Abstract
We generalize the concept of Nash equilibrium in mixed strategies for strategic form games to allow for ambiguity in the players' expectations. In contrast to other contributions, we model ambiguity by means of so-called lower probability measures or belief functions, which makes it possible to distinguish between a player's assessment of ambiguity and his attitude towards ambiguity. We also generalize the concept of trembling hand perfect equilibrium. Finally, we demonstrate that for certain attitudes towards ambiguity it is possible to explain cooperation in the one-shot Prisoner's Dilemma in a way that is in accordance with some recent experimental findings.
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REFERENCES
Bernheim, B.D.: 1984, ‘Rationalizable Strategic Behavior', Econometrica 52(4), 1007–1028.
Binmore, K.: 1992, ‘Suppose Everybody Behaved Like That', MS, Economics Department, University of Michigan, Ann Arbor.
Dow, J. and Werlang, C.: 1994, Nash Equilibrium under Knightian Uncertainty: Breaking Down Backward Induction', Journal of Economic Theory 64(2), 305–324.
Eichberger, J. and Kelsey, D.: 1995, ‘Non-additive Belief and Game Theory', Mimeo, University of Saarbrücken.
Ellsberg, D.: 1961, ‘Risk, Ambiguity and the Savage Axioms', Quarterly Journal of Economics 75, 643–669.
Fudenberg, D. and Levine, D.: 1993, ‘Self-Confirming Equilibrium;, Econometrica 61(3), 523–547.
Gilboa, I.: 1987, ‘Expected Utility with Purely Subjective Non-additive Probabilities', Journal of Mathematical Economics 16(1), 65–88.
Gilboa, I. and Schmeidler, D.: 1989, ‘Maximin Expected Utility with Non-unique Prior', Journal of Mathematical Economics 18(2), 141–153.
Greenberg, J.: 1995, ‘Stable (Incomplete) Contracts in Dynamic Games', mimeo, McGill University.
Harsanyi, J.: 1973, ‘Games with Randomly Distributed Payoffs: A New Rationale for Mixed Strategy Equilibrium Points', International Journal of Game Theory 3, 211–225.
Hendon, E.: 1995, ‘Properties of Various Representations of Preferences on Lower Probabilities', Ch. 8 in Fictitious Play in Games and Lower Probabilities in Decision Theories, Ph.D. Dissertation, University of Copenhagen.
Hendon, E., Jacobsen, H.J., Sloth, B. and Tranæs, T.: 1994, ‘Expected Utility with Lower Probabilities', Journal of Risk and Uncertainty 8(2), 197–216.
Hendon, E., Jacobsen, H.J., Sloth, B. and Tranæs, T.: 1996, ‘The Product of Capacities and Belief Functions', Mathematical Social Sciences, 32, 95–108.
Hofstadter, D.R.: 1983, ‘Metamagical Themes', Scientific American 248, 14–20.
Howard, J.: 1988, ‘Cooperation in the Prisoner's Dilemma', Theory and Decision 24, 203–213.
Hurwicz, L.: 1951, ‘Optimality Criteria for Decision Making under Ignorance', Cowles Commission Discussion Paper Statistics, No. 370.
Jaffray, J.-Y.: 1989, ‘Linear Utility Theory for Belief Functions', Operations Research Letters 8, 107–112.
Klibanoff, P.: 1993, ‘Uncertainty, Decision, and Normal-form Games', Mimeo, Stanford University.
Lo, K.C.: 1995, ‘Equilibrium in Beliefs under Uncertainty', Mimeo, University of Toronto.
Pearce, D.G.: 1984, ‘Rationalizable Strategic Behaviour and the Problem of Perfection', Econometrica 52(4), 1029–1050.
Rapoport, A.: 1966, Two-Person Game Theory, Ann Arbor, Michigan: University of Michigan Press.
Rubinstein, A.: 1991, ‘Comments on the Interpretation of Game Theory', Econometrica 59(4), 909–924.
Shafir D. and Tversky, A.: 1992, ‘Thinking through Uncertainty: Nonconsequential Reasoning and Choice', Cognitive Psychology 24, 449–474.
Sarin, R.K. and Wakker, P.: 1995, ‘On the Interpretation of Likelihood in Choquet Expected Utility', Mimeo.
Schmeidler, D.: 1972, ‘Cores of Exact Games, I', Journal of Mathematical Analysis and Applications 40, 214–225.
Schmeidler, D.: 1989, ‘Subjective Probability and Expected Utility without Additivity', Econometrica 57(3), 571–587.
Shafer, G.: 1976, A Mathematical Theory of Evidence, Princeton: Princeton University Press.
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Groes, E., Jørgen Jacobsen, H., Sloth, B. et al. Nash Equilibrium with Lower Probabilities. Theory and Decision 44, 37–66 (1998). https://doi.org/10.1023/A:1004962423985
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DOI: https://doi.org/10.1023/A:1004962423985