Journal of Risk and Uncertainty

, Volume 32, Issue 2, pp 101–130 | Cite as

Cumulative prospect theory's functional menagerie

  • Henry P. StottEmail author


Many different functional forms have been suggested for both the value function and probability weighting function of Cumulative Prospect Theory (Tversky and Kahneman, 1992). There are also many stochastic choice functions available. Since these three components only make predictions when considered in combination, this paper examines the complete pattern of 256 model variants that can be constructed from twenty functions. All these variants are fit to experimental data and their explanatory power assessed. Significant interaction effects are observed. The best model has a power value function, a risky weighting function due to Prelec (1998), and a Logit function.


Cumulative prospect theory Stochastic choice 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abdellaoui, Mohammed. (2000). “Parameter Free Elicitation of Utilities and Probability Weighting Functions,” Management Science 46, 1497–1512.CrossRefGoogle Scholar
  2. Abdellaoui, Mohammed, Frank Vossmann and Martin Weber. (2003). “Choice-Based Elicitation and Decomposition of Decision Weights for Gains and Losses Under Uncertainty,” Manuscript obtained from authors.Google Scholar
  3. Akaike, Hirotugu (1973). “Information Theory and an Extension of the Maximum Likelihood Principle,” in: B. N. Petrox and F. Caski (Eds.), Second International Symposium on Information Theory. Budapest, Hungary: Akademiai Kiado pp. 267–281.Google Scholar
  4. Allais, Maurice (1953). “Le Comportement de l'homme Rationel Devant le Risque, Critique des Postulates et Axiomes de l'ecole Americaine,” Econometrica 21, 503–546. [In French].Google Scholar
  5. Ballinger, T. Parker and Nathaniel T. Wilcox (1997). “Decisions, Error and Heterogeneity,” Economic Journal 107, 1090–1105.CrossRefGoogle Scholar
  6. Barberis, Nicholas and Ming Huang. (2005). “Stocks as Lotteries: The Implications of Probability Weighting for Securities Prices,” Manuscript Obtained from Authors.Google Scholar
  7. Becker, Gordon M, Maurice H. De Groot and Jacob Marschak. (1963a). “An Experimental Study of Some Stochastic Models for Wagers,” Behavioural Science 3, 199–202.Google Scholar
  8. Becker, Gordon M., Maurice H. De Groot and Jacob Marschak (1963b). Stochastic Models of Choice Behaviour. Behavioural Science 8, 41–55.CrossRefGoogle Scholar
  9. Bell, David E. (1995a). “Risk, Return and Utility,” Management Science 41(1), 23–30Google Scholar
  10. Bell, David E. (1995b). “A Contextual Uncertainty Condition for Behavior Under Risk,” Management Science 41(7), 1145–1150Google Scholar
  11. Bell, David E. and Peter C. Fishburn. (2000). “Utility Functions for Wealth, Journal of Risk and Uncertainty 20, 5–44.CrossRefGoogle Scholar
  12. Bell, David E. and Peter C. Fishburn (2001). “Strong One-Switch Utility,” Management Science 47(4), 601–604CrossRefGoogle Scholar
  13. Birnbaum, Michael H. and Alfredo Chavez. (1997). “Tests of Theories of Decision Making: Violations of Branch Independence and Distribution Independence,” Organizational Behavior and Human Decision Processes 71, 161–194.CrossRefGoogle Scholar
  14. Birnbaum, Michael H, Jamie N. Patton and Melissa K. Lott. (1999). “Evidence Against Rank Dependent Utility Theories: Tests of Cumulative Independence, Interval Dependence, Stochastic Dominance and Transitivity,” Organizational Behavior and Human Decision Processes 77, 44–83.CrossRefGoogle Scholar
  15. Bleichrodt, Hans and Jose, Luis Pinto. (2000). “A Parameter-Free Elicitation of the Probability Weighting Function in Medical Decision Analysis,” Management Science 46(11), 1485–1496CrossRefGoogle Scholar
  16. Blondel, Serge. (2002). “Testing Theories of Choice Under Risk: Estimation of Individual Functionals,” Journal of Risk and Uncertainty 24(3), 251–265CrossRefGoogle Scholar
