Skip to main content

Game logic and its applications I

Abstract

This paper provides a logic framework for investigations of game theoretical problems. We adopt an infinitary extension of classical predicate logic as the base logic of the framework. The reason for an infinitary extension is to express the common knowledge concept explicitly. Depending upon the choice of axioms on the knowledge operators, there is a hierarchy of logics. The limit case is an infinitary predicate extension of modal propositional logic KD4, and is of special interest in applications. In Part I, we develop the basic framework, and show some applications: an epistemic axiomatization of Nash equilibrium and formal undecidability on the playability of a game. To show the formal undecidability, we use a term existence theorem, which will be proved in Part II.

This is a preview of subscription content, access via your institution.

References

  1. Aumann, R. J., (1993), Backward Induction and Common Knowledge of Rationality, Mimeo.

  2. Bacharach, M., (1987), ‘A Theory of Rational Decision in Games’, Erkenntnis 27, 17–55.

    Google Scholar 

  3. Bacharach, M., (1994), ‘The Epistemic Structure of a Theory of a Game’, Theory and Decision 37, 7–48.

    Google Scholar 

  4. Balkenborg, D., and E. Winter, (1995), A Necessary and Sufficient Condition for Playing Backward Induction, Mimeo.

  5. Bubelis, V., (1979), ‘On Equilibria in Finite Games’, International Journal of Game Theory 8, 65–79.

    Google Scholar 

  6. Halpern, J. H., and Y. Moses, (1992), ‘A Guide to Completeness and Complexity for Modal Logics of Knowledge and Beliefs’, Artificial Intelligence 54, 319–379.

    Google Scholar 

  7. Herstein, I. N., and J. Milnor, (1953), ‘An Axiomatic Approach to Measurable Utility’, Econometrica 21, 291–297.

    Google Scholar 

  8. Hintikka, J., (1962), Knowledge and Belief, Cornell University Press.

  9. Kaneko, M., (1996a), Epistemic Considerations of Nash Equilibrium. Forthcoming.

  10. Kaneko, M., (1996b), Mere and Specific Knowledge of the Existence of a Nash Equilibrium. Forthcoming.

  11. Kaneko, M., and T. Nagashima, (1991), ‘Final Decisions, Nash Equilibrium and Solvability in Games with the Common Knowledge of Logical Abilities’, Mathematical Social Sciences 22, 229–255.

    Google Scholar 

  12. Kaneko, M., and T. Nagashima, (1994), ‘Axiomatic Indefinability of Common Knowledge in Finitary Logics’, To appear in Epistemic Logic and the Theory of Games and Decision, eds. M. Bacharach, L. A. Gerard-Varet, P. Mongin and H. Shin.

  13. Karp, C., (1964), Languages with Expressions of Infinite Length, North-Holland.

  14. Keisler, H. J., (1971), Model Theory for Infinitary Logic, North-Holland.

  15. Lemke, C. E., and J. T. Howson, (1964), ‘Equilibrium Points of Bimatrix Games’, SIAM Journal of Applied Mathematics 12, 412–423.

    Google Scholar 

  16. Lismont, L., and P. Mongin, (1994), ‘On the Logic of Common Belief and Common Knowledge’, Theory and Decision 37, 75–106.

    Google Scholar 

  17. Luce, R. D., and H. Raiffa, (1957), Games and Decisions, John Wiley Sons, Inc.

  18. Mendelson, E., (1973), Number Systems and the Foundations of Analysis, Academic Press.

  19. Mendelson, E., (1987), Introduction to Mathematical Logic, Wadsworth.

  20. Nash, J. F., (1951), ‘Noncooperative Games’, Annals of Mathematics 54, 286–295.

    Google Scholar 

  21. Rabin, M., (1977), ‘Decidable Theories’, Handbook of Mathematical Logic, ed. J. Barwise, 595–630.

  22. Segerberg, K., (1994), ‘A Model Existence Theorem in Infinitary Propositional Modal Logic’, Journal of Philosophical Logic 23, 337–367.

    Google Scholar 

  23. von Neumann, J., (1928), ‘Zur Theorie der Gesellschaftsspiele’, Mathematische Annalen 100, 295–320.

    Google Scholar 

  24. von Neumann, J., (1937), ‘Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes’, Ergebrnisse eines Mathematischen Kolloquiums 8, 73–83. English translation, ‘A Model of General Economic Equilibrium’, Review of Economic Studies 13 (1945), 1–9.

    Google Scholar 

  25. von Neumann, J., and O. Morgenstern, (1944), Theory of Games and Economic Behavior, Princeton University Press.

Download references

Author information

Authors and Affiliations

Authors

Additional information

The authors thank Hiroakira Ono for helpful discussions and encouragements from the early stage of this research project, and Philippe Mongin, Mitio Takano and a referee of this journal for comments on earlier versions of this paper. The first and second authors are partially supported, respectively, by Tokyo Center of Economic Research and Grant-in-Aids for Scientific Research 04640215, Ministry of Education, Science and Culture.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kaneko, M., Nagashima, T. Game logic and its applications I. Stud Logica 57, 325–354 (1996). https://doi.org/10.1007/BF00370838

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00370838

Key words

  • infinitary predicate KD4
  • common knowledge
  • Nash equilibrium
  • undecidability on playability