Theory and Decision

, Volume 37, Issue 3, pp 267–309 | Cite as

A challenge to the compound lottery axiom: A two-stage normative structure and comparison to other theories

  • Donald B. Davis
  • M.-Elisabeth Paté-Cornell


This paper examines preferences among uncertain prospects when the decision maker is uneasy about his assignment of subjective probabilities. It proposes a two-stage lottery framework for the analysis of such prospects, where the first stage represents an assessment of the vagueness (ambiguity) in defining the problem's randomness and the second stage represents an assessment of the problem for each hypothesized randomness condition. Standard axioms of rationality are prescribed for each stage, including weak ordering, continuity, and strong independence. The ‘Reduction of Compound Lotteries' axiom is weakened, however, so that the two lottery stages have consistent, but not collapsible, preference structures. The paper derives a representation theorem from the primitive preference axioms, and the theorem asserts that preference-consistent decisions are made as if the decision maker is maximizing a modified expected utility functional. This representation and its implications are compared to alternative decision models. Criteria for assigning the relative empirical power of the alternative models are suggested.


Ambiguity rationality decision compound lottery two-stage lottery 


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  1. 1.
    Allais, M.: 1979, ‘The so-called Allais paradox and rational decisions under uncetainty’, in M. Allais and O. Hagen (Eds.),Expected Utility Hypotheses and the Allais Paradox, Dordrecht, Holland: D. Reidel Publishing Co., pp. 437–681.Google Scholar
  2. 2.
    Allais, M.: 1983, ‘Frequency, probability, and chance’, in B. P. Stigum and F. Wenstop (Eds.),Foundations of Utility and Risk Theory with Applications, Theory and Decision Library, Dordrecht, Holland: D. Reidel Publishing Co., pp. 35–86.Google Scholar
  3. 3.
    Anscombe, F. J. and Aumann, R. J.: 1963, ‘A definition of subjective probability’,Annals of Mathematics and Statistics,34, 199–205.Google Scholar
  4. 4.
    Arrow, K. J.: 1970,Essays in the Theory of Risk-Bearing, Amsterdam: North-Holland.Google Scholar
  5. 5.
    Arrow, K. J.: 1982, ‘Risk perception in psychology and economics’,Economic Inquiry,20, 1–9.Google Scholar
  6. 6.
    Ashton, A. H.: 1984, ‘Ellsberg revisited: the distinction between risk and ambiguity in the real world’, unpublished manuscript, New York University, Graduate School of Business.Google Scholar
  7. 7.
    Becker, J. L. and Sarin, R. K.: 1987, ‘Lottery dependent utility’,Management Science,33, 1367–1382.Google Scholar
  8. 8.
    Becker, S. W. and Brownson, F. O.: 1964, ‘What price ambiguity? Or the role of ambiguity in decision-making’,Journal of Political Economics,72, 62–73.Google Scholar
  9. 9.
    Bell, D. E.: 1982, ‘Regret in decision making under uncertainty’,Operations Research,30, 961–981.Google Scholar
  10. 10.
    Bell, D. E.: 1985, ‘Disappointment in decision making under uncertainty’,Operations Research,33, 1–27.Google Scholar
  11. 11.
    Blyth, C. R.: 1972, ‘Some probability paradoxes in choice from among random alternatives’,Journal of the American Statistical Association,67, 366–373.Google Scholar
  12. 12.
    Boettger, R.: 1986, ‘Skew second-order probabilities, informational uncertainty, and the conjunction effect’, University of California Graduate School of Business Administration, Berkeley, CA.Google Scholar
  13. 13.
    Brewer, K. R. W.: 1963, ‘Decisions under uncertainty: comment’,Quarterly Journal of Economics,77, 159–161.Google Scholar
  14. 14.
    Brewer, K. R. W. and Fellner, W.: 1965; ‘The slanting of subjective probabilities — agreement on some essentials’,Quarterly Journal of Economics,79, 657–663.Google Scholar
  15. 15.
    Chew, S. H.: 1981, ‘A mixture set axiomatization of weighted utility’, Unpublished manuscript, University of Arizona.Google Scholar
  16. 16.
    Chew, S. H.: 1983, ‘A generalization of the quasilinear mean with applications to the measurement of inequality and the decision theory resolving the Allais paradox’,Econometrica,51, 1065–1092.Google Scholar
  17. 17.
