Skip to main content
SpringerLink
Log in
Book cover

Johan van Benthem on Logic and Information Dynamics pp 421–449Cite as

Logic and Game Theory

  • Chapter
  • First Online:

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 5))

Abstract

Johan van Benthem has highlighted in his work that many questions arising in the analysis of strategic interaction call for logical and computational analysis. These questions lead to both formal and conceptually illuminating answers, in that they contribute to clarifying some of the underlying assumptions behind certain aspects of game-theoretical reasoning. We focus on the insights of a part of the literature at the interface of game theory and mathematical logic that gravitates around van Benthem’s work. We discuss the formal questions raised by the perspective consisting in taking games as models for formal languages, in particular modal languages, and how eliminative reasoning processes and solution algorithms can be analyzed logically as epistemic dynamics, and discuss the role played by beliefs in game-theoretical analysis and how they should be modeled from a logical point of view. We give many pointers to the literature throughout the chapter.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    The following definition is Selten’s [84] reformulation of Kuhn’s [71] original property which was stated in terms of pure strategies.

  2. 2.

    A formula is valid in extensive form \(G\) if is true at every \(s\in S\) in every model based on \(G.\)

  3. 3.

    In the game theorist’s sense of the word, namely a function that assigns to every information set of player \(i\) an action at that information set. van Benthem calls such objects ‘uniform strategies’.

  4. 4.

    The dotted line in the extensive-form on the right represents an information set of Player 1.

References

  1. Alchourrón CE, Gärdenfors P, Makinson D (1985) On the logic of theory change: partial meet contraction and revision functions. J Symbolic Logic 50(2):510–530

    Article  Google Scholar 

  2. Alur R, Henzinger TA, Kupferman O (1997) Alternating-time temporal logic. In: Proceedings of the 38th IEEE symposium on foundations of computer science

    Google Scholar 

  3. Aumann RJ (1976) Agreeing to disagree. Ann Stat 4(6):1236–1239

    Article  Google Scholar 

  4. Aumann RJ (1995) Backward induction and common knowledge of rationality. Games Econ Behav 8(1):6–19

    Article  Google Scholar 

  5. Aumann RJ (1999) Interactive epistemology I: knowledge. Int J Game Theory 28(3):263–300

    Google Scholar 

  6. Bacharach M (1994) The epistemic structure of a theory of a game. Theory Decis 37:7–48

    Article  Google Scholar 

  7. Baltag A, Moss LS (2004) Logics for epistemic programs. Synthese 139(2):165–224

    Article  Google Scholar 

  8. Baltag A, Smets S (2006) Dynamic belief revision over multi-agent plausibility models. In: Bonanno G, van der Hoek W, Wooldridge M (eds) Logic and the foundations of game and decision theory: proceedings of LOFT’06, TLG, AUP, pp 11–24

    Google Scholar 

  9. Baltag A, Smets S (2009) Group belief dynamics under iterated revision: fixed points and cycles of joint upgrades. In: Heifetz A (ed) TARK, pp 41–50

    Google Scholar 

  10. Baltag A, Moss LS, Solecki S (1998) The logic of public announcements, common knowledge, and private suspicions. In: TARK ’98: proceedings of the 7th conference on theoretical aspects of rationality and knowledge, Morgan Kaufmann Publishers Inc., San Francisco, pp 43–56

    Google Scholar 

  11. Baltag A, Smets S, Zvesper JA (2009) Keep ’hoping’ for rationality: a solution to the backward induction paradox. Synthese 169(2):301–333

    Article  Google Scholar 

  12. Baltag A, Gierasimczuk N, Smets S (2011) Belief revision as a truth-tracking process. In: Apt KR (ed) TARK, ACM, pp 187–190

    Google Scholar 

  13. Battigalli P, Bonanno G (1997) Synchronic information, knowledge and common knowledge in extensive games. In: Bacharach M, Gérard-Varet LA, Mongin P, Shin H (ed) Epistemic logic and the theory of games and decisions, Theory and decision library, Kluwer Academic, pp 235–263. Reprinted in (Battigalli and Bonanno, 1999b)

    Google Scholar 

  14. Battigalli P, Bonanno G (1999a) Recent results on belief, knowledge and the epistemic foundations of game theory. Res Econ 53(2):149–225

    Google Scholar 

  15. Battigalli P, Bonanno G (1999b) Synchronic information, knowledge and common knowledge in extensive games. Res Econ 53(1):77–99

