Abstract
This article discusses relations between several notions of continuity in rank-dependent utility, and in the generalized version of rank-dependent utility as initiated by Segal. Primarily, examples are given to show logical independencies between these notions of continuity. This also leads to counterexamples to Segal's (1989) characterizing theorem 1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chateauneuf, A. (1990). “On the Use of Comonotonicity in the Axiomatization of EURDP Theory for Arbitrary Consequences,” CERMSEM, University of Paris I; extended abstract presented at Fifth International Conference on the Foundations and Applications of Utility, Risk and Decision Theory (FUR-90).
Chew, S.H. and L.G. Epstein. (1989). “A Unifying Approach to Axiomatic Non-Expected Utility Theories,”Journal of Economic Theory 49, 207–240.
Chew, S.H., L.G. Epstein, and P.P. Wakker. (1991). “A Unifying Approach to Axiomatic Non-Expected Utility Theories: Corrigenda,”Journal of Economic Theory, forthcoming.
Chew, S.H. and P.P. Wakker. (1991). “Generalizing Choquet Expected Utility by Weakening Savage's Sure-Thing Principle,” University of California, Irvine Research Unit in Mathematical Behavioral Sciences, MBS 91-16, Irvine, CA, USA.
Fishburn, P.C. (1986). “The Axioms of Subjective Probability,”Statistical Science 1, 335–358.
Gorman, W.M. (1968). “The Structure of Utility Functions,”Review of Economic Studies 35, 367–390.
Green, J. and B. Jullien. (1988). “Ordinal Independence in Non-Linear Utility Theory,”Journal of Risk and Uncertainty 1, 355–387. (“Erratum,” 2 (1989, 119).)
Karn, E. and D. Schmeidler. (1990). “Utility Theory with Uncertainty.” In W. Hildenbrand and H. Sonnenschein (eds.),Handbook of Mathematical Economics, Vol. 4. Amsterdam: North-Holland.
Puppe, C. (1990). “The Irrelevance Axiom, Relative Utility and Choice under Risk,” Department of Statistics and Mathematical Economics, University of Karlsruhe, Karlsruhe, Germany.
Quiggin, J. (1982). “A Theory of Anticipated Utility,”Journal of Economic Behaviour and Organization 3, 323–343.
Quiggin, J. and P.P. Wakker. (1992). “The Axiomatic Basis of Anticipated Utility; A Clarification,” University of Nijmegen, NICI, Nijmegen, The Netherlands.
Royden, H.L. (1963).Real Analysis. New York: MacMillan.
Schmeidler, D. (1989). “Subjective Probability and Expected Utility without Additivity,”Econometrica 57, 571–587.
Segal, U. (1986). “On Lexicographic Probability Relations,”Mathematical Social Sciences 11, 195–199.
Segal, U. (1989). “Anticipated Utility: A Measure Representation Approach,”Annals of Operations Research 19, 359–373.
Segal, U. (1990). “Two-Stage Lotteries without the Reduction Axiom,”Econometrica 58, 349–377.
Segal, U. (1992). “The Measure Representation: A Correction,”Journal of Risk and Uncertainty, this issue.
Wakker, P.P. (1990a). “Continuity, Absolute Continuity, and Weak Convergence in Anticipated-Utility Representations,” Fuqua School of Business, Duke University.
Wakker, P.P. (1990b). “Separating Marginal Utility and Probabilistic Risk Aversion,” working paper FSB-9005, Fuqua School of Business, Duke University, Durham, NC, USA.
Wakker, P.P. (1991a). “Additive Representations on Rank-Ordered Sets. I. The Algebraic Approach,”Journal of Mathematical Psychology 35, 501–531.
Wakker, P.P. (1991b). “Additive Representations on Rank-Ordered Sets. II. The Topological Approach,”Journal of Mathematical Economics, forthcoming.
Author information
Authors and Affiliations
Additional information
This article is a rewritten version of Wakker (1990a). Puppe (1990) independently discovered that Segal's (1989) theorem 1 is not correct. This research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.
Rights and permissions
About this article
Cite this article
Wakker, P. Counterexamples to Segal's measure representation theorem. J Risk Uncertainty 6, 91–98 (1993). https://doi.org/10.1007/BF01065352
Issue Date:
DOI: https://doi.org/10.1007/BF01065352