    Chew, S. H. and MacCrimmon, K. R.: 1979, ‘Alpha-nu choice theory: a generalization of expected utility theory’, Working Paper No. 669, University of British Columbia, Vancouver.Google Scholar
  18. 18.
    Chew, S. and Waller, W.: 1986, ‘Empirical tests of weighted utility theory’,Journal of Mathematical Psychology,30, 55–72.Google Scholar
  19. 19.
    Chipman, J. S.: 1960, ‘Stochastic choice and subjective probability’, in D. Willner (Ed.),Decisions, Values and Groups, New York: Pergamon Press, pp. 70–95.Google Scholar
  20. 20.
    Curley, S. P. and Yates, J. F.: 1985, ‘The center and range of the probability interval as factors affecting ambiguity preferences’,Organizational Behavior and Human Decision Processes,36, 273–287.Google Scholar
  21. 21.
    Davis, D. B.: 1990,The effect of ambiguity on lottery preferences, Ph.D. Dissertation, Department of Engineering-Economic Systems, Stanford University, Stanford, CA.Google Scholar
  22. 22.
    Debreu, G.: 1954, ‘Representation of a preference ordering by a numerical function’, in R. M. Thrall, C. H. Coombs and R. C. Davis (Eds.),Decision Processes, New York: Wiley, pp. 159–165.Google Scholar
  23. 23.
    de Finetti, B.: 1977, ‘Probabilities of probabilities: a real problem or a misunder-standing?’, in A. Aykac and C. Brumat (Eds.),New Developments in the Application of Bayesian Methods, Amsterdam: North-Holland, pp. 1–10.Google Scholar
  24. 24.
    de Finetti, B.: 1977, ‘Probability: beware of falsifications’, in A. Aykac and C. Brumat (Eds.),New Developments in the Applications of Bayesian Methods, New York: North Holland, pp. 347–379.Google Scholar
  25. 25.
    Dekel, E.: 1986, ‘An axiomatic characterization of preferences under uncertainty: weakening the independence axiom’,Journal of Economic Theory,40, 304–318.Google Scholar
  26. 26.
    Dempster, A. P.: 1968, ‘A generalization of Bayesian inference (with discussion)’,Journal of Royal Statistical Society Series B,30, 205–247.Google Scholar
  27. 27.
    Dreze, J. H.: 1974, ‘Axiomatic theories of choice, cardinal utility and subjective probability: a review’, in J. H. Dreze (Ed.),Allocation Under Uncertainty: Equilibrium and Optimality, New York: Wiley, pp. 3–23.Google Scholar
  28. 28.
    Einhorn, H. J. and Hogarth, R. M.: 1985, ‘Ambiguity and uncertainty in probabilistic inference’,Psychological Review,92, 433–461.Google Scholar
  29. 29.
    Einhorn, H. J. and Hogarth, R. M.: 1986, ‘Decision making under ambiguity’,Journal of Business,59(4), 225–249.Google Scholar
  30. 30.
    Ellsberg, D.: 1961, ‘Risk, ambiguity, and the Savage axioms’,Quarterly Journal of Economics,75, 643–669.Google Scholar
  31. 31.
    Ellsberg, D.: 1963, ‘Risk, ambiguity, and the Savage axioms: a reply’,Quarterly Journal of Economics,77, 336–342.Google Scholar
  32. 32.
    Fellner, W.: 1961, ‘Distortion of subjective probabilities as a reaction to uncertainty’,Quarterly Journal of Economics,75, 670–689.Google Scholar
  33. 33.
    Fellner, W.: 1963, ‘Slanted subjective probabilities and randomization: Reply to Howard Raiffa and K. R. W. Brewer’,Quarterly Journal of Economics,77, 676–690.Google Scholar
  34. 34.
    Fischbeck, P. S.: 1991,Epistemic uncertainty in rational decision making, Ph.D. Dissertation, Department of Industrial Engineering, Stanford University, Stanford, CA.Google Scholar
  35. 35.
    Fishburn, P. C.: 1969, ‘A general theory of subjective probabilities and expected utilities’,Annals of Mathematics and Statistics,40, 1419–1429.Google Scholar
  36. 36.
    Fishburn, P. C.: 1970,Utility Theory for Decision Making, New York: John Wiley & Sons.Google Scholar
  37. 37.
    Fishburn, P. C.: 1982, ‘Nontransitive measurable utility’,Journal of Mathematical Psychology,26, 31–67.Google Scholar
  38. 38.