  17. Brandstätter, Eduard, Anton Kühberger and Friedrich Schneider. (2002). “A Cognitive-Emotional Account of the Shape of the Probability Weighting Function,” Journal of Behavioral Decision Making 15 , 79–100.CrossRefGoogle Scholar
  18. Burnham, Kenneth P. and David R. Anderson. (2002). Model Selection and Multimodel Inference. New York: Springer-Verlag.Google Scholar
  19. Buschena, David and David Zilberman (2000). “Generalized Expected Utility, Heteroscedastic Error, and Path Dependence in Risky Choice,” Journal of Risk and Uncertainty 20, 67–88.CrossRefGoogle Scholar
  20. Camerer, Colin F. (1989). “An Experimental Test of Several Generalized Utility Theories,” Journal of Risk and Uncertainty 2, 61–104.CrossRefGoogle Scholar
  21. Camerer, Colin F. and Robin M. Hogarth (1999). “The Effects of Financial Incentives in Experiments: A Review and Capital-Labor-Production Framework,” Journal of Risk and Uncertainty 19, 7–42.CrossRefGoogle Scholar
  22. Camerer, Colin F. and Teck-Hua Ho. (1994). “Violations of the Betweeness Axiom and Nonlineraity in Probability,” Journal of Risk and Uncertainty 8, 167–196.CrossRefGoogle Scholar
  23. Carbone, Enrica. (1997). “Investigation of Stochastic Preference Theory Using Experimental Data,” Economic Letters 57(3), 305–311CrossRefGoogle Scholar
  24. Carbone, Enrica and John D. Hey. (1994). “Discriminating Between Preference Functionals: A Preliminary Monte Carlo Study,” Journal of Risk and Uncertainty 8, 223–242.CrossRefGoogle Scholar
  25. Carbone, Enrica and John D. Hey. (1995). “A Comparison of the Estimates of EU and Non-EU Preference Functionals Using Data from Pairwise Choice and Complete Ranking Experiments,” Geneva Papers on Risk and Insurance Theory 20, 111–133.CrossRefGoogle Scholar
  26. Carbone, Enrica and John D. Hey. (2000). “Which Error Story is Best,” Journal of Risk and Uncertainty 20, 161–176.CrossRefGoogle Scholar
  27. Chechile, Richard A. and Alan D. J. Cooke. (1997). “An Experimental Test of a General Class of Utility Models: Evidence for Context Dependency,” Journal of Risk and Uncertainty 14, 75–93.CrossRefGoogle Scholar
  28. Debreu, Gerard. (1958). “Stochastic Choice and Cardinal Utility,” Econometrica 26(3), 440–444Google Scholar
  29. Doctor, Jason, N. et al. (2004). “A New and More Robust Test of QALYs,” Journal of Health Economics 23, 353–367.CrossRefGoogle Scholar
  30. Fennema, Hein and Marcel van Assen. (1998). “Measuring the Utility of Losses by Means of the Trade-Off Method,” Journal of Risk and Uncertainty 17, 277–295.CrossRefGoogle Scholar
  31. Fennema, Hein and Peter, Wakker. (1997). “Original and Cumulative Prospect Theory: A Discussion of Empirical Differences,” Journal of Behavioral Decision Making 10, 53–64.CrossRefGoogle Scholar
  32. Fishburn, Peter C. and Gary A. Kochenberger. (1979). “Two Piece Von Neumann-Morgernstern Utility Functions,” Decision Sciences 10(4), 503–518Google Scholar
  33. Goldstein, William M. and Hillel J. Einhorn. (1987). “Expression Theory and the Preference Reversal Phenomena,” Psychological Review 94, 236–254.CrossRefGoogle Scholar
  34. Gonzalez, Richard and George Wu. (1999). “On the Shape of Probability Weighting Function,” Cognitive Psychology 38, 129–166.CrossRefGoogle Scholar
  35. Guthrie, Chris. (2003). “Prospect Theory, Risk Preference, and the Law,” Northwestern University Law Review 97(3), 1115–1163Google Scholar
  36. Grünwald, Peter. (2000). “Model Selection Based on Minimum Description Length.” Journal of Mathematical Psychology 44, 133–152.CrossRefGoogle Scholar
  37. Harless, David W. and Colin F. Camerer. (1994). “The Predictive Utility of Generalized Expected Utility Theories,” Econometrica 62(6), 1251–1290Google Scholar
  38. Herrnstein, Richard J. (1997). The Matching Law: Papers in Psychology and Economics. H. Rachlin and D. I. Laibson (Eds.). New York: Russell Sage Foundation.Google Scholar
