Abstract
This paper shows how notions of vagueness in preferences and judgments of personal probabilities can be accommodated within an axiomatization of subjective expected utility by the use of extraneous scaling probabilities and gambles on consequences. The representational form obtained says that the subjective expected utility of one act exceeds the subjective expected utility of a second act whenever the first is preferred to the second. The paper also explores the possibility of obtaining this representational form under Savage's formulation, which does not use extraneous probabilities, and discusses difficulties encountered in this approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Bibliography
Allais, M., ‘Le comportement de l'homme rationnel devant le risque: Critique des postulats et axiomes de l'école Américaine’, Econometrica 21 (1953) 503–546.
Anscombe, F. J. and Aumann, R. J., ‘A Definition of Subjective Probability’, Annals of Mathematical Statistics 34 (1963) 199–205.
Armstrong, W. E., ‘A Note on the Theory of Consumer's Behavior’, Oxford Economic Papers 2 (1950) 119–122.
Balch, M. S. and Fishburn, P. C., ‘Subjective Expected Utility for Conditional Primitives’, in Essays on Economic Behavior under Uncertainty (ed. by M. S. Balch, D. L. McFadden and S. Y. Wu), North-Holland, Amsterdam, 1974.
Ellsberg, D., ‘Risk, Ambiguity, and the Savage Axioms’, Quarterly Journal of Economics 75 (1961) 643–669.
Fishburn, P. C., Decision and Value Theory, Wiley, New York, 1964.
Fishburn, P. C., ‘A General Theory of Subjective Probabilities and Expected Utilities’, Annals of Mathematical Statistics 40 (1969) 1419–1429.
Fishburn, P. C., ‘Intransitive Indifference in Preference Theory: A Survey’, Operations Research 18 (1970) 207–228.
Fishburn, P. C., Utility Theory for Decision Making, Wiley, New York, 1970.
Fishburn, P. C., ‘Minimal Preferences on Lotteries’, Journal of Mathematical Psychology 9 (1972) 294–305.
Fishburn, P. C., ‘Subjective Expected Utility with Mixture Sets and Boolean Algebras’, Annals of Mathematical Statistics 43 (1972) 917–927.
Fishburn, P. C., ‘A Mixture-Set Axiomatization of Conditional Subjective Expected Utility’, Econometrica 41 (1973) 1–25.
Fishburn, P. C., ‘Binary Choice Probabilities: on the Varieties of Stochastic Transitivity’, Journal of Mathematical Psychology 10 (1973) 327–352.
Fishburn, P. C., ‘On the Foundations of Decision Making under Uncertainty’, in Essays on Economic Behavior under Uncertainty (ed. by M. S. Balch, D. L. McFadden and S. Y. Wu), North-Holland, Amsterdam, 1974.
Fishburn, P. C., ‘Bounded One-Way Expected Utility’, Econometrica, forthcoming.
Fishburn, P. C., ‘Unbounded Utility Functions in Expected Utility Theory’, Quarterly Journal of Economics, forthcoming.
Fishburn, P. C., ‘Unbounded Expected Utility’, Annals of Statistics, forthcoming.
Fishburn, P. C., ‘Weak Comparative Probability with Infinite Sets’, Annals of Probability, forthcoming.
Hausner, M. and Wendel, J. G., ‘Ordered Vector Spaces’, Proceedings of The American Mathematical Society 3 (1952) 977–982.
Luce, R. D., ‘Semiorders and a Theory of Utility Discrimination’, Econometrica 24 (1956) 178–191.
Luce, R. D., Individual Choice Behavior, Wiley, New York, 1959.
Luce, R. D. and Krantz, D. H., ‘Conditional Expected Utility’, Econometrica 39 (1971) 253–272.
Luce, R. D. and Suppes, P., ‘Preference, Utility, and Subjective Probability’, in Handbook of Mathematical Psychology, Vol. III (ed. by R. D. Luce, R. R. Bush, and E. Galanter), Wiley, New York, 1965.
Marley, A. A. J., ‘The Relation between Discard and Regularity Conditions for Choice Probabilities’, Journal of Mathematical Psychology 2 (1965) 242–253.
Marley, A. A. J., ‘Some Probabilistic Models of Simple Choice and Ranking’, Journal of Mathematical Psychology 5 (1968) 311–332.
May, K. O., ‘Intransitivity, Utility, and the Aggregation of Preference Patterns’, Econometrica 22 (1954) 1–13.
Pratt, J. W., Raiffa, H., and Schlaifer, R., ‘The Foundations of Decision under Uncertainty: An Elementary Exposition’, Journal of the American Statistical Association 59 (1964) 353–375.
Ramsey, F. P., ‘Truth and Probability’, in The Foundations of Mathematics and Other Logical Essays (ed. by F. P. Ramsey), Harcourt, Brace and Co., New York, 1931.
Savage, L. J., The Foundations of Statistics, Wiley, New York, 1954.
Savage, L. J., ‘Bayesian Statistics’, in Recent Developments in Information and Decision Processes (ed. by R. E. Machol and P. Gray), Macmillan, New York, 1962.
Savage, L. J., ‘Implications of Personal Probability for Induction’, Journal of Philosophy 64 (1967) 593–607.
Savage, L. J., ‘Elicitation of Personal Probabilities and Expectations’, Journal of the American Statistical Association 66 (1971) 783–801.
Suppes, P., ‘The Role of Subjective Probability and Utility in Decision-Making’, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955 5 (1956) 61–73.
Tversky, A., ‘Elimination by Aspects: A Theory of Choice’, Psychological Review 79 (1972) 281–299.
Tversky, A., ‘Choice by Elimination’, Journal of Mathematical Psychology 9 (1972) 341–367.
von Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior (2nd ed.). Princeton University Press, Princeton, N.J., 1947.
Author information
Authors and Affiliations
Additional information
This paper was prepared for the MSSB Workshop on Uncertainty, Political Processes, and Public Policy, August, 1974, San Diego, California. The author is indebted to Raymond Burros for stimulating exchanges on the incorporation of vagueness into the axioms for subjective expected utility. The research was supported by the Office of Naval Research.
Rights and permissions
About this article
Cite this article
Fishburn, P.C. A theory of subjective expected utility with vague preferences. Theor Decis 6, 287–310 (1975). https://doi.org/10.1007/BF00136199
Issue Date:
DOI: https://doi.org/10.1007/BF00136199