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Differentiating ambiguity and ambiguity attitude. (English) Zbl 1112.91021
Summary: The objective of this paper is to show how ambiguity, and a decision maker (DM)’s response to it, can be modelled formally in the context of a general decision model. We introduce a relation derived from the DM’s preferences, called “unambiguous preference”, and show that it can be represented by a set of probabilities. We provide such set with a simple differential characterization,and argue that it is a behavioral representation of the ”ambiguity” that the DM may perceive. Given such revealed ambiguity, we provide a representation of ambiguity attitudes. We also characterize axiomatically a special case of our decision model, the “\(\alpha\)-maxmin” expected utility model.

MSC:
91B16 Utility theory
91B06 Decision theory
91B08 Individual preferences
91B10 Group preferences
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[1] Anscombe, F.J.; Aumann, R.J., A definition of subjective probability, Ann. of math. stat., 34, 199-205, (1963) · Zbl 0114.07204
[2] K.J. Arrow, Exposition of a theory of choice under uncertainty, in: Essays in the Theory of Risk-Bearing, North-Holland, Amsterdam, 1974. (Chapter 2). (Part of the Yriö Jahnssonin Säätio lectures in Helsinki, 1965.)
[3] Bewley, T., Knightian decision theorypart I, Decisions econom. finance, 25, 79-110, (2002), (First version: 1986.) · Zbl 1041.91023
[4] Carlier, G.; Dana, R.A., Core of convex distortions of a probability, J. econom. theory, 113, 119-222, (2003) · Zbl 1078.28003
[5] E. Castagnoli, F. Maccheroni, M. Marinacci, Expected utility with multiple priors, in: J.-M. Bernard, T. Seidenfeld and M. Zaffalon (Eds.), Proceedings of the Third International Symposium on Imprecise Probabilities and their Applications Carleton Scientific, Waterloo, 2003, 121-132.
[6] A. Chateauneuf, F. Maccheroni, M. Marinacci, J.-M. Tallon, Monotone continuous multiple-priors, Econom. Theory, forthcoming. · Zbl 1116.91317
[7] Christensen, J.P.R., On sets of Haar measure zero in abelian Polish groups, Israel J. math., 13, 255-260, (1972)
[8] Christensen, J.P.R., Topology and Borel structure, (1974), North-Holland Amsterdam
[9] Clarke, F.H., Optimization and nonsmooth analysis, (1983), Wiley New York · Zbl 0727.90045
[10] Dunford, N.; Schwartz, J.T., Linear operators; part I: general theory, (1958), Wiley New York
[11] Ellsberg, D., Risk, ambiguity, the savage axioms, Quart. J. econom., 75, 643-669, (1961) · Zbl 1280.91045
[12] Epstein, L.G., A definition of uncertainty aversion, Rev. econom stud., 66, 579-608, (1999) · Zbl 0953.91002
[13] Epstein, L.G.; Wang, T., Intertemporal asset pricing under Knightian uncertainty, Econometrica, 62, 283-322, (1994) · Zbl 0799.90016
[14] Epstein, L.G.; Zhang, J., Subjective probabilities on subjectively unambiguous events, Econometrica, 69, 265-306, (2001) · Zbl 1020.91048
[15] Fox, C.R.; Tversky, A., Ambiguity aversion and comparative ignorance, Quart. J. econom., 110, 585-603, (1995) · Zbl 0836.90004
[16] P. Ghirardato, J.N. Katz, Indecision theory: quality of information and voting behavior, Social Science Working Paper 1106R, Caltech, 2002. .
[17] P. Ghirardato, F. Maccheroni, M. Marinacci, Certainty independence and the separation of utility and beliefs, J. Econom. Theory, forthcoming. · Zbl 1080.91506
[18] P. Ghirardato, F. Maccheroni, M. Marinacci, Revealed ambiguity and its consequences, Mimeo, Università Bocconi, Università di Torino, 2003. · Zbl 1159.91344
[19] Ghirardato, P.; Maccheroni, F.; Marinacci, M.; Siniscalchi, M., A subjective spin on roulette wheels, Econometrica, 17, 1897-1908, (2003) · Zbl 1152.91404
[20] Ghirardato, P.; Marinacci, M., Risk, ambiguity and the separation of utility and beliefs, Math. oper. res., 26, 864-890, (2001) · Zbl 1082.91513
[21] Ghirardato, P.; Marinacci, M., Ambiguity made precisea comparative foundation, J. econom. theory, 102, 251-289, (2002) · Zbl 1019.91015
[22] Gilboa, I.; Schmeidler, D., Maxmin expected utility with a non-unique prior, J. math. econom., 18, 141-153, (1989) · Zbl 0675.90012
[23] Heath, C.; Tversky, A., Preference and belief: ambiguity and competence in choice under uncertainty, J. risk uncertainty, 4, 5-28, (1991) · Zbl 0729.90713
[24] Kopylov, I., Procedural rationality in the multiple priors model, (2001), Mimeo University of Rochester
[25] Luce, R.D., Utility of gains and losses: measurement-theoretical and experimental approaches, (2000), Lawrence Erlbaum London · Zbl 0997.91500
[26] Machina, M.J., Almost-objective uncertainty, (2001), Mimeo UC San Diego
[27] Marinacci, M., Probabilistic sophistication and multiple priors, Econometrica, 70, 755-764, (2002) · Zbl 1103.91333
[28] M. Marinacci, L. Montrucchio, A characterization of the core of convex games through Gateaux derivatives, J. Econom. Theory, forthcoming. · Zbl 1117.91310
[29] Matouskova, E., Convexity and Haar null sets, Proc. amer. math. soc., 125, 1793-1799, (1997) · Zbl 0871.46005
[30] Nehring, K., Capacities and probabilistic beliefsa precarious coexistence, Math. soc. sci., 38, 197-213, (1999) · Zbl 1073.91571
[31] Nehring, K., Ambiguity in the context of probabilistic beliefs, (2001), Mimeo UC Davis
[32] Nishimura, K.G.; Ozaki, H., Search and Knightian uncertainty, (2001), Mimeo University of Tokyo, Tohoku University · Zbl 1099.91070
[33] Rockafellar, R.T., Convex analysis, (1970), Princeton University Press Princeton, New Jersey · Zbl 0229.90020
[34] Schmeidler, D., Subjective probability and expected utility without additivity, Econometrica, 57, 571-587, (1989) · Zbl 0672.90011
[35] Shapley, L.S., Cores of convex games, Int. J. game theory, 1, 11-26, (1971) · Zbl 0222.90054
[36] Siniscalchi, M., A behavioral characterization of plausible priors, (2002), Mimeo Northwestern University · Zbl 1151.91034
[37] Thibault, L., On generalized differentials and subdifferentials of Lipschitz vector-valued functions, Nonlinear anal., 6, 1037-1053, (1982) · Zbl 0492.46036
[38] Villegas, C., On qualitative probability σ-algebras, Ann. math. stat., 35, 1787-1796, (1964) · Zbl 0127.34807
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