Abstract
Self-reflecting signed orders on a set A and its anti-set A * were introduced previously as a way to account for negative as well as positive feelings about the inclusion of items in A in potential subsets of choice. The present paper extends the notion of signed orders to lotteries on A ∪ A *, describes reflection axioms for the lottery context, and shows how these axioms simplify utility representations for preference between lotteries. The simplified representations are then used to guide procedures for extending preferences from A ∪ A * and its lotteries to preferences between subsets of items.
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References
Bossert, W.: 1989, ‘On the Extension of Preferences over a Set to the Power Set: An Axiomatic Characterization of a Quasi-Ordering’, Journal of Economic Theory 49, 84–92.
Bossert, W.: 1993, ‘Preference Extension Rules for Ranking Sets of Alternatives with a Fixed Cardinality’, Preprint, Department of Economics, University of Waterloo, Waterloo, Ontario.
Chew, S.H.: 1983, ‘A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox’, Econometrica 51, 1065–1092.
Chew, S.H. and MacCrimmon, K.R.: 1979, ‘Alpha-Nu Choice Theory: A Generalization of Expected Utility Theory’, Working Paper No. 669, University of British Columbia, Vancouver.
Ellingsen, T.: 1994, ‘Cardinal Utility: A History of Hedonimetry’, in: Allais, M. and Hagen, O. (eds.), Cardinalism, Kluwer Academic, Dordrecht, pp. 105–165.
Fishburn, P.C.: 1970, Utility Theory for Decision Making, Wiley, New York.
Fishburn, P.C.: 1972, ‘Even-Chance Lotteries in Social Choice Theory’, Theory and Decision 3, 18–40.
Fishburn, P.C.: 1976, ‘Cardinal Utility: An Interpretive Essay’, International Review of Economics and Business 23, 1102–1114.
Fishburn, P.C.: 1982a, The Foundations of Expected Utility, Reidel, Dordrecht.
Fishburn, P.C.: 1982b, ‘Nontransitive Measurable Utility’, Journal of Mathematical Psychology 26, 31–67.
Fishburn, P.C.: 1983, ‘Transitive Measurable Utility’, Journal of Economic Theory 31, 293–317.
Fishburn, P.C.: 1988, Nonlinear Preference and Utility Theory, Johns Hopkins University Press, Baltimore.
Fishburn, P.C.: 1992a, ‘Signed Orders and Power Set Extensions’, Journal of Economic Theory 56, 1–19.
Fishburn, P.C.: 1992b, ‘Multiattribute Signed Orders’, Journal of Multi-Criteria Decision Analysis 1, 3–16.
Fishburn, P.C. and LaValle, I.H.: 1993, ‘Subset Preferences in Linear and Nonlinear Utility Theory’, Journal of Mathematical Psychology 37, 611–623.
Gärdenfors, P.: 1976, ‘Manipulation of Social Choice Functions’, Journal of Economic Theory 13, 217–228.
Hausner, M.: 1954, ‘Multidimensional Utilities’, in: Thrall, R.M., Coombs, C.H. and Davis, R.L. (eds.), Decision Processes, Wiley, New York, pp. 167–180.
Herstein, I.N. and Milnor, J.: 1953, ‘An Axiomatic Approach to Measurable Utility’, Econometrica 21, 291–297.
Kannai, Y. and Peleg, B.: 1984, ‘A Note on the Extension of an Order on a Set to the Power Set’, Journal of Economic Theory 32, 172–175.
Krantz, D.H., Luce, R.D., Suppes, P. and Tversky, A.: 1971, Foundations of Measurement, Vol. I, Academic Press, New York.
Kreweras, G.: 1961, ‘Sur une possibilité de rationaliser les intransitivités’, La Décision, Colloques Internationaux CNRS, Paris, pp. 27–32.
Pattanaik, P.K. and Peleg, B.: 1984, ‘An Axiomatic Characterization of the Lexicographic Maximin Extension of an Ordering over a Set to the Power Set’, Social Choice and Welfare 1, 113–122.
von Neumann, J. and Morgenstern, O.: 1944, Theory of Games and Economic Behavior, Princeton University Press, Princeton, NJ.
Wakker, P.P.: 1989, Additive Representations of Preferences, Kluwer, Dordrecht.
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Fishburn, P.C., La Valle, I.H. Signed orders in linear and nonlinear utility theory. Theor Decis 40, 79–101 (1996). https://doi.org/10.1007/BF00133161
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DOI: https://doi.org/10.1007/BF00133161