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Journal of Risk and Uncertainty

, Volume 8, Issue 2, pp 167–196 | Cite as

Violations of the betweenness axiom and nonlinearity in probability

  • Colin F. Camerer
  • Teck-Hua Ho
Article

Abstract

Betweenness is a weakened form of the independence axiom, stating that a probability mixture of two gambles should lie between them in preference. Betweenness is used in many generalizations of expected utility and in applications to game theory and macroeconomics. Experimental violations of betweenness are widespread. We rule out intransitivity as a source of violations and find that violations are less systematic when mixtures are presented in compound form (because the compound lottery reduction axiom fails empirically). We also fit data from nine studies using Gul's disappointment-aversion theory and two variants of EU, which weight separate or cumulative probabilities nonlinearly. The three theories add only one parameter to EU and fit much better.

Key words

expected utility generalized utility risk-aversion prospect theory Allais paradox betweenness disappointment-aversion 

JEL codes

D81 C91 

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References

  1. 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.Google Scholar
  2. Battalio, Ray C., John H. Kagel, and Jiranyakul Komain. (1990). “Testing Between Alternative Models of Choice Under Uncertainty: Some Initial Results,”Journal of Risk and Uncertainty 3, 25–50.Google Scholar
  3. Becker, Gordon, Morris DeGroot, and Jacob Marschak. (1963). “An Experimental Study of Some Stochastic Models for Wagers,”Behavioral Science 3, 199–202.Google Scholar
  4. Bernasconi, Michele. (1994). “Nonlinear Preference and Two-Stage Lotteries: Theories and Evidence,”The Economic Journal 104, 54–70.Google Scholar
  5. Bordley, Robert. (1992). “An Intransitive Expectations-Based Bayesian Variant of Prospect Theory,”Journal of Risk and Uncertainty 5, 127–144.Google Scholar
  6. Bordley, Robert, and Gordon Hazen. (1991). “SSB and Weighted Linear Utility as Expected Utility with Suspicion,”Management Science 37, 396–408.Google Scholar
  7. Camerer, Colin F. (1989). “An Experimental Test of Several Generalized Utility Theories,”Journal of Risk and Uncertainty 2, 61–104.Google Scholar
  8. Camerer, Colin F. (1992). “Recent Tests of Generalized Utility Theories.” In W. Edwards (ed.),Utility: Measurement, Theories, and Applications. Dordrecht: Kluwer Academic, pp. 207–251.Google Scholar
  9. Chew, Soo-Hong, and Kenneth R. MacCrimmon. (1979). “Alpha-nu Choice Theory: An Axiomatization of Expected Utility,” Faculty of Commerce Working Paper No. 669, University of British Columbia.Google Scholar
  10. Chew, Soo-Hong. (1983). “A Generalization of the Quasilinear Mean With Applications to the Measurement of the Income Inequality and Decision Theory Resolving the Allais Paradox,”Econometrica 53, 1065–1092.Google Scholar
  11. Chew, Soo-Hong, and William S. Waller. (1986). “Empirical Tests of Weighted Utility Theory,”Journal of Mathematical Psychology 30, 55–72.Google Scholar
  12. Chew, Soo-Hong. (1989). “Axiomatic Utility Theories With the Betweenness Property,”Annals of Operations Research 19, 273–298.Google Scholar
  13. Chew, Soo-Hong, and Larry G. Epstein. (1990). “A Unifying Approach to Axiomatic Non-Expected Utility Theories,”Journal of Economic Theory 49, 207–240.Google Scholar
  14. Chew, Soo-Hong, Larry G. Epstein, and Uzi Segal. (1991). “Mixture Symmetry and Quadratic Utility,”Econometrica 59, 139–163.Google Scholar
  15. Chew, Soo-Hong, Edi Kami, and Zvi Safra. (1987). “Risk Aversion in the Theory of Expected Utility with Rank Dependent Probabilities,”Journal of Economic Theory 42, 370–381.Google Scholar
  16. Conlisk, John. (1987). “Verifying the Betweenness Axiom With Questionnaire Evidence, or Not: Take Your Pick,”Economics Letter, 25, 319–322.Google Scholar
  17. Coombs, Clyde, and Lily Huang. (1976). “Tests of the Betweenness Property of Expected Utility,”Journal of Mathematical Psychology 13, 323–337.Google Scholar
