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Non-additive beliefs and strategic equilibria. (English) Zbl 0962.91002
The authors introduce a notion of equilibrium under uncertainty for \(n\)-player games generalizing the notion for \(2\)-player games introduced by J. Dow and S. Ribeiro da Costa Werlang [J. Econ. Theory 64, 305-324 (1994; Zbl 0813.90132)]. In this framework, beliefs are modelled by (non-additive) capacities. For a capacity \(\nu\), we have two notions measuring the degree of uncertainty: The degree of confidence and the degree of ambiguity.
The authors show some facts about their notion: (a) mixed-strategy Nash equilibria are equilibria under uncertainty, (b) pure-strategy maximin strategy combinations are equilibria under uncertainty, (c) if the degree of confidence is sufficiently small, then an equilibrium under uncertainty induces maximin play, and (d) if the beliefs of all players are independent and consistent, then every sequence of equilibria under uncertainty whose degrees of ambiguity converge to zero converges to a Nash equilibrium.

MSC:
91A06 \(n\)-person games, \(n>2\)
91A10 Noncooperative games
28A12 Contents, measures, outer measures, capacities
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[1] Blume, L.; Brandenburger, A.; Dekel, E., Lexicographic probabilities and choice under uncertainty, Econometrica, 59, 61-79, (1991) · Zbl 0732.90005
[2] Blume, L.; Brandenburger, A.; Dekel, E., Lexicographic probabilities and equilibrium refinements, Econometrica, 59, 81-98, (1991) · Zbl 0729.90096
[3] Crawford, V.P., Adaptive dynamics in coordination games, Econometrica, 63, 103-143, (1995) · Zbl 0827.90143
[4] Crawford, V.P., Equilibrium without independence, J. econom. theory, 50, 127-154, (1990) · Zbl 0689.90075
[5] Dow, J.; Werlang, S.R.d.C., Nash equilibrium under Knightian uncertainty: breaking down backward induction, J. econom. theory, 64, 305-324, (1994) · Zbl 0813.90132
[6] Dow, J.; Werlang, S.R.d.C., Uncertainty aversion, risk aversion, and the optimal choice of portfolio, Econometrica, 60, 197-204, (1992) · Zbl 0756.90002
[7] Dow, J, and, Werlang, S. R. d. C. 1991, Nash Equilibrium under Knightian Uncertainty: Breaking Down Backward Induction, working paper, London Business School. · Zbl 0813.90132
[8] Eichberger, J.; Haller, H.; Milne, F., Naive Bayesian learning in 2×2 matrix games, J. econom. behav. organization, 22, 69-90, (1993)
[9] Eichberger, J.; Kelsey, D., E-capacities and the ellsberg paradox, Theory and decision, 46, 107-140, (1999) · Zbl 1044.91505
[10] Eichberger, J, and, Kelsey, D. 1999b, Free Riders Do Not Like Uncertainty, working paper, University of Birmingham.
[11] Eichberger, J, and, Kelsey, D. 1997, Signalling Games with Uncertainty, working paper, University of Birmingham. · Zbl 0996.91027
[12] Eichberger, J.; Kelsey, D., Uncertainty aversion and preference for randomisation, J. econom. theory, 71, 31-43, (1996) · Zbl 0864.90007
[13] Ellsberg, D., Risk, ambiguity and the savage axioms, Quart. J. econom., 75, 643-669, (1961) · Zbl 1280.91045
[14] Fudenberg, D.; Tirole, J., Game theory, (1991), MIT Press Cambridge · Zbl 1339.91001
[15] Ghirardato, P., Some remarks on capacities on product spaces and Fubini’s theorem, J. econom. theory, 73, 261-291, (1997)
[16] Gilboa, I., Expected utility theory with purely subjective probabilities, J. math. econom., 16, 65-88, (1987) · Zbl 0632.90008
[17] Gilboa, I.; Schmeidler, D., Additive representations of non-additive measures and the Choquet integral, Ann. oper. res., 52, 43-65, (1994) · Zbl 0814.28010
[18] Gilboa, I.; Schmeidler, D., Maxmin expected utility with non-unique priors, J. math. econom., 18, 141-153, (1989) · Zbl 0675.90012
[19] Groes, E.; Jacobsen, H.J.; Sloth, B.; Tranæs, T., Nash equilibrium with lower probabilities, Theory and decision, 44, 37-66, (1998) · Zbl 0895.90183
[20] Haller, H., Non-additive beliefs in solvable games, Theory and decision, (1997)
[21] Hendon, E.; Jacobsen, H.J.; Sloth, B.; Tranæs, T., The product of capacities and belief functions, Math. social sci., 32, 95-108, (1996) · Zbl 0917.90292
[22] Kelsey, D., Maxmin expected utility and weight of evidence, Oxford econom. papers, 46, 425-444, (1994)
[23] Klibanoff, P. 1996, Uncertainty, Decision, and Normal Form Games, working paper, Northwestern University.
[24] Lo, K.Ch., Equilibrium in beliefs under uncertainty, J. econom. theory, 71, 443-484, (1996) · Zbl 0877.90092
[25] Lo, K.Ch., Nash equilibrium without mutual knowledge of rationality, Econom. theory, (1995)
[26] Marinacci, M. 1997, Vague Probabilities and the Bayesian Foundations of Perfection, working paper, Northwestern University.
[27] Marinacci, M., Ambiguous games, Games and econom. behav., (1996) · Zbl 0966.91002
[28] Moulin, H., Game theory for the social sciences, (1986), New York Univ. Press New York
[29] Raiffa, H., Risk, ambiguity and the savage axioms: comment, Quart. J. econom., 75, 690-695, (1961)
[30] Ryan, M. 1997, Supports and the Updating of Capacities, working paper, University of Auckland, NZ.
[31] Sarin, R.; Wakker, P., A simple axiomatization of non-additive expected utility, Econometrica, 60, 1255-1272, (1992) · Zbl 0772.90030
[32] Savage, L.J., The foundations of statistics, (1954), Wiley New York · Zbl 0121.13603
[33] Schmeidler, D., Subjective probability and expected utility without additivity, Econometrica, 57, 571-587, (1989) · Zbl 0672.90011
[34] Van Huyck, J.; Battalio, R.; Beil, R., Tacit coordination games, strategic uncertainty, and coordination failure, Amer. econom. rev., 80, 234-248, (1990)
[35] Von Neumann, J.; Morgenstern, O., Theory of games and economic behavior, (1944), Princeton Univ. Press Princeton · Zbl 0063.05930
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