Economic Theory

, Volume 58, Issue 1, pp 59–98 | Cite as

Decision making in phantom spaces

  • Yehuda Izhakian
  • Zur Izhakian
Research Article


This paper introduces a new model of decision making under uncertainty. Aiming to provide a more realistic depiction of decision making, it generalizes the von Neumann–Morgenstern theory by including additional tiers of uncertainty. In this model, beliefs about the probabilities of events are ambiguous and their consequential utilities are vague; both are naturally formulated in the phantom space using phantom numbers. The degree of uncertainty, determined by the decision maker’s beliefs, is distinguished from the attitude toward uncertainty, which is drawn from her preferences. Decision making under ambiguity is a particular case of our model in which probabilities are ambiguous, but resulting utilities of events are knowable.


Phantom probability Decision making under uncertainty   Expected utility Imprecise risk Ambiguity Uncertainty Ellsberg paradox 

JEL Classification

C65 D81 D83 


  1. Ahn, D.S.: Ambiguity without a state space. Rev. Econ. Stud. 75, 3–28 (2008)CrossRefGoogle Scholar
  2. Amarante, M.: Ambiguity, measurability and multiple priors. Econ. Theory 26, 995–1006 (2005)CrossRefGoogle Scholar
  3. Arrow, K.J.: Aspects of the Theory of Risk Bearing. Yrjo Jahnssonin Saatio, Helsinki (1965)Google Scholar
  4. Bewley, T.F.: Knightian decision theory and econometric inferences. J. Econ. Theory 146, 1134–1147 (2011)CrossRefGoogle Scholar
  5. Boyle, P.P., Garlappi, L., Uppal, R., Wang, T.: Keynes meets markowitz: the tradeoff between familiarity and diversification. Manag. Sci. 5, 1–20 (2011)Google Scholar
  6. Brenner, M., Izhakian, Y.: Asset Prices and Ambiguity: Empirical Evidence. Stern School of Business, Finance Working Paper Series, FIN-11-010 (2011)Google Scholar
  7. Brenner, M., Izhakian, Y.: Pricing Systematic Ambiguity in Capital Markets. Stern School of Business, Finance Working Paper Series, FIN-12-008 (2012)Google Scholar
  8. Casadesus-Masanell, R., Klibanoff, P., Ozdenoren, E.: Maxmin expected utility over Savage acts with a set of priors. J. Econ. Theory 92, 35–65 (2000)CrossRefGoogle Scholar
  9. Castro, L., Chateauneuf, A.: Ambiguity aversion and trade. Econ. Theory 48, 243–273 (2011)CrossRefGoogle Scholar
  10. Cerreia-Vioglio, S., Ghirardato, P., Maccheroni, F., Marinacci, M., Siniscalchi, M.: Rational preferences under ambiguity. Econ. Theory 48, 341–375 (2011)CrossRefGoogle Scholar
  11. Chateauneuf, A., Eichberger, J., Grant, S.: Choice under uncertainty with the best and worst in mind: neo-additive capacities. J. Econ. Theory 137(1), 538–567 (2007)CrossRefGoogle Scholar
  12. Chen, Z., Epstein, L.: Ambiguity, risk, and asset returns in continuous time. Econometrica 70, 1403–1443 (2002)CrossRefGoogle Scholar
  13. Chew, S.H., Sagi, J.S.: Small worlds: modeling attitudes toward sources of uncertainty. J. Econ. Theory 139, 1–24 (2008)CrossRefGoogle Scholar
  14. Cohen, M., Meilijson, I.: Preference for safety under the choquet model: in search of a characterization. Econ. Theory 1–24 (2013). doi: 10.1007/s00199-013-0762-2
  15. Debreu, G.: Representation of a preference ordering by a numerical function. In: Thrall, R., Coombs, C., Davies, R. (eds.) Decision Processes, pp. 159–175. Wiley, New York (1954)Google Scholar
  16. Dow, J., Werlang, SRdC: Uncertainty aversion, risk aversion, and the optimal choice of portfolio. Econometrica 60(1), 197–204 (1992)CrossRefGoogle Scholar
  17. Einhorn, H.J., Hogarth, R.M.: Decision making under ambiguity. J. Bus. 59, 225–250 (1986)CrossRefGoogle Scholar
  18. Ellsberg, D.: Risk, ambiguity, and the savage axioms. Q. J. Econ. 75(4), 643–669 (1961)CrossRefGoogle Scholar
  19. Epstein, L.G.: A definition of uncertainty aversion. Rev. Econ. Stud. 66(3), 579–608 (1999)CrossRefGoogle Scholar
  20. Epstein, L.G., Schneider, M.: Recursive multiple-priors. J. Econ. Theory 113, 1–31 (2003)CrossRefGoogle Scholar