  18. Cox, James, Vernon L. Smith, and James M. Walker. (1983). “A Test That Discriminates Between Two Models of the Dutch-first Auction Non-isomorphism,”Journal of Economic Behavior and Organization 4, 205–219.Google Scholar
  19. Crawford, Vincent P. (1988). “Stochastic Choice with Quasiconcave Preference Functions.” University of California-San Diego Department of Economics Working Paper.Google Scholar
  20. Crawford, Vincent P. (1990). “Equilibrium Without Independence,”Journal of Economic Theory 50, 127–154.Google Scholar
  21. Debreu, Gerard. (1952). “A Social Equilibrium Existence Theorem,”Proceedings of the National Academy of Sciences 38, 886–893. Reprinted inMathematical Economics: Twenty Papers of Gerard Debreu. (1983). New York: Cambridge University Press.Google Scholar
  22. Dekel, Eddie. (1986). “An Axiomatic Characterization of Preference Under Uncertainty: Weakening the Independence Axiom,”Journal of Economic Theory 40, 304–318.Google Scholar
  23. Epstein, Larry G., and Stanley E. Zin. (1989). “Substitution, Risk Aversion and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework,”Econometrica 57, 937–969.Google Scholar
  24. Epstein, Larry G., and Stanley E. Zin. (1991a). “Substitution, Risk Aversion and the Temporal Behavior of Consumption and Asset Returns: An Empirical Analysis,”Journal of Political Economy 99, 263–286.Google Scholar
  25. Epstein, Larry G., and Stanley E. Zin. (1991b). “The Independence Axiom and Asset Returns,” Department of Economics, University of Toronto.Google Scholar
  26. Fishburn, Peter. (1982). “Nontransitive Measurable Utility,”Journal of Mathematical Psychology 26, 31–67.Google Scholar
  27. Fishburn, Peter. (1983). “Transitive Measurable Utility,”Journal of Economic Theory 31, 293–317.Google Scholar
  28. Fishburn, Peter. (1988).Nonlinear Preference and Utility Theory. Baltimore: Johns Hopkins University Press.Google Scholar
  29. Gigliotti, Gary, and Barry Sopher. (1993). “A Test of Generalized Expected Utility Theory,”Theory and Decision 35, 75–106.Google Scholar
  30. Green, Jerry, and Bruno Jullien. (1988). “Ordinal Independence in Nonlinear Utility Theory,”Journal of Risk and Uncertainty 1, 355–387. (erratum in 2(1988), 119).Google Scholar
  31. Gul, Faruk, and Otto Lantto. (1990). “Betweenness Satisfying Preferences and Dynamic Choice,”Journal of Economic Theory 52, 162–177.Google Scholar
  32. Gul, Faruk. (1991). “A Theory of Disappointment Aversion,”Econometrica, 59 667–686.Google Scholar
  33. Handa, Jagdish. (1977). “Risk, Probabilities, and a New Theory of Cardinal Utility,”Journal of Political Economy 85, 97–122.Google Scholar
  34. Harless, David. (1992). “Experimental Tests of Quasi-Bayesian Generalizations of Expected Utility Theory,” Virginia Commonwealth University Department of Economics Working Paper.Google Scholar
  35. Harless, David, and Colin Camerer. (in press). “The Predictive Utility of Generalized Utility Theories,”Econometrica.Google Scholar
  36. Hey, John and Chris Orme. (in press). “Investigating Parsimonious Generalizations of Expected Utility Theory Using Experimental Data,”Econometrica.Google Scholar
  37. Kahneman, Daniel, and Amos Tversky. (1979). “Prospect Theory: An Analysis of Decision Under Risk,”Econometrica 47, 263–291.Google Scholar
  38. Karmarkar, Uday S. (1978). “Subjective Weighted Utility: A Descriptive Extension of the Expected Utility Model,”Organizational Behavior and Human Performance 21, 61–72.Google Scholar
  39. Karni, Edi, and Zvi Safra. (1989a). “Ascending Bid Auctions with Behaviorally Consistent Bidders,”Annals of Operations Research 19, 435–446.Google Scholar
  40. Karni, Edi, and Zvi Safra. (1989b). “Dynamic Consistency, Revelations in Auctions and the Structure of Preferences,”Review of Economic Studies 56, 421–434.Google Scholar
  41. Kreps, David M., and Evan L. Porteus. (1978). “Temporal Resolution of Uncertainty and Dynamic Choice Theory,”Econometrica 46, 185–200.Google Scholar
  42. Kreps, David M., and Evan L. Porteus. (1979). “Temporal von Neumann-Morgenstern and Induced Preferences, ”Journal of Economic Theory 20, 81–109.Google Scholar
  43. Lattimore, P.M., J.R. Baker, and A. Dryden Witte. (1992). “The Influence of Probability on Risky Choice,”Journal of Economic Behavior and Organization 17, 377–400.Google Scholar