    Google Scholar 

  16. Belnap N, Perloff M (2001) Facing the future: agents and choices in our indeterminist world. Oxford University Press, Oxford

    Google Scholar 

  17. van Benthem J (1983) Modal logic and classical logic. Bibliopolis, Naples

    Google Scholar 

  18. van Benthem J (1997) Exploring logical dynamics. Center for the Study of Language and Information, Stanford

    Google Scholar 

  19. van Benthem J (2001) Games in dynamic-epistemic logic. Bull Econ Res 53(4):219–248

    Article  Google Scholar 

  20. van Benthem J (2002) Extensive games as process models. J Logic Lang Inform 11(3):289–313

    Article  Google Scholar 

  21. van Benthem J (2006) One is a lonely number. In: Chatzidakis Z, Koepke P, Pohlers W (eds) Logic Colloquium ’02, ASL & A.K. Peters, Wellesley

    Google Scholar 

  22. van Benthem J (2007a) Five questions on game theory, Automatic Press, pp 9–19

    Google Scholar 

  23. van Benthem J (2007b) Rational dynamics and epistemic logic in games. Int Game Theory Rev 9(1): 377–409 (Erratum reprint, Volume 9(2), 2007, 377–409)

    Google Scholar 

  24. van Benthem J (2007c) Dynamic logic for belief change. J Appl Non-Classical Logics 17(2): 129–155

    Google Scholar 

  25. van Benthem J, Dégremont C (2010) Bridges between dynamic doxastic and doxastic temporal logics. In: Bonanno et al (2008) pp 151–173

    Google Scholar 

  26. van Benthem J, Gheerbrant A (2010) Game solution, epistemic dynamics and fixed-point logics. Fundamenta Informaticae 100(1–4):19–41

    Google Scholar 

  27. van Benthem J, Liu F (2004) Diversity of logical agents in games. Philosophia Scientiae 8:163–178

    Article  Google Scholar 

  28. van Benthem J, Pacuit E (2006) The tree of knowledge in action: towards a common perspective. In: Hodkinson I, Governatori G, Venema Y (eds) Advances in modal logic, vol 6. College Publications

    Google Scholar 

  29. van Benthem J, van Otterlo S, Roy O (2005) Preference logic, conditionals and solution concepts in games. In: Lindström S, Lagerlund H, Sliwinski R (eds) Modality matters. Festschrift for Krister Segerberg, University of Uppsala, pp 61–76 (also available on the ILLC prepublication repository: PP-2005-28)

    Google Scholar 

  30. van Benthem J, Gerbrandy J, Hoshi T, Pacuit E (2009) Merging frameworks for interaction: DEL and ETL. J Phil Logic 38(5):491–526

    Google Scholar 

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

    Article  Google Scholar 

  32. van Benthem J (2014) Logic in Games. Cambridge, MA: MIT Press

    Google Scholar 

  33. Bernheim BD (1984) Rationalizable strategic behavior. Econometrica 52(4):1007–1028

    Article  Google Scholar 

  34. Blume L, Brandenburger A, Dekel E (1991) Lexicographic probabilities and choice under uncertainty. Econometrica 59(1):61–79

    Article  Google Scholar 

  35. Board O (2004) Dynamic interactive epistemology. Games Econ Behav 49:49–80

    Article  Google Scholar 

  36. Bonanno G (1992a) Set-theoretic equivalence of extensive-form games. Int J Game Theory 20(4):429–447

    Google Scholar 

  37. Bonanno G (1992b) Players’ information in extensive games. Math Soc Sci 24(1):35–48

    Google Scholar 

  38. Bonanno G (2003) A syntactic characterization of perfect recall in extensive games. Res Econ 57(3):201–217

    Article  Google Scholar 

  39. Bonanno G (2007) Axiomatic characterization of the AGM theory of belief revision in a temporal logic. Artif Intell 171(2–3):144–160

    Google Scholar 

  40. Bonanno G (2010) A syntactic approach to rationality in games with ordinal payoffs. In: Bonanno et al (2008) pp 59–86

    Google Scholar 

  41. Bonanno G (2013) A dynamic epistemic characterization of backward induction without counterfactuals. Games Econ Behav 78:31–43

    Article  Google Scholar 

  42. Bonanno G, Löwe B, van der Hoek W (eds) (2008) Logic and the foundations of game and decision theory—LOFT 8, 8th International conference, Amsterdam, 3–5 July 2008 (Revised Selected Papers, volume 6006 of Lecture Notes in Computer Science, 2010. Springer.)

