Skip to main content
SpringerLink
Log in

On the use (and abuse) of Logic in Game Theory

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Notes

  1. A (biased) sampling from my bookshelf: Shoenfield’s Mathematical Logic: “Logic is the study of reasoning; and mathematical logic is the study of the type of reasoning done by mathematicians”; Enderton’s A Mathematical Introduction of Logic: “Symbolic logic is a mathematical model of deductive thought”; and Chiswell and Hodges Mathematical Logic: “In this course we shall study some ways of proving statements.”

  2. See [16, 54, 56, 57] for general overviews of logic in game theory.

  3. That is, if, according to player i’s beliefs, strategy s is optimal for player j, then i cannot rule out all states where player j follows strategy s.

  4. I am assuming that the set of states is finite, so that individual states can be assigned a non-zero probability.

  5. I assume that the reader is familiar with the standard formulation of common knowledge: An event E is commonly known provided everyone knows E, everyone knows that everyone knows E, everyone knows that everyone knows E, and so on ad infinitum.

  6. Note that the framework used in [20] differs in small but important ways from the epistemic-probability models introduced in this paper. These technical details are not important for the main point I am making here, and, indeed, this argument can be made more formal using epistemic probability models.

  7. This follows from the well-known fact that a strategy is weakly dominated iff it does not maximize expected utility with respect to any probability that assigns non-zero probability to all of the opponent’s choices.

  8. Well, I do restrict attention to games with a finite set of players and and finite sets of actions for each player.

References

  1. Abramsky, S. (2007). A compositional game semantics for multi-agent logics of partial information. Volume 1 of Texts in Logic and Games, pp. 11–48. Amsterdam University Press.

  2. Apt, K., & Zvesper, J. (2010). Public announcements in strategic games with arbitrary strategy sets In Proceedings of LOFT 2010.

  3. Arrow, K. (1951). Mathematical models in the social sciences. In Lerner, D. and Lasswell, H. (Eds.) The Policy Sciences. Stanford University Press.

  4. Asheim, G., & Dufwenberg, M. (2003). Admissibility and common belief. Game and Economic Behavior, 42, 208–234.

    Article  Google Scholar 

  5. Aumann, R. (1985). What is game theory trying to accomplish? In Arrow, K. and Honkapohja, S. (Eds.), Frontiers of Economics Oxford.

  6. Aumann, R., & Brandenburger, A. (1995). Epistemic conditions for Nash equilibrium. Econometrica, 63, 1161–1180.

    Article  Google Scholar 

  7. Bacharach, M. (1993). Variable universe games, In BInmore, K. Kirman, A. and Tami, P. (Eds.), Frontiers of Game Theory, The MIT Press.

  8. Bacharach, M. (2006). Beyond Individual Choices Teams and Frames in Game Theory. Princeton University Press.

  9. Bacharach, M.O.L. (1997). The epistemic structure of a theory of a game. In Bacharach, M. Gerard-Varet, L. Mongin, P. and Shin, H. (Eds.), em Epistemic Logic and the Theory of Games and Decisions, Kluwer Academic Publishers.

  10. Baltag, A., Smets, S., & Zvesper, J. (2009). Keep ‘hoping’ for rationality: a solution to the backwards induction paradox. Synthese, 169, 301–333.

    Article  Google Scholar 

  11. Bardsley, N., Mehta, J., Starmer, C., & Sugden, R. (2010). The nature of salience revisited: Cognitve hierarchy theory versus team reasoning. Economic Journal, 120(543), 40–79.

    Article  Google Scholar 

  12. Bicchieri, C. (1993). Rationality and Coordination. Cambridge University Press.

  13. Binmore, K. (1987). Modeling rational players: Part I. Economics and Philosophy, 3, 179–214.

    Article  Google Scholar 

  14. Binmore, K. (2009). Rational Decisions. Princeton University Press.

  15. Bonanno, G., & Battigalli, P. (1999). Recent results on belief, knowledge and the epistemic foundations of game theory. Research in Economics, 53(2), 149–225.

    Article  Google Scholar 

  16. Bonanno, G., & Dégremont, C. (2014). Logic and game theory. In Trends in Logic: Papers on the work of Johan van Benthem, Volume Forthcoming. Springer.

  17. Brandenburger, A. (2007). The power of paradox: some recent developments in interactive epistemology. International Journal of Game Theory, 35, 465–492.

    Article  Google Scholar 

  18. Bratman, M. (2007). Structures of Agency. Oxford University Press.

  19. Cubitt, R., & Sugden, R. (1994). Rationally justifiable play and the theory of non-cooperative games. The Economic Journal, 104, 798–803.

    Article  Google Scholar 

  20. Cubitt, R., & Sugden, R. (2011). Common reasoning in games A Lewisian analysis of common knowledge of rationality. CeDEx Discussion Paper.

  21. Cubitt, R.P., & Sugden, R. (175). Common Knowledge, Salience and Convention: A Reconstruction of David Lewis’ Game Theory. Economics and Philosophy, 19(2).

  22. Fitting, M. (2011). Reasoning about games. Studia Logica, 99(1-3), 143–169.

    Article  Google Scholar 

  23. Geanakoplos, J., Pearce, D., & Stacchetti, E. (1999). Psychological games and sequential rationality. Games and Economic Behavior, 1(1), 60–80.

    Article  Google Scholar 

  24. Ghosh, S., & Ramanujam, R. (2011). Strategies in games: A logic-automata study. In Lectures on Logic and Computation, Volume LNCS 7388.

  25. Goranko, V., & Jamroga, W. (2004). Comparing semantics of logics for multi-agent systems. Synthese: Knowledge. Rationality, and Action, 139(2), 241–280.