  39. Hertwig, Raph and Andreas Ortmann. (2001). “Experimental Practices in Economics: A Methodological Challenge to Psychologists?,” Behavioral and Brain Sciences 24, 383–451.Google Scholar
  40. Hey, John D. (1995). “Experimental Investigations of Errors in Decision Making Under Risk,” European Economic Review 39, 633–640.CrossRefGoogle Scholar
  41. Hey, John D. and Chris Orme. (1994). “Investigating Generalizations of the Expected Utility Theory Using Experimental Data,” Econometrica 62(6), 1291–1326Google Scholar
  42. Hoeting, Jennifer A. et al. (1999). “Bayesian Model Averaging: A Tutorial,” Statistical Science 14, 382–401.CrossRefGoogle Scholar
  43. Ingersoll Jonathan E, Jr. (1987). Theory of Financial Decision Making. Savage, MD: Rowman and Littlefield.Google Scholar
  44. Isaac, R. Mark and Duncan James. (2000). “Just Who Are You Calling Risk Averse?,” Journal of Risk and Uncertainty 20(2), 177–187CrossRefGoogle Scholar
  45. Kachelmeier, Steven J. and Mohamed Shehata. (1992). “Examining Risk Preferences Under High Monetary Incentives: Experimental Evidence from the People's Republic of China,” American Economic Review 82(5), 1120–1141Google Scholar
  46. Karmarkar, Uday. (1978). “Subjectively Weighted Utility: A Descriptive Extension of the Expected Utility Model,” Organizational Behavior and Human Performance 21, 61–72.CrossRefGoogle Scholar
  47. Karmarkar, Uday. (1979). “Subjectively Weighted Utility and Allais Paradox,” Organizational Behavior and Human Performance 24, 67–72.CrossRefGoogle Scholar
  48. Lattimore, Pamela K, Joanna R. Baker and Ann D. Witte. (1992). “The Influence of Probability on Risky Choice,” Journal of Economic Behaviour and Organization 17, 377–400.CrossRefGoogle Scholar
  49. Lichtenstein, Sarah and Paul Slovic. (1971). “Reversals of Preferences Between Bids and Choices in Gambling Decisions,” Journal of Experimental Psychology 89, 46–55.Google Scholar
  50. Loomes, Graham and Robert Sugden. (1995). “Incorporating a Stochastic Element into Decision Theories,” European Economic Review 39, 641–648.CrossRefGoogle Scholar
  51. Loomes, Graham and Robert Sugden. (1998). “Testing Different Stochastic Specifications of Risky Choice,” Economica 65, 581–598.CrossRefGoogle Scholar
  52. Loomes, Graham, Peter Moffat and Robert Sugden. (2002). “A Microeconometric Test of Alternative Stochastic Theories of Risk Choice,” Journal of Risk and Uncertainty 24, 103–130.CrossRefGoogle Scholar
  53. Luce, R. Duncan. (1959). Individual Choice Behavior. New York: John Wiley & Sons.Google Scholar
  54. Luce, R. Duncan. (1991). “Rank and Sign Dependent Linear Utility Models for Binary Gambles,” Journal of Economic Theory 53, 75–100.CrossRefGoogle Scholar
  55. Luce, R. Duncan. (2000). Utility of Gains and Losses: Measurement-Theoretical and Experimental Approaches. Mahwah, NJ: Erlbaum.Google Scholar
  56. Luce, R. Duncan (2001). “Reduction Invariance and Prelec's Weighting Functions,” Journal of Mathematical Psychology 45(1), 167–179CrossRefGoogle Scholar
  57. Luce, R. Duncan and Peter C. Fishburn. (1991). “Rank and Sign Dependent Linear Utility Models of Finite First-Order Gambles,” Journal of Risk and Uncertainty 4, 29–59.CrossRefGoogle Scholar
  58. Luce, R. Duncan and Peter C. Fishburn. (1995). “A Note on Deriving Rank-Dependent Utility Using Additive Joint Receipts,” Journal of Risk and Uncertainty 11, 5–16.CrossRefGoogle Scholar
  59. Luce, R. Duncan, Barbara, A. Mellers and Shi-jie, Chang. (1993). “Is Choice the Correct Primitive? On Using Certainty Equivalents and Reference Levels to Predict Choices Amoung Gambles,” Journal of Risk and Uncertainty 6, 115–143.CrossRefGoogle Scholar
  60. Mellers, Barbara A. et al. (1992). “Preferences, Prices, and Ratings in Risky Decision Making,” Journal of Experimental Psychology: Human Perception and Performance 18, 347–361.CrossRefGoogle Scholar
  61. Mosteller, Frederick and Philip Nogee. (1951). “An Experimental Measurement of Utility,” Journal of Political Economy 59, 371–404.CrossRefGoogle Scholar