  21. Epstein, L.G., Zhang, J.: Subjective probabilities on subjectively unambiguous events. Econometrica 69, 265–306 (2001)CrossRefGoogle Scholar
  22. Ergin, H., Gul, F.: A theory of subjective compound lotteries. J. Econ. Theory 144, 899–929 (2009)CrossRefGoogle Scholar
  23. Fishburn, P.: Utility Theory for Decision Making. Wiley, New York (1970)Google Scholar
  24. Gajdos, T., Hayashi, T., Tallon, J.M., Vergnaud, J.C.: Attitude toward imprecise information. J. Econ. Theory 140(1), 27–65 (2008)CrossRefGoogle Scholar
  25. Gajdos, T., Tallon, J.M., Vergnaud, J.C.: Decision making with imprecise probabilistic information. J. Math. Econ. 40, 647–681 (2004)CrossRefGoogle Scholar
  26. Ghirardato, P., Maccheroni, F., Marinacci, M.: Differentiating ambiguity and ambiguity attitude. J. Econ. Theory 118, 133–173 (2004)CrossRefGoogle Scholar
  27. Ghirardato, P., Marinacci, M.: Ambiguity made precise: a comparative foundation. J. Econ. Theory 102(2), 251–289 (2002)CrossRefGoogle Scholar
  28. Gilboa, I.: Expected utility with purely subjective non-additive probabilities. J. Math. Econ. 16, 65–88 (1987)CrossRefGoogle Scholar
  29. Gilboa, I., Schmeidler, D.: Maxmin expected utility with non-unique prior. J. Math. Econ. 18, 141–153 (1989)CrossRefGoogle Scholar
  30. Giraud, R.: Objective Imprecise Probabilistic Information, Second Order Beliefs and Ambiguity Aversion: An Axiomatization. Working Paper Series, University of Franche-Comté (2006)Google Scholar
  31. Giraud, R.: Second order beliefs models of choice under imprecise risk: non-additive second order beliefs vs. nonlinear second order utility. Theor. Econ. (2013, forthcoming)Google Scholar
  32. Giraud, R., Tallon, J.M.: Are beliefs a matter of taste? A case for objective imprecise information. Theory Decis. 71, 23–31 (2011)CrossRefGoogle Scholar
  33. Hansen, L.P., Sargent, T.J.: Robust control and model uncertainty. Am. Econ. Rev. 91, 60–66 (2001)CrossRefGoogle Scholar
  34. Hansen, L.P., Sargent, T.J., Tallarini, T.D.: Robust permanent income and pricing. Rev. Econ. Stud. 66, 873–907 (1999)CrossRefGoogle Scholar
  35. Hayashi, T., Miao, J.: Intertemporal substitution and recursive smooth ambiguity preferences. Theor. Econ. 6, 423–472 (2011)CrossRefGoogle Scholar
  36. Iwaki, H., Osaki, Y.: The dual theory of the smooth ambiguity model. Econ. Theory. 1–15 (2013). doi:  10.1007/s00199-013-0779-6
  37. Izhakian, Y.: A theoretical foundation of ambiguity measurement. Stern School of Business, Economics Working Paper Series, ECN-12-01 (2012a)Google Scholar
  38. Izhakian, Y.: Capital asset pricing under ambiguity. Stern School of Business, Economics Working Paper Series, ECN-12-02 (2012b)Google Scholar
  39. Izhakian, Y.: Does ambiguity diversification pay? Stern School of Business, Economics Working Paper Series, ECN-12-11 (2012c)Google Scholar
  40. Izhakian, Y., Izhakian, Z.: Phantom probability. Preprint at arXiv:0901.0902 (2009)
  41. Jaffray, J.Y.: Linear utility theory for belief functions. Oper. Res. Lett. 8, 107–112 (1989)CrossRefGoogle Scholar
  42. Ju, N., Miao, J.: Ambiguity, learning, and asset returns. Econometrica 80, 559–591 (2012)CrossRefGoogle Scholar
  43. Klibanoff, P., Marinacci, M., Mukerji, S.: A smooth model of decision making under ambiguity. Econometrica 73, 1849–1892 (2005)CrossRefGoogle Scholar
  44. Klibanoff, P., Marinacci, M., Mukerji, S.: Definitions of ambiguous events and the smooth ambiguity model. Econ. Theory 48(2–3), 399–424 (2011)CrossRefGoogle Scholar
  45. Kopylov, I.: A Parametric Model of Hedging Under Ambiguity. Mimeo, UC Irvine. [829] (2006)Google Scholar
  46. Maccheroni, F., Marinacci, M., Ruffino, D.: Alpha as ambiguity: Robust mean-variance portfolio analysis. Econometrica 81, 10751113 (2013)Google Scholar
  47. Maccheroni, F., Marinacci, M., Rustichini, A.: Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74, 1447–1498 (2006)CrossRefGoogle Scholar
  48. Machina, M.J.: Event-separability in the ellsberg urn. Econ. Theory 48, 425–436 (2011)CrossRefGoogle Scholar