    Google Scholar 

  26. Grädel, E. (2011). Back and Forth Between Logics and Games. In Lectures in Game Theory for Computer Scientists, Springer.

  27. Halpern, J. (1999). Set-theoretic completeness for epistemic and conditional logic. Annals of Mathematics and Artificial Intelligence, 26, 1–27.

    Article  Google Scholar 

  28. Halpern, J., Bjorndahl, A., & Pass, R. (2013). Language-based games. In Proceedings of the Fourteenth Conference on Theoretical Aspects of Rationality and Knowledge, ACM Press.

  29. Harman, G. (1999). Reasoning Meaning and Mind Chapter Rationality. Oxford University Press.

  30. Harsanyi, J. (1975). The tracing procedure: A bayesian approach to defining a solution for n-person noncooperative games. International Journal of Game Theory, 4, 61–94.

    Article  Google Scholar 

  31. Hodges, W. (2013). Logic and games, In Zalta, E.N. (Ed.), The Stanford Encyclopedia of Philosophy (Spring 2013 ed.)

  32. Hu, H., & Stuart, H.W. (2001). An epistemic analysis of the harsanyi transformation. International Journal of Game Theory, 30, 517–525.

    Article  Google Scholar 

  33. Huber, F. (2013). Formal representations of belief. In Zalta, E.N. (Ed.), The Stanford Encyclopedia of Philosophy (Summer 2013 ed.)

  34. Kreps, D. (1990). Game Theory and Economic Modelling. Clarendon Press.

  35. Lorini, E., & Schwarzentruber, F. (2010). A modal logic of epistemic games. Games, 1(4), 478–526.

    Article  Google Scholar 

  36. McClennen, E (1990). Rationality and Dynamic Choice Foundational Explorations. Cambridge University Press.

  37. Pacuit, E. (2013a). Dynamic epistemic logic I: Modeling knowledge and belief. Philosophy Compass, 8(9), 798–814.

    Article  Google Scholar 

  38. Pacuit, E. (2013b). Dynamic epistemic logic II: Logics of information change. Philosophy Compass, 8(9), 815–833.

    Article  Google Scholar 

  39. Pacuit, E. (2014). Strategic reasoning in games. Manuscript.

  40. Pacuit, E., & Roy, O. (2015). Epistemic game theory. In Zalta, E.N. (Ed.), Stanford Encyclopedia of Philosophy.

  41. Parikh, R. (1983). Propositional game logic In Proceedings of FOCS.

  42. Parikh, R. (2002). Social software. Synthese, 132(3).

  43. Rabinowicz, W. (1992). Tortous labyrinth: Noncooperative normal-form games between hyperrational players, In Bicchieri, C. Chiara, M.L.D. (Eds.), Knowledge, Belief and Strategic Interaction.

  44. Ramanujam, R., & Simon, S.E. (2008). Dynamic logic on games with structured strategies In Proceedings of Knowledge Representation and Reasoning.

  45. Schelling, T. (1960). The Strategy of Conflict. Harvard University Press.

  46. Skyrms, B. (1990). The Dynamics of Rational Deliberation. Harvard University Press.

  47. Stalnaker, R. (1999). Extensive and strategic forms: Games and models for games. Research in Economics, 53, 293–319.

    Article  Google Scholar 

  48. Sugden, R. (2003). The logic of team reasoning. Philosophical Explorations, 63, 165–181.

    Article  Google Scholar 

  49. van Benthem, J. (1996). Exploring Logical Dynamics. CSLI Press.

  50. van Benthem, J. (2005). Where is logic going and should it. Topoi, 25, 117–122.

    Article  Google Scholar 

  51. van Benthem, J (2007). Rational dynamics and epistemic logic in games. International Game Theory Review, 9(1), 13–45.

    Article  Google Scholar 

  52. van Benthem, J. (2008). Logic and reasong: do the facts matter. Studia Logica, 88(1), 67–84.

    Article  Google Scholar 

  53. van Benthem, J. (2011). Logical Dynamics of Information and Interaction. Cambridge University Press.

  54. van Benthem, J. (2014). Logic in Games. The MIT Press.

  55. van Benthem, J., & Gheerbrant, A. (2010). Game solution, epistemic dynamics and fixed-point logics. Fund. Inform, 100, 1–23.

    Google Scholar 

  56. van Benthem, J., Pacuit, E., & Roy, O. (2011). Towards a theory of play: A logical perspective on games and interaction. Games, 2(1), 52–86.

    Article  Google Scholar 

  57. van der Hoek, W., & Pauly, M. (2006). Modal logic for games and information. In Blackburn, P. van Benthem, J. and Wolter, F. (Eds.), em Handbook of Modal Logic, Volume 3 of Studies in Logic, Elsevier.

  58. van der Hoek, W., & Wooldridge, M. (2003). Towards a logic of rational agency. Logic Journal of the IGPL, 11(2), 135–160.

    Article  Google Scholar 

  59. van Ditmarsch, H., van der Hoek, W., & Kooi, B. (2007). Dynamic Epistemic Logic. Synthese Library. Springer.

  60. van Eijck, J., & Vergrugge, R. (Eds.) (2009). Discourses on Social Software. Texts in Logic and Games.

  61. Vestergaard, R. (2006). A constructive approach to sequential Nash equilibria. Information Processes Letter, 97(2), 46–51.

    Article  Google Scholar 

  62. Wooldridge, M. (2000). Reasoning about Rational Agents. The MIT Press.

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy, University of Maryland, College Park, MD, 20742, USA

    Eric Pacuit

Authors
  1. Eric Pacuit
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric Pacuit.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pacuit, E. On the use (and abuse) of Logic in Game Theory. J Philos Logic 44, 741–753 (2015). https://doi.org/10.1007/s10992-015-9356-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-015-9356-8

Keywords

Search

Navigation