Theory and Decision

, Volume 44, Issue 1, pp 37–66 | Cite as

Nash Equilibrium with Lower Probabilities

  • Ebbe Groes
  • Hans Jørgen Jacobsen
  • Birgitte Sloth
  • Torben Tranaes
Article

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.

Knightian uncertainty ambiguity mixed strategy Nash equilibrium lower probabilities belief functions prisoner's dilemma 

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REFERENCES

  1. Bernheim, B.D.: 1984, ‘Rationalizable Strategic Behavior', Econometrica 52(4), 1007–1028.Google Scholar
  2. Binmore, K.: 1992, ‘Suppose Everybody Behaved Like That', MS, Economics Department, University of Michigan, Ann Arbor.Google Scholar
  3. Dow, J. and Werlang, C.: 1994, Nash Equilibrium under Knightian Uncertainty: Breaking Down Backward Induction', Journal of Economic Theory 64(2), 305–324.Google Scholar
  4. Eichberger, J. and Kelsey, D.: 1995, ‘Non-additive Belief and Game Theory', Mimeo, University of Saarbrücken.Google Scholar
  5. Ellsberg, D.: 1961, ‘Risk, Ambiguity and the Savage Axioms', Quarterly Journal of Economics 75, 643–669.Google Scholar
  6. Fudenberg, D. and Levine, D.: 1993, ‘Self-Confirming Equilibrium;, Econometrica 61(3), 523–547.Google Scholar
  7. Gilboa, I.: 1987, ‘Expected Utility with Purely Subjective Non-additive Probabilities', Journal of Mathematical Economics 16(1), 65–88.Google Scholar
  8. Gilboa, I. and Schmeidler, D.: 1989, ‘Maximin Expected Utility with Non-unique Prior', Journal of Mathematical Economics 18(2), 141–153.Google Scholar
  9. Greenberg, J.: 1995, ‘Stable (Incomplete) Contracts in Dynamic Games', mimeo, McGill University.Google Scholar
  10. Harsanyi, J.: 1973, ‘Games with Randomly Distributed Payoffs: A New Rationale for Mixed Strategy Equilibrium Points', International Journal of Game Theory 3, 211–225.Google Scholar
  11. 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.Google Scholar
  12. 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.Google Scholar
  13. Hendon, E., Jacobsen, H.J., Sloth, B. and Tranæs, T.: 1996, ‘The Product of Capacities and Belief Functions', Mathematical Social Sciences, 32, 95–108.Google Scholar
  14. Hofstadter, D.R.: 1983, ‘Metamagical Themes', Scientific American 248, 14–20.Google Scholar
  15. Howard, J.: 1988, ‘Cooperation in the Prisoner's Dilemma', Theory and Decision 24, 203–213.Google Scholar
  16. Hurwicz, L.: 1951, ‘Optimality Criteria for Decision Making under Ignorance', Cowles Commission Discussion Paper Statistics, No. 370.Google Scholar
  17. Jaffray, J.-Y.: 1989, ‘Linear Utility Theory for Belief Functions', Operations Research Letters 8, 107–112.Google Scholar
  18. Klibanoff, P.: 1993, ‘Uncertainty, Decision, and Normal-form Games', Mimeo, Stanford University.Google Scholar
  19. Lo, K.C.: 1995, ‘Equilibrium in Beliefs under Uncertainty', Mimeo, University of Toronto.Google Scholar
  20. Pearce, D.G.: 1984, ‘Rationalizable Strategic Behaviour and the Problem of Perfection', Econometrica 52(4), 1029–1050.Google Scholar
  21. Rapoport, A.: 1966, Two-Person Game Theory, Ann Arbor, Michigan: University of Michigan Press.Google Scholar
  22. Rubinstein, A.: 1991, ‘Comments on the Interpretation of Game Theory', Econometrica 59(4), 909–924.Google Scholar
  23. Shafir D. and Tversky, A.: 1992, ‘Thinking through Uncertainty: Nonconsequential Reasoning and Choice', Cognitive Psychology 24, 449–474.Google Scholar
  24. Sarin, R.K. and Wakker, P.: 1995, ‘On the Interpretation of Likelihood in Choquet Expected Utility', Mimeo.Google Scholar
  25. Schmeidler, D.: 1972, ‘Cores of Exact Games, I', Journal of Mathematical Analysis and Applications 40, 214–225.Google Scholar
  26. Schmeidler, D.: 1989, ‘Subjective Probability and Expected Utility without Additivity', Econometrica 57(3), 571–587.Google Scholar
  27. Shafer, G.: 1976, A Mathematical Theory of Evidence, Princeton: Princeton University Press.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Ebbe Groes
    • 1
  • Hans Jørgen Jacobsen
    • 1
  • Birgitte Sloth
    • 1
  • Torben Tranaes
    • 1
  1. 1.Institute of Economics, University of CopenhagenCopenhagen K.Denmark

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