Theory and Decision

, Volume 58, Issue 1, pp 3–76 | Cite as

The Likelihood Method for Decision under Uncertainty

Article

Abstract

This paper introduces the likelihood method for decision under uncertainty. The method allows the quantitative determination of subjective beliefs or decision weights without invoking additional separability conditions, and generalizes the Savage–de Finetti betting method. It is applied to a number of popular models for decision under uncertainty. In each case, preference foundations result from the requirement that no inconsistencies are to be revealed by the version of the likelihood method appropriate for the model considered. A unified treatment of subjective decision weights results for most of the decision models popular today. Savage’s derivation of subjective expected utility can now be generalized and simplified. In addition to the intuitive and empirical contributions of the likelihood method, we provide a number of technical contributions: We generalize Savage’s nonatomiticy condition (“P6”) and his assumption of (sigma) algebras of events, while fully maintaining his flexibility regarding the outcome set. Derivations of Choquet expected utility and probabilistic sophistication are generalized and simplified similarly. The likelihood method also reveals a common intuition underlying many other conditions for uncertainty, such as definitions of ambiguity aversion and pessimism.

Keywords

Likelihood method Subjective expected utility Probabilistic sophistication Choquet expected utility Rank dependence Ambiguity Belief measurement 

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References

  1. Abdellaoui, Mohammed 2000Parameter-free elicitation of utilities and probability weighting functionsManagement Science4614971512CrossRefGoogle Scholar
  2. Abdellaoui, Mohammed 2002A genuine rank-dependent generalization of the von Neumann–Morgenstern expected utility theoremEconometrica70717736CrossRefGoogle Scholar
  3. Anscombe Francis, J., Aumann Robert, J. 1963A definition of subjective probabilityAnnals of Mathematical Statistics34199205Google Scholar
  4. Blackorby, Charles, Davidson, Russell, Donaldson, David 1977A homiletic exposition of the expected utility hypothesisEconomica44351358Google Scholar
  5. Casadesus-Masanell, Ramon, Klibanoff, Peter, Ozdenoren, Emre 2000Maxmin expected utility over Savage acts with a set of priorsJournal of Economic Theory923565CrossRefGoogle Scholar
  6. Chateauneuf, Alain 1991On the use of capacities in modeling uncertainty aversion and risk aversionJournal of Mathematical Economics20343369CrossRefGoogle Scholar
  7. Chateauneuf, Alain 1999axioms and rank-dependent expected utility theory for arbitrary consequencesJournal of Mathematical Economics322145CrossRefGoogle Scholar
  8. Chew Soo, Hong, Karni, Edi 1994Choquet expected utility with a finite state space: commutativity and act-independenceJournal of Economic Theory62469479CrossRefGoogle Scholar
  9. Chew Soo, Hong, Sagi, Jacob 2004"Event Exchangeability: Small Worlds Probabilistic Sophistication without Continuity or Monotonicity"Haas School of Business, University of CaliforniaBerkeley, CAGoogle Scholar
  10. Cohen, Michèle, Gilboa, Itzhak, Jaffray, Jean-Yves, Schmeidler, David 2000An experimental study of updating ambiguous beliefsRisk, Decision, and Policy5123133Google Scholar
  11. Davidson, Donald, Suppes, Patrick 1956A finitistic axiomatization of utility and subjective probabilityEconometrica24264275Google Scholar
  12. de Finetti, Bruno (1937), La prévision: ses lois logiques, ses sources subjectives, Annales de l’Institut Henri Poincaré 7, 1–68. Translated into English by Henry E. Kyburg Jr., Foresight: Its Logical Laws, its Subjective Sources, in Henry E. Kyburg Jr. and Howard E. Smokler (1964, eds.). Studies in Subjective Probability, Wiley, New York; pp. 93–158, 2nd edition 1980, pp. 53–118, Krieger, New York.Google Scholar