    Google Scholar 

  43. Brandenburger A, Friedenberg A, Keisler HJ (2008) Admissibility in games. Econometrica 76(2):307–352

    Google Scholar 

  44. Broersen J, Herzig A, Troquard N (2007) Normal coalition logic and its conformant extension. In: Samet D (ed) TARK’07, PUL, pp 91–101

    Google Scholar 

  45. de Bruin B (2008) Common knowledge of rationality in extensive games. Notre Dame J Formal Logic 49(3):261–280

    Article  Google Scholar 

  46. Chellas BF (1980) Modal logic: an Introduction. Cambridge University Press, New York

    Google Scholar 

  47. Dalkey N (1953) Equivalence of information patterns and essentially determinate games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games, vol II, Annals of mathematics studies, no. 28. Princeton University Press, NJ, pp 217–243

    Google Scholar 

  48. Dawar A, Grädel E, Kreutzer S (2004) Inflationary fixed points in modal logic. ACM Trans Comput Logic 5(2):282–315

    Article  Google Scholar 

  49. Dégremont C (2010) The temporal mind. Observations on the logic of belief change in interactive systems. ILLC, ILLC Dissertation Series DS-2010-03, PhD thesis, Universiteit van Amsterdam

    Google Scholar 

  50. Dégremont C, Roy O (2012) Agreement theorems in dynamic-epistemic logic. J Philos Logic 41(4):735–764

    Article  Google Scholar 

  51. van Ditmarsch H (2005) Prolegomena to dynamic logic for belief revision. Synthese 147:229–275

    Article  Google Scholar 

  52. Ebbinghaus H-D, Flum J, Thomas W (1994) Mathematical logic, 2nd edn. Springer, Undergraduate texts in mathematics

    Book  Google Scholar 

  53. Ehrenfeucht A (1961) An application of games to the completeness problem for formalized theories. Fundamenta Mathematicae 49:129–141

    Google Scholar 

  54. van Eijck J, Stokhof M (2006) The gamut of dynamic logics. In: Gabbay D, Woods J (eds) Handbook of the history of logic, vol 6—Logic and the modalities in the 20th century, Elsevier, pp 499–600

    Google Scholar 

  55. Fagin R, Halpern JY, Moses Y, Vardi MY (1995) Reasoning about knowledge. MIT Press, Cambridge

    Google Scholar 

  56. Fraïssé R (1954) Sur quelques classifications des systèmes de relations. Publ Sci Univ Alger Sér A. 1:35–182

    Google Scholar 

  57. Friedman N, Halpern JY (1997) Modeling belief in dynamic systems, part I: foundations. Artif Intell 95(2):257–316

    Google Scholar 

  58. Gerbrandy J (1999) Bisimulations on Planet Kripke. PhD thesis, ILLC, Universiteit van Amsterdam, 1999. ILLC Dissertation Series DS-1999-01

    Google Scholar 

  59. Gheerbrant A (2010) Fixed-Point logics on trees. PhD thesis, ILLC Dissertation Series DS-2010-08, University of Amsterdam

    Google Scholar 

  60. Grädel E, Kreutzer S (2003) Will deflation lead to depletion? on non-monotone fixed point inductions. In: LICS, IEEE Computer Society, ISBN 0-7695-1884-2, p 158

    Google Scholar 

  61. Grove A (1988) Two modellings for theory change. J Philos Logic 17:157–170

    Article  Google Scholar 

  62. Gurevich Y, Shelah S (1986) Fixed-point extensions of first-order logic. Ann Pure Appl Logic 32:265–280

    Article  Google Scholar 

  63. Halpern JY, Vardi MY (1989) The complexity of reasoning about knowledge and time. I. Lower bounds. J Comput Syst Sci 38(1):195–237

    Google Scholar 

  64. Harel D (1985) Recurring dominoes: making the highly undecidable highly understandable. In: Topics in the theory of computation, ISBN 0-444-87647-2, Elsevier, New York, pp 51–71

    Google Scholar 

  65. Hendricks V, de Jongh D, Gierasimczuk N (2014) Logic and Learning. In: Baltag A, Smets S (eds) Johan van Benthem on Logic and Information Dynamics. Springer, Dordrecht, pp 267–288 (Chapter 10 in this volume)