    Fishburn, P. C.: 1983, ‘Ellsberg revisited: a new look at comparative probability’,Annual Statistics,11, 1047–1059.Google Scholar
  39. 39.
    Fishburn, P. C.: 1983, ‘Transitive measurable utility’,Journal of Economic Theory,31, 293–317.Google Scholar
  40. 40.
    Fishburn, P. C.: 1984, ‘SSB utility theory: an economic perspective’,Mathematical Society of Science,8, 63–94.Google Scholar
  41. 41.
    Fishburn, P. C.: 1984, ‘SSB utility theory and decision making under uncertainty’,Mathematical Society of Science,8, 253–285.Google Scholar
  42. 42.
    Fishburn, P. C.: 1986, ‘Nontransitive measurable utility for decision under uncertainty’, AT&T Bell Laboratories, Murray Hill, NJ.Google Scholar
  43. 43.
    Fishburn, P. C.: 1989, ‘Foundations of decision analysis: along the way’,Management Science,35, 387–405.Google Scholar
  44. 44.
    Gärdenfors, P.: 1979, ‘Forecasts, decisions and uncertain probabilities’,Erkenntnis,14, 159–181.Google Scholar
  45. 45.
    Gärdenfors, P. and Sahlin, N.-E.: 1982, ‘Unreliable probabilities, risk taking, and decisions making’,Synthese,53, 361–386.Google Scholar
  46. 46.
    Gärdenfors, P. and Sahlin, N.-E.: 1982, ‘Reply to Levi’,Synthese,53, 433–438.Google Scholar
  47. 47.
    Gärdenfors, P. and Sahlin, N.-E.: 1983, ‘Decision making with unreliable probabilities’,British Journal of Mathematical Statistical Psychology,36, 240–251.Google Scholar
  48. 48.
    Georgescu-Roegen, N.: 1954, ‘Choice, expectations and measurability’,Quarterly Journal of Economics,LXVIII, 503–534.Google Scholar
  49. 49.
    Goldsmith, R. W. and Sahlin, N.-E.: 1982, ‘The role of second-order probabilities in decision making’, in P. C. Humphreys, O. Svenson and A. Vari (Eds.),Analysing and Aiding Decision Processes, Amsterdam: North-Holland, pp. 455–467.Google Scholar
  50. 50.
    Good, I. J.: 1960, ‘Subjective probability as the measure of a non-measurable set’, inProceedings, International Congress for Logic, Methodology and Philosophy of Science, Stanford, CA: Stanford University Press, pp. 319–329.Google Scholar
  51. 51.
    Grandmont, J.-M.: 1972, ‘Continuity properties of a von Neumann-Morgenstern utility’,Journal of Economic Theory,4, 45–57.Google Scholar
  52. 52.
    Handa, J.: 1977, ‘Risk, probabilities, and a new theory of cardinal utility’,Journal of Political Economics,85, 97–122.Google Scholar
  53. 53.
    Harsanyi, J. C.: 1986, ‘Practical certainty and the acceptance of empirical statements’, in L. Daboni, A. Montesano and M. Lines (Eds.),Recent Developments in the Foundations of Utility and Risk Theory, Dordrecht, Holland: D. Reidel Publishing Co., pp. 27–41.Google Scholar
  54. 54.
    Herstein, I. and Milnor, J.: 1953, ‘An axiomatic approach to measurable utility’,Econometrica,21, 291–297.Google Scholar
  55. 55.
    Hogarth, R. M. and Kunreuther, H.: 1984, ‘Risk, ambiguity, and insurance’, Center for Risk and Decision Processes, WP 84-10-05, The Wharton School, University of Pennsylvania, Philadelphia.Google Scholar
  56. 56.
    Hogarth, R. M. and Kunreuther, H.: 1985, ‘Ambiguity and insurance decisions (with discussion)’,American Economic Association Papers and Proceedings,75, 386–394.Google Scholar
  57. 57.
    Howard, R. A.: 1970, ‘Decision analysis: perspectives on inference, decision, and experimentation’, inProceedings of the IEEE,58(5). Also in Howard, R. A.: 1984,Readings on the Principles and Applications of Decision Analysis, Vol. II. Menlo Park, CA.Google Scholar
  58. 58.
    Kahneman, D. and Tversky, A.: 1979, ‘Prospect theory: an analysis of decision under risk’,Econometrica,47, 263–291.Google Scholar
  59. 59.