  13. Finetti, Bruno 1974Theory of Probability. Vol. IWileyNew YorkGoogle Scholar
  14. Ellsberg, Daniel 1961Risk, ambiguity and the Savage axiomsQuarterly Journal of Economics75643669Google Scholar
  15. Epstein Larry, G., Breton, Michel 1993Dynamically consistent beliefs must be BayesianJournal of Economic Theory61122CrossRefGoogle Scholar
  16. Epstein Larry, G., Zhang, Jiangkang 2001Subjective probabilities on subjectively unambiguous eventsEconometrica69265306CrossRefGoogle Scholar
  17. Ergin, Haluk, Gul, Faruk 2004A Subjective Theory of Compound LotteriesMITCambridge, MAGoogle Scholar
  18. Fishburn Peter, C. 1986The axioms of subjective probabilityStatistical Science1335358Google Scholar
  19. Fishburn Peter, C. 1988Nonlinear Preference and Utility TheoryJohns Hopkins University PressBaltimore, MDGoogle Scholar
  20. Fox Craig, R., Tversky, Amos 1995Ambiguity aversion and comparative ignoranceQuarterly Journal of Economics110585603Google Scholar
  21. Ghirardato, Paolo, Marinacci, Massimo 2001Risk, ambiguity, and the separation of utility and beliefsMathematics of Operations Research26864890CrossRefGoogle Scholar
  22. Ghirardato, Paolo, Maccheroni, Fabio, Marinacci, Massimo, Siniscalchi, Marciano 2003A subjective spin on roulette wheelsEconometrica7118971908CrossRefGoogle Scholar
  23. Gilboa, Itzhak 1987Expected utility with purely subjective non-additive probabilitiesJournal of Mathematical Economics166588CrossRefGoogle Scholar
  24. Gilboa, Itzhak eds. 2004Uncertainty in Economic Theory: Essays in Honor of David Schmeidler’s 65th BirthdayRoutledgeLondonGoogle Scholar
  25. Gilboa, Itzhak, Schmeidler, David 1989Maxmin expected utility with a non-unique priorJournal of Mathematical Economics18141153CrossRefGoogle Scholar
  26. Gilboa, Itzhak, Schmeidler, David 2004Subjective distributionsTheory and Decision56345357CrossRefGoogle Scholar
  27. Gonzalez, Richard, Wu, George 1999On the shape of the probability weighting functionCognitive Psychology38129166CrossRefPubMedGoogle Scholar
  28. Grant, Simon 1995Subjective probability without eventwise montonicity: or: how Machina’s mom may also be probabilistically sophisticatedEconometrica63159189Google Scholar
  29. Grodal, Birgit (1978), Some further results on integral representation of utility functions, Institute of Economics, University of Copenhagen, Copenhagen. Appeared in rewritten form in Chapter 12 of Karl Vind (2003). Independence, Additivity, Uncertainty. With contributions by B. Grodal. Springer, Berlin.Google Scholar
  30. Gul Faruk (1992), Savage’s theorem with a finite number of states. Journal of Economic Theory 57, 99–110. (Erratum, 1993. Journal of Economic Theory 61, 184).Google Scholar
  31. Karni, Edi 1993Subjective expected utility with state-dependent preferencesJournal of Economic Theory60428438CrossRefGoogle Scholar
  32. Keynes John, Maynard 1921A Treatise on ProbabilitysecondMcMillanLondon1948Google Scholar
  33. Knight Frank, H. 1921Risk, Uncertainty, and ProfitHoughton MifflinNew YorkGoogle Scholar
  34. Köbberling, Veronika, Wakker Peter, P. 2003Preference foundations for nonexpected utility: a generalized and simplified techniqueMathematics of Operations Research28395423CrossRefGoogle Scholar
  35. Kopylov, Igor 2004Subjective Probabilities on “Small” Domains, Institute for Mathematical Behavioral SciencesUniversity of CaliforniaIrvine CAGoogle Scholar
  36. Krantz David, H., Luce , Duncan R., Suppes, Patrick, Tversky, Amos 1971Foundations of Measurement, Vol. I (Additive and Polynomial Representations)Academic PressNew YorkGoogle Scholar
  37. Luce, Duncan R. 1967Sufficient conditions for the existence of a finitely additive probability measureThe Annals of Mathematical Statistics38780786Google Scholar
  38. Luce, Duncan R. 2000Utility of Gains and Losses: Measurement-Theoretical and Experimental ApproachesLawrence Erlbaum PublishersLondonGoogle Scholar
  39. MacCrimmon Kenneth, R., Larsson, Stig 1979

    Utility Theory: Axioms versus “Paradoxes”