  62. Myung, In Jae (2000). “Importance of Complexity in Model Selection.” Journal of Mathematical Psychology 44(1), 190–204CrossRefGoogle Scholar
  63. Myung, In Jae and Mark A. Pitt (1997). “Applying Occam's Razor in Modeling Cognition: A Bayesian Approach,” Psychonomic Bulletin and Review 4(1), 79–95Google Scholar
  64. Myung, In Jae, Cheongtag Kim, and Mark A. Pitt (2000). “Towards an Explanation of the Power Law Artifact: Insights From Response Surface Analysis,” Memory and Cognition 28(5), 832–840Google Scholar
  65. Prelec, Drazen. (1998). “The Probability Weighting Function,” Econometrica 66(3), 497–528Google Scholar
  66. Quiggin, John (1982). “A Theory of Anticipated Utility.” Journal of Economic Behaviour and Organization 3, 323–343.CrossRefGoogle Scholar
  67. Rich, Elaine and Kevin Knight. (1991). Artificial Intelligence. New York, NY: McGraw-Hill.Google Scholar
  68. Segal, Uzi. (1989). “Anticipated Utility: A Measure Representation Approach,” Annals of Operations Research 19, 359–373.CrossRefGoogle Scholar
  69. Slovic, Paul. (1972). “Psychological Study of Human Judgement: Implications for Investment Decision Making,” Journal of Finance 27, 779–799.Google Scholar
  70. Smith, Vernon L. and James M. Walker. (1993). “Monetary Rewards and Decision Cost in Experimental Economics,” Economic Inquiry 31(2), 245–261CrossRefGoogle Scholar
  71. Sneddon, Robert and R. Duncan Luce. (2001). “Empirical Comparisons of Bilinear and Nonbilinear Utility Theories,” Organizational Behavior and Human Decision Processes 84(1), 71–94CrossRefGoogle Scholar
  72. Starmer, Chris. (2000). “Development in Non-Expected Utility Theory: The Hunt for a Descriptive Theory of Choice Under Risk,” Journal of Economic Literature 38, 332–382.Google Scholar
  73. Starmer, Chris and Robert Sugden. (1989a). “Violations of the Independence Axiom in Common Ratio Problems: An Experimental Test,” Annals of Operations Research 19, 79–102.CrossRefGoogle Scholar
  74. Starmer, Chris and Robert Sugden. (1989b). “Probability and Juxtaposition Effects: An Experimental Investigation of the Common Ratio Effect,” Journal of Risk and Uncertainty 2, 159–178.CrossRefGoogle Scholar
  75. Stevens, S. Smitty. (1957). “On the Psychophysical Law,” Psychological Review 64, 153–181.Google Scholar
  76. Stewart, Neil et al. (2002). “Prospect Relativity: How Choice Options Influence Decision Under Risk,” Journal of Experimental Psychology: General 132(1), 23–46CrossRefGoogle Scholar
  77. Tversky, Amos and Craig R. Fox (1995). “Weighing Risk and Uncertainty.” Psychological Review 102(2), 269–283CrossRefGoogle Scholar
  78. Tversky, Amos and Daniel Kahneman. (1992). “Advances in Prospect Theory: Cumulative Representation of Uncertainty,” Journal of Risk and Uncertainty 5, 297–323.CrossRefGoogle Scholar
  79. Von Neumann John and Oskar Morgenstern. (1944). Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press.Google Scholar
  80. Wakker, Peter and Amos Tversky. (1993). “An Axiomatization of Cumulative Prospect Theory,” Journal of Risk and Uncertainty 7, 147–176.CrossRefGoogle Scholar
  81. Wakker, Peter, Ido Erev and Elke Weber. (1994). “Comonotonic Independence: The Critical Test Between Classical and Rank-Dependent Utility Theories,” Journal of Risk and Uncertainty 9, 195–230.CrossRefGoogle Scholar
  82. Weber, Elke U. and Britt Kirsner. (1997). “Reasons for Rank-Dependent Utility Evaluation,” Journal of Risk and Uncertainty 14, 41–61.CrossRefGoogle Scholar
  83. Wu, George and Richard Gonzalez. (1996). “Curvature of the Probability Weighting Function,” Management Science 42(12), 1676–1690CrossRefGoogle Scholar
  84. Yaari, Menahem E. (1987). “The Dual Theory of Choice Under Risk,” Econometrica 55(1), 95–115Google Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Department of PsychologyUniversity of WarwickCoventryEngland

Personalised recommendations