Abstract
This paper continues a study of event ambiguity as a primitive concept. Axioms are described for a comparative ambiguity relation on an arbitrary event set that are necessary and sufficient for a representation of the relation by a functional that is nonnegative, vanishes at the empty event, and satisfies complementary equality and submodularity. Uniqueness characteristics of representing functionals are discussed. The theory is extended to multifactor events, where marginal ambiguity and additive representations arise.
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References
Curley, S.P. and Yates, J.F.: 1989, ‘An empirical evaluation of descriptive models of ambiguity reactions in choice situations’,Journal of Mathematical Psychology 33, 397–427.
Einhorn, H.J. and Hogarth, R.M.: 1986, ‘Decision making under ambiguity’,Journal of Business 59 (2), S225-S250.
Ellsberg, D.: 1961, ‘Risk, ambiguity, and the Savage axioms’,Quarterly Journal of Economics 75, 643–669.
Fishburn, P.C.: 1970,Utility Theory for Decision Making, Wiley, New York.
Fishburn, P.C.: 1988,Nonlinear Preference and Utility Theory, Johns Hopkins University Press, Baltimore, MD.
Fishburn, P.C.: 1991, ‘On the theory of ambiguity’,International Journal of Information and Management Sciences 2 (2), 1–16.
Fishburn, P.C.: 1992a, ‘On nonstandard nontransitive additive utility’,Journal of Economic Theory 56, 426–433.
Fishburn, P.C.: 1992b, ‘A general axiomatization of additive measurement with applications’,Naval Research Logistics 39, 741–755.
Fishburn, P.C. and LaValle, I.H.: 1991, ‘Nonstandard nontransitive utility on mixture sets’,Mathematical Social Sciences 21, 233–244.
Hazen, G.B.: 1989, ‘Ambiguity aversion and ambiguity content in decision making under uncertainty’,Annals of Operations Research 19, 415–434.
Heath, C. and Tversky, A.: 1991, ‘Preference and belief: ambiguity and competence in choice under uncertainty’,Journal of Risk and Uncertainty 4, 5–28.
Herstein, I.N. and Milnor, J.: 1953, ‘An axiomatic approach to measurable utility’,Econometrica 21, 291–297.
Hogarth, R.M.: 1989, ‘Ambiguity and competitive decision making: some implications and tests’,Annals of Operations Research 19, 31–50.
Kelley, J.L.: 1955,General Topology, American Book Company, New York.
Keynes, J.M.: 1921,A Treatise on Probability, Macmillian, New York.
Knight, F.H.: 1921,Risk, Uncertainty and Profit, Houghton Mifflin, Boston.
Krantz, D.H., Luce, R.D., Suppes, P. and Tversky, A.: 1971,Foundations of Measurement, Volume I, Academic Press, New York.
Narens, L.: 1985,Abstract Measurement Theory, MIT Press, Cambridge, MA.
Roberts, F.S.: 1979,Measurement Theory, Addison-Wesley, Reading, MA.
Savage, L.J.: 1954,The Foundations of Statistics, Wiley, New York.
Schmeidler, D.: 1989, ‘Subjective probability and expected utility without additivity’,Econometrica 57, 571–587.
Suppes, P., Krantz, D.H., Luce, R.D. and Tversky, A.: 1989,Foundations of Measurement, Volume II, Academic Press, San Diego.
Wakker, P.P.: 1989,Additive Representations of Preferences, Kluwer Academic, Dordrecht.
Wallsten, T.S.: 1990, ‘The costs and benefits of vague information’, in: Hogarth, R.M. (ed.),Insights in Decision Making: A Tribute to Hillel J. Einhorn, University of Chicago Press, Chicago, pp. 28–43.
Wong, S.K.M., Yao, Y.Y., Bollmann, P. and Bürger, H.C.: 1991, ‘Axiomatization of qualitative believe structure’,IEEE Transactions on Systems, Man, and Cybernetics 21, 726–734.
Wong, S.K.M., Yao, Y.Y. and Lingras, P.: 1992, ‘Comparative beliefs and their measurements’,International Journal of General Systems (in press).
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Fishburn, P.C. The axioms and algebra of ambiguity. Theor Decis 34, 119–137 (1993). https://doi.org/10.1007/BF01074898
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DOI: https://doi.org/10.1007/BF01074898