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Separable and additive representations of binary gambles of gains. (English) Zbl 1068.91522

Summary: Two approaches are taken to a new utility representation of binary gambles that is called “ratio rank-dependent utility.” Both are based on known axiomatizations of a ranked-additive representation of consequence pairs \((x,y)\) in binary gambles \((x,C;y)\) of gains with \(C\) held fixed and of a separable one of the special gambles \((x,C;e)\), where \(e\) denotes the status quo. The axiomatized version imposes the condition of status-quo event commutativity to get a functional equation that leads to the result. The other assumes, but does not axiomatize, a separable representation of the \((C;y)\) portion of the gamble. These assumptions lead to two difficult functional equations that are solved in the mathematical literature, but the former only under the assumption that the function is twice differentiable. Three behavioral conditions are shown to force this new utility representation to reduce to the standard rank-dependent utility one for gains. They are co-monotonic trade-off consistency, ranked bisymmetry, and segregation, the latter requiring the addition of an operation of joint receipt.

MSC:

91B26 Auctions, bargaining, bidding and selling, and other market models
90B30 Production models
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[1] Aczél, J., Lectures on Functional Equations and Their Applications (1966), Academic Press: Academic Press New York · Zbl 0139.09301
[2] Aczél, J., Maksa, Gy., 2000. A functional equation generated by event commutativity in separable and additive utility theory. Submitted for publication.; Aczél, J., Maksa, Gy., 2000. A functional equation generated by event commutativity in separable and additive utility theory. Submitted for publication.
[3] Aczél, J., Maksa, Gy., Ng, C.T., Páles, Z., 2000. A functional equation arising from ranked additive and separable utility. Proc. Am. Math. Soc. In press.; Aczél, J., Maksa, Gy., Ng, C.T., Páles, Z., 2000. A functional equation arising from ranked additive and separable utility. Proc. Am. Math. Soc. In press.
[4] Chew, S.H., 1989. The rank-dependent quasilinear mean. Department of Economics, University of California, Irvine, CA.; Chew, S.H., 1989. The rank-dependent quasilinear mean. Department of Economics, University of California, Irvine, CA.
[5] Chung, N.-K.; von Winterfeldt, D.; Luce, R. D., An experimental test of event commutativity in decision making under uncertainty, Psychol. Sci., 5, 394-400 (1994)
[6] Feller, W., An Introduction to Probability Theory and its Applications (1950), John Wiley: John Wiley New York · Zbl 0039.13201
[7] Hayes, W. H., Statistics for Psychologists (1963), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York
[8] Kahneman, D.; Tversky, A., Prospect theory: An analysis of decision under risk, Econometrica, 47, 263-291 (1979) · Zbl 0411.90012
[9] Keeney, R. L.; Raiffa, H., Decisions with Multiple Objectives: Preferences and Value Tradeoffs (1976), John Wiley: John Wiley New York · Zbl 0488.90001
[10] Luce, R. D., When four distinct ways to measure utility are the same, J. Math. Psychol., 40, 297-317 (1996) · Zbl 0887.92041
[11] Luce, R. D., Associative joint receipts, Math. Soc. Sci., 34, 51-74 (1997) · Zbl 0918.90017
[12] Luce, R. D., Coalescing, event commutativity, and theories of utility, J. Risk Uncertainty, 16, 87-114 (1998) · Zbl 0911.90038
[13] Luce, R.D., 2000. Utility of Gains and Losses: Measurement-Theoretical and Experimental Approaches. Lawrence Erlbaum Associates, Mahwah, NJ. In press.; Luce, R.D., 2000. Utility of Gains and Losses: Measurement-Theoretical and Experimental Approaches. Lawrence Erlbaum Associates, Mahwah, NJ. In press. · Zbl 0997.91500
[14] Luce, R. D.; Fishburn, P. C., Rank- and sign-dependent linear utility models for finite first-order gambles, J. Risk Uncertainty, 4, 25-59 (1991) · Zbl 0743.90009
[15] Luce, R. D.; Fishburn, P. C., A note on deriving rank-dependent utility using additive joint receipts, J. Risk Uncertainty, 11, 5-16 (1995) · Zbl 0847.90024
[16] Nakamura, Y., Subjective expected utility with non-additive probabilities on finite state spaces, J. Econ. Theory, 51, 346-366 (1990) · Zbl 0724.90005
[17] Nakamura, Y., Multi-symmetric structures and non-expected utility, J. Math. Psychol., 36, 375-395 (1992) · Zbl 0761.90010
[18] Parzen, E., Modern Probability Theory and its Applications (1960), John Wiley: John Wiley New York · Zbl 0089.33701
[19] Rotor, V., Probability Theory (1997), World Scientific: World Scientific Singapore
[20] Wakker, P. P., Additive Representations of Preferences: A New Foundation of Decision Analysis (1989), Kluwer Academic: Kluwer Academic Dordrecht, The Netherlands · Zbl 0668.90001
[21] Wakker, P. P., Additive representations on rank-ordered sets. I: The algebraic approach, J. Math. Psychol., 35, 501-531 (1991) · Zbl 0763.92013
[22] Wakker, P. P., Additive representations on rank-ordered sets. II: The topological approach, J. Math. Econ., 22, 1-26 (1993) · Zbl 0894.92041
[23] Wakker, P. P.; Tversky, A., An axiomatization of cumulative prospect theory, J. Risk Uncertainty, 7, 147-175 (1993) · Zbl 0785.90004
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