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Logical Player Types for a Theory of Play

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Johan van Benthem on Logic and Information Dynamics

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

Abstract

In theory of play, a player needs to reason about other players’ types that could conceivably explain how play has reached a particular node of the extensive form game tree. Notions of rationalizability are relevant for such reasoning. We present a logical description of such player types and show that the associated type space is constructible (by a Turing machine).

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Notes

  1. 1.

    By automata, we refer only to finite state devices here, though probablistic polynomial time Turing machines are a natural class to consider as well [13].

  2. 2.

    A surprise move by an opponent is perhaps much harder for an automaton to digest than for a human player.

  3. 3.

    This observation can be formalized by defining an indistinguishability relation \(\sim _i\) on player \(i\) nodes, but we do not pursue this here, since we do not attempt an axiomatization of the type space.

  4. 4.

    Note that \( B _i\) is a “model changing” operator and thus ‘dynamic’, in the spirit of [8].

  5. 5.

    We refer the reader to [8] for a more detailed justification.

  6. 6.

    A formal characterization of this process as a recursive function on the tree is in progress, but there are many technical challenges.

References

  1. Ågotnes T, Walther D (2009) A logic of strategic ability under bounded memory. J Logic Lang Inf 18:55–77

    Article  Google Scholar 

  2. Aumann R, Dreze JH (2005) When all is said and done, how should you play and what should you expect? http://www.ma.huji.ac.il/raumann/pdf/dp-387.pdf

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

    Article  Google Scholar 

  4. Battigalli P, Sinisalchi M (1999) Hierarchies of conditional beliefs and interactive epistemology in dynamic games. J Econ Theory 88:188–230

    Article  Google Scholar 

  5. van Benthem J (2009) Decisions, actions, and games: a logical perspective. In: Third Indian conference on logic and applications, pp 1–22

    Google Scholar 

  6. van Benthem J (2011) Exploring a theory of play. In: TARK, pp 12–16

    Google Scholar 

  7. van Benthem J (2012) In praise of strategies. In: Games, actions, and social software. LNCS, vol 7010. Springer, Berlin, pp 96–116

    Google Scholar 

  8. van Benthem J (2013) Logic in games. Forthcoming book

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  11. de Bruin B (2010) Explaining games: the epistemic program in game theory. Synthese Library. Springer, New York

    Book  Google Scholar 

  12. Büchi JR, Landweber LH (1969) Solving sequential conditions by finite-state strategies. Trans Am Math Soc 138:295–311

    Article  Google Scholar 

  13. Fortnow L, Whang D (1994) Optimality and domination in repeated games with bounded players. In: Proceedings of the twenty-sixth annual ACM symposium on theory of computing. ACM, pp 741–749

    Google Scholar 

  14. Ghosh S (2008) Strategies made explicit in dynamic game logic. In: Proceedings of the workshop on logic and intelligent interaction, ESSLLI, pp 74–81

    Google Scholar 

  15. Grädel E, Thomas W, Wilke T (ed) (2002) Automata, logics and infinite games. Lecture notes in computer science, vol 2500. Springer, Berlin

    Google Scholar 

  16. Halpern J (2001) Substantive rationality and backward induction. Games Econ Behav 37:425–435

    Article  Google Scholar 

  17. van der Hoek W, Jamroga W, Wooldridge M (2005) A logic for strategic reasoning. In: Proceedings of the fourth international joint conference on autonomous agents and multi-agent systems, pp 157–164

    Google Scholar 

  18. Hoshi T, Isaac A (2011) Taking mistakes seriously: equivalence notions for game scenarios with off equilibrium play. In: LORI-III, pp 111–124

    Google Scholar 

  19. Jamroga W, van der Hoek W (2004) Agents that know how to play. Fundamenta Informaticae 63(2–3):185–219

    Google Scholar 

  20. Liu F, Wang Y (2013) Reasoning about agent types and the hardest logic puzzle ever. Minds Mach 23(1):123–161

    Google Scholar 

  21. Parikh R (1985) The logic of games and its applications. Ann Discrete Math 24:111–140

    Google Scholar 

  22. Paul S, Ramanujam R, Simon S (2009) Stability under strategy switching. In: Proceedings of the 5th conference on computability in Europe, CiE 2009. LNCS, vol 5635. Springer, Berlin, pp 389–398

    Google Scholar 

  23. Pearce DG (1984) Rationalizable strategic behaviour and the problem of perfection. Econometrica 52:1029–1050

    Article  Google Scholar 

  24. Perea A (2010) Backward induction versus forward induction reasoning. Games 1:168–188

    Article  Google Scholar 

  25. Perea A (2012) Epistemic game theory. Reasoning and choice. Cambridge University Press, Cambridge

    Book  Google Scholar 

  26. Ramanujam R, Simon S (2008) A logical structure for strategies. In: Logic and the foundations of game and decision theory (loft 7), volume 3 of texts in logic and games. Amsterdam University Press, pp. 183–208 (an initial version appeared in the proceedings of the 7th conference on logic and the foundations of game and decision theory (LOFT06) under the title Axioms for composite, strategies, pp 189–198)

    Google Scholar 

  27. Ramanujam R, Simon S (2008) Dynamic logic on games with structured strategies. In: Proceedings of the 11th international conference on principles of knowledge representation and reasoning (KR-08). AAAI Press, pp 49–58

    Google Scholar 

  28. Rubinstein A (1991) Comments on the interpretation of game theory. Econometrica 59:909–924

    Article  Google Scholar 

  29. Walther D, van der Hoek W, Wooldridge M (2007) Alternating-time temporal logic with explicit strategies. In: Proceedings of the 11th conference on theoretical aspects of rationality and knowledge (TARK-2007), pp 269–278

    Google Scholar 

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Acknowledgments

A part of this article was written when I was visiting Peking University. I thank Yanjing Wang for stimulating discussions and warm hospitality at Beijing. I also thank Johan van Benthem for thoughtful and encouraging remarks that greatly helped in the formulation.

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Authors and Affiliations

  1. Institute of Mathematical Sciences, Chennai, India

    R. Ramanujam

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  1. R. Ramanujam
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Correspondence to R. Ramanujam .

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Editors and Affiliations

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

    Alexandru Baltag

  2. Amsterdam, The Netherlands

    Sonja Smets

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Ramanujam, R. (2014). Logical Player Types for a Theory of Play. 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_18

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