    Maurice, AllaisOle, Hagen eds. Expected Utility Hypotheses and the Allais ParadoxReidelDordrecht, The Netherlands333409
    Google Scholar
  40. Machina Mark, J., Schmeidler, David 1992A more robust definition of subjective probabilityEconometrica60745780Google Scholar
  41. Nakamura, Yutaka 1990Subjective expected utility with non-additive probabilities on finite state spacesJournal of Economic Theory51346366CrossRefGoogle Scholar
  42. Nakamura, Yutaka 1995Rank dependent utility for arbitrary consequence spacesMathematical Social Sciences29103129CrossRefGoogle Scholar
  43. Nau Robert, F. 1995Coherent decision analysis with inseparable probabilities and utilitiesJournal of Risk and Uncertainty107191CrossRefGoogle Scholar
  44. Nehring, Klaus D.O. (2001), Ambiguity in the context of probabilistic beliefs, mimeo.Google Scholar
  45. Pfanzagl, Johann 1968Theory of MeasurementPhysica-VerlagViennaGoogle Scholar
  46. Quiggin, John 1981Risk perception and risk aversion among Australian farmersAustralian Journal of Agricultural Economics25160169Google Scholar
  47. Sarin Rakesh, K., Wakker Peter, P. 1992A simple axiomatization of nonadditive expected utilityEconometrica6012551272Google Scholar
  48. Sarin Rakesh, K., Wakker Peter, P. 1998Revealed likelihood and Knightian uncertaintyJournal of Risk and Uncertainty16223250CrossRefGoogle Scholar
  49. Sarin Rakesh, K., Wakker Peter, P. 2000Cumulative dominance and probabilistic sophisticationMathematical Social Sciences40191196CrossRefGoogle Scholar
  50. Savage Leonard, J. 1954The Foundations of StatisticsWileyNew York(Second ed. 1972, Dover Publications, New York)Google Scholar
  51. Schmeidler, David 1989Subjective Probability and expected utility without additivityEconometrica57571587Google Scholar
  52. Tversky, Amos, Kahneman, Daniel 1992Advances in prospect theory: cumulative representation of uncertaintyJournal of Risk and Uncertainty5297323CrossRefGoogle Scholar
  53. Tversky, Amos, Wakker Peter, P. 1995Risk attitudes and decision weightsEconometrica6312551280Google Scholar
  54. Wakker Peter, P. 1989Additive Representations of Preferences, A New Foundation of Decision AnalysisKluwer Academic PublishersDordrecht, The NetherlandsGoogle Scholar
  55. Wakker Peter, P. 1990Under stochastic dominance Choquet-expected utility and anticipated utility are identicalTheory and Decision29119132CrossRefGoogle Scholar
  56. Wakker Peter, P. 1991Additive representation for equally spaced structuresJournal of Mathematical Psychology35260266CrossRefGoogle Scholar
  57. Wakker Peter, P. 1993Unbounded utility for Savage’s “foundations of statistics,” and other modelsMathematics of Operations Research18446485Google Scholar
  58. Wakker Peter, P. 2001Testing and characterizing properties of nonadditive measures through violations of the sure-thing principleEconometrica6910391059CrossRefGoogle Scholar
  59. Wakker Peter, P. 2004On the composition of risk preference and beliefPsychogical Review111236241CrossRefGoogle Scholar
  60. Wakker Peter, P. 2005Decision-foundations for properties of nonadditive measures for general state spaces or for general outcome spacesGames and Economic Behavior50107125CrossRefGoogle Scholar
  61. Wald, Abraham 1950Statistical Decision FunctionsWileyNew YorkGoogle Scholar
  62. Wu, George, Gonzalez, Richard 1999Nonlinear decision weights in choice under uncertaintyManagement Science457485Google Scholar
  63. Yaari Menahem, E. 1987The dual theory of choice under riskEconometrica5595115Google Scholar
  64. Zank, Horst 2004Deriving rank-dependent expected utility through probabilistic consistency, School of Economic StudiesThe University of ManchesterManchester, UKGoogle Scholar
  65. Zhang, Jiangkang 1999Qualitative probabilities on lambda-systemsMathematical Social Sciences381120CrossRefGoogle Scholar
  66. Zhang, Jiangkang 2002Subjective ambiguity, expected utility and Choquet expected utilityEconomic Theory20159181CrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.GRIDMaison de la Recherche de l’ESTPCachanFrance
  2. 2.CREED, Dept. of EconomicsUniversity of AmsterdamAmsterdamThe Netherlands

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