    Karmarkar, U. S.: 1978, ‘Subjectively weighted utility: a descriptive extension of the expected utility model’,Organizational Behavior and Human Performance,21, 61–72.Google Scholar
  60. 60.
    Karmarkar, U. S.: 1979, ‘Subjective weighted utility and the Allais paradox’,Organizational Behavior and Human Performance,24, 67–72.Google Scholar
  61. 61.
    Kreps, D. M. and Porteus, E. L.: 1978, ‘Temporal resolution of uncertainty and dynamic choice theory’,Econometrica,46, 185–200.Google Scholar
  62. 62.
    Kreps, D. M. and Porteus, E. L.: 1979, ‘Temporal von Neumann-Morgenstern and induced preferences’,Journal of Economic Theory,20, 81–109.Google Scholar
  63. 63.
    Levi, I.: 1974, ‘On indeterminate probabilities’,Journal of Philosophy,71, 391–418.Google Scholar
  64. 64.
    Levi, I.: 1982, ‘Ignorance, probability, and rational choice’,Synthese,53, 387–417.Google Scholar
  65. 65.
    Loomes, G. and Sugden, R.: 1982, ‘Regret theory: an alternative theory of rational choice under uncertainty’,Economics Journal,92, 805–824.Google Scholar
  66. 66.
    MacCrimmon, K. R.: 1968, ‘Descriptive and normative implications of the decision-theory postulates (with discussion)’, in K. Borch and J. Mossin (Eds.),Risk and Uncertainty: Proceedings of a Conference Held by the International Economic Association, London: Macmillan & Co., pp. 3–32.Google Scholar
  67. 67.
    MacCrimmon, K. R. and Larsson, S.: 1979, ‘Utility theory: axioms versus paradoxes’, in M. Allais and O. Hagen (Eds.),Expected Utility Hypothesis and the Allais Paradox, Dordrecht, Holland: D. Reidel Publishing Co., pp. 333–409.Google Scholar
  68. 68.
    Machina, M. J.: 1983, ‘The economic theory of individual behavior toward risk: theory, evidence, and new directions’, Stanford University Institute for Mathematical Studies in the Social Sciences, Technical Report No. 433.Google Scholar
  69. 69.
    Machina, M. J.: 1985, ‘Stochastic choice functions generated from deterministic preferences over lotteries’,Economics Journal,95, 575–594.Google Scholar
  70. 70.
    Machina, M.: 1987, ‘Choice under uncertainty: problems solved and unsolved’,Journal of Economic Perspective,1, 121–154.Google Scholar
  71. 71.
    Malinvaud, E.: 1952, ‘Note on von Neumann-Morgenstern's strong independence axiom’,Econometrica,20, 679.Google Scholar
  72. 72.
    March, J. G.: 1978, ‘Bounded rationality, ambiguity, and the engineering of choice’,Bell Journal of Economics,9, 587–608.Google Scholar
  73. 73.
    Marschak, J.: 1950, ‘Rational behavior, uncertain prospects, and measurable utility’,Econometrica,18, 111–141. (‘Errata’,Econometrica,18, 312.)Google Scholar
  74. 74.
    Marschak, J.: 1964, ‘Actual vs consistent decision behavior’,Behavioral Science,9, 103–110.Google Scholar
  75. 75.
    Marschak, J.: 1977, ‘Are norms unique: a brief remark’, in A. Aykac and C. Brumat (Eds.),New Developments in the Applications of Bayesian Methods, New York: North Holland, pp. 39–41.Google Scholar
  76. 76.
    Packard, D. J.: 1982, ‘Cyclical preference logic’,Theory and Decision,14, 415–426.Google Scholar
  77. 77.
    Paté-Cornell, M. E.: 1989, Unpublished communication, Stanford, CA.Google Scholar
  78. 78.
    Pratt, J. W., Raiffa, H. and Schlaifer, R.: 1964, ‘The foundations of decisions under uncertainty: an elementary exposition’,Journal of American Statistical Association,59, 353–375.Google Scholar
  79. 79.
    Quiggin, J.: 1982, ‘A theory of anticipated utility’,Journal of Economic Behavior and Organization Statistical Association,3, 323–343.Google Scholar
  80. 80.
    Raiffa, H.: 1961, ‘Risk, ambiguity, and the Savage axioms: comment’,Quarterly Journal of Economics,75, 690–694.Google Scholar
  81. 81.