  49. Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, Oxford (1995)Google Scholar
  50. Meyer, P.A.: Quantum Probability for Probabilists, 2nd edn. Springer, Berlin (1995)Google Scholar
  51. Mukerji, S.: Understanding the nonadditive probability decision model. Econ. Theory 9, 23–46 (1997)CrossRefGoogle Scholar
  52. Nascimento, L., Riella, G.: Second-order ambiguous beliefs. Econ. Theory 52, 1005–1037 (2013)CrossRefGoogle Scholar
  53. Nau, R.F.: Uncertainty aversion with second-order utilities and probabilities. Manag. Sci. 52, 136–145 (2006)CrossRefGoogle Scholar
  54. Nehring, K.D.: Bernoulli Without Bayes: A Theory of Utility-Sophisticated Preferences Under Ambiguity. Mimeo. University of California, Davis (2007)Google Scholar
  55. Nehring, K.D.: Imprecise probabilistic beliefs as a context for decision-making under ambiguity. J. Econ. Theory 144, 1054–1091 (2009)CrossRefGoogle Scholar
  56. Neilson, W.S.: Ambiguity Aversion: An Axiomatic Approach Using Second Order Probabilities. Mimeo. A&M University, Texas (1993)Google Scholar
  57. Olszewski, W.: Preferences over sets of lotteries. Rev. Econ. Stud. 74, 567–595 (2007)CrossRefGoogle Scholar
  58. Pratt, J.W.: Risk aversion in the small and in the large. Econometrica 32, 122–136 (1964)CrossRefGoogle Scholar
  59. Quiggin, J., Chambers, R.G.: Invariant risk attitudes. J. Econ. Theory 117, 96–118 (2004)CrossRefGoogle Scholar
  60. Rébillé, Y.: Decision making over necessity measures through the choquet integral criterion. Fuzzy Sets Syst. 157, 3025–3039 (2006)CrossRefGoogle Scholar
  61. Roberts, K.W.S.: Interpersonal comparability and social choice theory. Rev. Econ. Stud. 47, 421–439 (1980)CrossRefGoogle Scholar
  62. Rothschild, M., Stiglitz, J.E.: Increasing risk: I. A definition. J. Econ. Theory 2, 225–243 (1970)CrossRefGoogle Scholar
  63. Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)CrossRefGoogle Scholar
  64. Seo, K.: Ambiguity and second-order belief. Econometrica 77, 1575–1605 (2009)CrossRefGoogle Scholar
  65. Siniscalchi, M.: A behavioral characterization of plausible priors. J. Econ. Theory 128, 91–135 (2006)CrossRefGoogle Scholar
  66. Siniscalchi, M.: Vector expected utility and attitudes toward variation. Econometrica 77, 801–855 (2009)CrossRefGoogle Scholar
  67. Stinchcombe, M.: Choice and Games with Ambiguity as Sets of Probabilities. Mimeo, University of Texas, Austin (2003)Google Scholar
  68. Takesaki, M.: Theory of Operator Algebras I, Volume Operator Algebras and Non-commutative Geometry of Encyclopaedia of Mathematical Sciences, vol. 124. Springer, Berlin (2000)Google Scholar
  69. Tversky, A., Kahneman, D.: Advances in prospect theory: cumulative representation of uncertainty. J. Risk Uncertain 5, 297–323 (1992)CrossRefGoogle Scholar
  70. Ui, T.: The ambiguity premium vs. the risk premium under limited market participation. Rev. Finance 15, 245–275 (2011)CrossRefGoogle Scholar
  71. Von-Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)Google Scholar
  72. Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)CrossRefGoogle Scholar
  73. Wang, T.: A Class of Multi-priors Preferences. University of British Columbia, British Columbia (2003)Google Scholar
  74. Wang, Z., Klir, G.J.: Fuzzy Measure Theory. Plenum Press, New York (1991)Google Scholar
  75. Yaari, M.E.: The dual theory of choice under risk. Econometrica 55, 95–115 (1987)CrossRefGoogle Scholar
  76. Youssef, S.: Quantum mechanics as bayesian complex probability theory. Mod. Phys. Lett. 9(28), 2571–2586 (1994)Google Scholar
  77. Zhang, J.: Subjective Ambiguity, Expected Utility and Choquet Expected Utility. Carleton Economic Papers 99–19, Carleton University, Department of Economics (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Economics and Finance, Zicklin School of BusinessBaruch CollegeNew York USA
  2. 2.Department of MathematicsUniversity of BremenBremenGermany

Personalised recommendations