  44. Loomes, Graham, and Robert Sugden. (1987). “Some Implications of a More General Form of Regret Theory,”Journal of Economic Theory 41, 270–287.Google Scholar
  45. Luce, R. Duncan. (1990). “Rational Versus Plausible Accounting Equivalences in Preference Judgments,”Psychological Science 1, 225–234.Google Scholar
  46. Luce, R. Duncan. (1991). “Rank- and Sign-Dependent Linear Utility Models for Binary Gambles,”Journal of Economic Theory 53, 75–100.Google Scholar
  47. Luce, R. Duncan, and Peter C. Fishburn. (1991). “Rank- and Sign-Dependent Linear Utility Models for Finite First-Order Gambles,”Journal of Risk and Uncertainty 4, 29–59.Google Scholar
  48. Luce, R. Duncan, and Patrick Suppes. (1965). “Preference, Utility, and Subjective Probability.” In R.D. Luce, R.B. Bush, and E. Galanter (eds.),Handbook of Mathematical Psychology, Vol. 3. New York: Wiley, pp. 249–410.Google Scholar
  49. Machina, Mark. (1982). “ ‘Expected Utility’ Analysis without the Independence Axiom,”Econometrica 50, 277–323.Google Scholar
  50. Machina, Mark. (1985). “Stochastic Choice Functions Generated from Deterministic Preferences Over Lotteries,”Economic Journal 95, 575–594.Google Scholar
  51. Machina, Mark. (1987). “Choice Under Uncertainty: Problems Solved and Unsolved,”Journal of Economic Perspectives 1, 121–154.Google Scholar
  52. Machina, Mark. (1989). “Dynamic Consistency and Non-Expected Utility Models of Choice Under Uncertainty,”Journal of Economic Literature 26, 1622–1668.Google Scholar
  53. Marschak, Jacob. (1950). “Rational Behavior, Uncertain Prospects, and Measurable Utility,”Econometrica 18, 111–141.Google Scholar
  54. Prelec, Drazen. (1990). “A ‘Pseudo-endowment’ Effect, and its Implications for Some Recent Nonexpected Utility Models,”Journal of Risk and Uncertainty 3, 247–259.Google Scholar
  55. Quiggin, John. (1982). “A Theory of Anticipated Utility,”Journal of Economic Behavior and Organization 3, 323–343.Google Scholar
  56. Röell, Alissa. (1987). “Risk Aversion in Quiggin and Yaari's Rank-Order Model of Choice under Uncertainty,”Economic Journal 97, 143–159.Google Scholar
  57. Schmeidler, David. (1989). “Subjective Expected Utility without Additivity,”Econometrica, 57 571–587.Google Scholar
  58. Segal, Uzi. (1989). “Anticipated Utility: A Measure Representation Approach,”Annals of Operations Research 19, 359–373.Google Scholar
  59. Segal, Uzi. (1990). “Two-stage Lotteries Without the Reduction Axiom,”Econometrica 58, 349–377.Google Scholar
  60. Tversky, Amos, and Daniel Kahneman. (1986). “Rational Choice and the Framing of Decisions,”Journal of Business 59, S251-S278. Reprinted in R. Hogarth and M. Reder (eds.),Rational Choice: The Contrast between Economics and Psychology. (1987). Chicago: University of Chicago Press.Google Scholar
  61. Tversky, Amos, Paul Slovic, and Daniel Kahneman. (1990). “The Causes of Preference Reversal,”The American Economic Review 80, 204–217.Google Scholar
  62. Tversky, Amos, and Daniel Kahneman. (1992). “Advances in Prospect Theory: Cumulative Representations of Uncertainty,”Journal of Risk and Uncertainty 5, 297–323.Google Scholar
  63. Viscusi, W. Kip. (1989). “Prospective Reference Theory: Toward An Explanation of the Paradoxes,”Journal of Risk and Uncertainty 2, 235–264.Google Scholar
  64. Weber, Robert J. (1982). “The Allais Paradox, Dutch Auctions, and Alpha-utility Theory,” MEDS Department Discussion Paper No. 536, Northwestern University.Google Scholar
  65. Weber, Martin, and Colin Camerer. (1987). “Recent Developments in Modelling Preferences Under Risk,”OR Spektrum 9, 129–151.Google Scholar
  66. Wu, George. (1993). “Editing and Prospect Theory: Ordinal Independence and Outcome Cancellation,” Harvard Business School Managerial Economics Working Paper.Google Scholar
  67. Yaari, Menahem E. (1987). “The Dual Theory of Choice Under Risk,”Econometrica 55, 95–115.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Colin F. Camerer
    • 1
  • Teck-Hua Ho
    • 2
  1. 1.Graduate School of BusinessUniversity of ChicagoUSA
  2. 2.Department of Decision SciencesUniversity of PennsylvaniaUSA

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