    Google Scholar 

  66. van der Hoek W, Pauly M (2006) Modal logic for games and information. In: Blackburn P, van Benthem J, Wolter F (eds) The handbook of modal logic, Elsevier, pp 1077–1148

    Google Scholar 

  67. Hoshi T (2009) Epistemic dynamics and protocol information. PhD thesis, Stanford

    Google Scholar 

  68. Kohlberg E, Mertens J-F (1986) On the strategic stability of equilibria. Econometrica 54(5):1003–1037

    Article  Google Scholar 

  69. Krentel WD, McKinsey JCC, Quine WV (1951) A simplification of games in extensive form. Duke Math J 18:885–900

    Article  Google Scholar 

  70. Kreutzer S (2004) Expressive equivalence of least and inflationary fixed-point logic. Ann Pure Appl Logic 130(1–3):61–78

    Article  Google Scholar 

  71. Kuhn HW (1953) Extensive games and the problem of information. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games, vol II, Annals of mathematics studies, no 28, Princeton University Press, NJ, pp 193–216

    Google Scholar 

  72. Miller JS, Moss LS (2005) The undecidability of iterated modal relativization. Studia Logica 79(3):373–407

    Article  Google Scholar 

  73. Moschovakis YM (1974) Elementary Induction on Abstract Structures. North-Holland, New York

    Google Scholar 

  74. Myerson RB (1991) Game theory: analysis of conflict. Harvard University Press, Cambridge

    Google Scholar 

  75. Osborne M (2004) An introduction to game theory. Oxford University Press, New York

    Google Scholar 

  76. Papadimitriou CH (1994) Computational Complexity. Addison Wesley, Reading

    Google Scholar 

  77. Pauly M (2002) A modal logic for coalitional power in games. J Logic Comput 12(1):149–166

    Article  Google Scholar 

  78. Pauly M, Parikh R (2003) Game logic—an overview. Studia Logica 75(2):165–182

    Article  Google Scholar 

  79. Pearce DG (1984) Rationalizable strategic behavior and the problem of perfection. Econometrica 52(4):1029–1050

    Google Scholar 

  80. Perea A (2007) Epistemic foundations for backward induction: an overview. In: van Benthem J, Gabbay D, Löwe B (eds) Interactive logic. Proceedings of the 7th Augustus de Morgan workshop, vol 1 of texts in logic and games, Amsterdam University Press, pp 159–193

    Google Scholar 

  81. Plaza JA (1989) Logics of public communications. In: Emrich ML, Pfeifer MS, Hadzikadic M, Ras ZW (eds) Proceedings of the 4th international symposium on methodologies for intelligent systems: poster session program, Oak Ridge National Laboratory, pp 201–216

    Google Scholar 

  82. Quesada A (2001) On expressing maximum information in extensive games. Math Soc Sci 42(2):161–167

    Article  Google Scholar 

  83. Reny PJ (1993) Common belief and the theory of games with perfect information. J Econ Theory 59(2):257–274

    Article  Google Scholar 

  84. Selten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive games. Int J Game Theory 4:25–55

    Google Scholar 

  85. Stalnaker R (1994) On the evaluation of solution concepts. Theor Decis 37:49–73

    Article  Google Scholar 

  86. Susan E, Reny PJ (1994) On the strategic equivalence of extensive form games. J Econ Theory 62(1):1–23

    Article  Google Scholar 

  87. Thompson FB (1952) Equivalence of games in extensive form. RAND research memorandum 759, The RAND Corporation

    Google Scholar 

  88. Zvesper JA (2010) Playing with information. ILLC, ILLC Dissertation Series DS-2010-02, PhD thesis, Universiteit van Amsterdam

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Economics, University of California at Davis, Davis, USA

    Giacomo Bonanno

  2. IRIT, Université Paul Sabatier, Toulouse, France

    Cédric Dégremont

Authors
  1. Giacomo Bonanno
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cédric Dégremont
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giacomo Bonanno .

Editor information

Editors and Affiliations

  1. University of Amsterdam Inst Logic,Language & Computation, Amsterdam, The Netherlands

    Alexandru Baltag

  2. Amsterdam, The Netherlands

    Sonja Smets

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Bonanno, G., Dégremont, C. (2014). Logic and Game Theory. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-06025-5_15

Download citation

Publish with us

Policies and ethics

Search

Navigation