    Ramsey, F. P.: 1931, ‘Truth and probability’, inThe Foundations of Mathematics and Other Logical Essays by F.P. Ramsey, New York: Harcourt, Brace, pp. 156–198. Reprinted in H. E. Kyburg and H. E. Smokler (Eds.): 1980,Studies in Subjective Probability, New York: Krieger, 23–52.Google Scholar
  82. 82.
    Roberts, H.: 1963, ‘Risk, ambiguity, and the Savage axioms: comment’,Quarterly Journal of Economics,77, 327–336.Google Scholar
  83. 83.
    Sahlin, N. E.: 1983, ‘On second order probabilities and the notion of epistemic risk’, in B. P. Stigum and F. Wenstop (Eds.),Foundations of Utility and Risk Theory with Applications, Theory and Decision Library, Dordrecht, Holland: D. Reidel.Google Scholar
  84. 84.
    Savage, L. J.: 1954,The Foundations of Statistics, New York: John Wiley & Sons. (Revised and enlarged version of this work is published by Dover Publications, New York, 1972.)Google Scholar
  85. 85.
    Savage, L. J.: 1964, ‘The foundations of statistics reconsidered’, inProceedings of the Fourth Berkeley Symposium, Berkeley: University of California Press, pp. 575–586.Google Scholar
  86. 86.
    Schmeidler, D.: 1984, ‘Subjective probability and expected utility without additivity’, Preprint Serie No. 84, University of Minnesota, Minneapolis.Google Scholar
  87. 87.
    Segal, U.: 1987, ‘The Ellsberg paradox and risk aversion: anticipated utility approach’,International Economics Review,28, 175–202.Google Scholar
  88. 88.
    Segal, U.: 1987, ‘Some remarks on Quiggin's anticipated utility, with response by Quiggin’,Journal of Economic Behavior and Organization Statistical Association,8, 145–154.Google Scholar
  89. 89.
    Sen, A.: 1986, ‘Rationality and uncertainty’, in L. Daboni, A. Montesano and M. Lines (Eds.),Recent Developments in the Foundations of Utility and Risk Theory, Dordrecht, Holland: D. Reidel Publishing Co., pp 3–25.Google Scholar
  90. 90.
    Shafer, G.: 1978, ‘Non-additive probabilities in the work of Bernoulli and Lambert’, inArchive for History of Exact Science, University of Kansas, Lawrence, pp. 309–370.Google Scholar
  91. 91.
    Sherman, R.: 1974, ‘The psychological difference between ambiguity and risk’,Quarterly Journal of Economics,88, 166–169.Google Scholar
  92. 92.
    Simon, H. A.: 1979, ‘Rational decision making in business organizations’,American Economics Review,69, 493–513.Google Scholar
  93. 93.
    Simon, H. A.: 1986, ‘Rationality in psychology and economics’,Journal of Business,59(4), S209–224.Google Scholar
  94. 94.
    Slovic, P. and Tversky, A.: 1974, ‘Who accepts Savage's axiom?’,Behavioral Science,19, 368–373.Google Scholar
  95. 95.
    Smith, V. L.: 1969, ‘Measuring nonmonetary utilities in uncertain choices: the Ellsberg urn’,Quarterly Journal of Economics,83, 324–329.Google Scholar
  96. 96.
    Sugden, R.: 1985, ‘Why be consistent?’,Economica,52, 167–183.Google Scholar
  97. 97.
    Thaler, R.: 1980, ‘Toward a positive theory of consumer choice’,Journal of Economic Behavior and Organization Statistical Association,1, 39–60.Google Scholar
  98. 98.
    Tversky, A. and Kahneman, D.: 1986, ‘Rational choice and the framing of decisions’,Journal of Business,59(4), S251–278.Google Scholar
  99. 99.
    von Neumann, J. and Morgenstern, O.: 1947,Theory of Games and Economic Behavior, Second Edition. Princeton: Princeton University Press. First Edition, 1944; Third Edition, 1953.Google Scholar
  100. 100.
    Walley, P.: 1987,Rationality and Vagueness, London: Chapman & Hall.Google Scholar
  101. 101.
    Yates, J. and Zukowski, L.: 1976, ‘Characterization of ambiguity in decision making’,Behavioral Science,21, 19–25.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Donald B. Davis
    • 1
  • M.-Elisabeth Paté-Cornell
    • 2
  1. 1.Lockheed Missiles and Space Co.SunnyvaleUSA
  2. 2.Department of Industrial Engineering and Engineering ManagementStanford UniversityStanfordUSA

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