization, 3, 323–343.
Rabin, M. (2000). Risk aversion and expected-utility theory: a calibration theorem. Econometrica, 68, 1281–1292.
Rachlin, H., & Green, L. (1972). Commitment, choice and self-control. Journal of the Experimental Analysis of Behavior, 17, 15–22.
Rosenblatt-Wisch, R. (2008). Loss aversion in aggregate macroeconomic time series. European Economic Review, 52, 1140–1159.
Savage, L.J. (1954). The foundations of statistics. New York: Wiley (Second edition 1972, New York: Dover Publications.)
Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57, 571–587.
Siniscalchi, M. (2006). A behavioral characterization of plausible priors. Journal of Economic Theory, 128, 91–135.
Siniscalchi, M. (2009). Vector expected utility and attitudes toward variation. Econometrica, 77, 801–855.
Sipos, J. (1979). Integral with respect to a pre-measure. Math. Slovaca, 29, 141–155.
Sopher, B., & Gigliotti, G. (1993). A test of generalized expected utility theory. Theory and Decision, 35, 75–106.
Starmer, C. (1992). Testing new theories of choice under uncertainty using the common consequence effect. Review of Economic Studies, 59, 813–830.
Strzalecki, T. (2011). Axiomatic foundations of multiplier preferences. Econometrica, 79, 47–73.
Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5, 297–323.
Wakker, P.P. (2010). Prospect theory for risk and ambiguity. Cambridge: Cambridge University Press.
Wald, A. (1950). Statistical decision functions. New York: Wiley.
Wilcox, N.T. (2008). Stochastic models for binary discrete choice under risk: a critical primer and econometric comparison In J.C. Cox, & G.W. Harrison (Eds.), Risk aversion in experiments; Research in experimental economics (Vol. 12, pp. 197–292). Emerald, Bingley.
Yechiam, E., Stout, J.C., Busemeyer, J.R., Rock, S.L., Finn, P.R. (2005). Individual differences in the response to forgone payoffs: an examination of high functioning drug abusers. Journal of Behavioral Decision Making, 18, 97–110.
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We are grateful to John Hey, Gianna Lotito, and Anna Maffioletti for providing us with their data set, their analyses, and many explanations.
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Kothiyal, A., Spinu, V. & Wakker, P.P. An experimental test of prospect theory for predicting choice under ambiguity. J Risk Uncertain 48, 1–17 (2014). https://doi.org/10.1007/s11166-014-9185-0
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DOI: https://doi.org/10.1007/s11166-014-9185-0