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Logic of Strategies: What and How?

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Book cover Models of Strategic Reasoning

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8972))

Abstract

This piece is not a paper reporting on original research, but rather a slightly expanded write-up of some notes for a concluding discussion at the 2012 Workshop on ‘Modeling Strategic Reasoning’ at the Lorentz Center in Leiden, an interdisciplinary meeting on the importance of strategies in many fields, from game theory to linguistics, computer science, and cognitive science, that was the incubator for the present volume on the logic-based analysis of strategies and how we reason with, and about them. My modest purpose here is to highlight a few general, somewhat unresolved, decision points about this proposed program that seemed to resonate with the audience at the Workshop, but that may also present food for thought to a more general reader of this book. The emphasis in the presentation that follows is on logic, a view of strategies that figures prominently in my own work on logic and games, cf. [9]. Still, there are certainly other equally viable and illuminating formal viewpoints on the study of strategies, coming, for instance, from automata theory or dynamical systems, cf. [16, 31].

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Notes

  1. 1.

    I will leave the further ‘modeling’ layer in the title out of consideration altogether.

  2. 2.

    For this level distinction in dynamic-epistemic logics of agency, and what is ‘easy’ and ‘hard’ for acting agents versus for theorists, see also the discussion in [6].

  3. 3.

    Sometimes this is seen as high rationality, but sometimes also as self-serving and not really nice, mirroring the pejorative meaning of ‘strategizing’ in common parlance.

  4. 4.

    In fact, this is exactly what happens when certain sub-communities of game theory, computer science, or AI do attach fixed meanings to some of these terms. Cf. [52].

  5. 5.

    I freely use even further terms, such as ‘plans’ as being more open-ended than strategies, and also, as something one is aware of and commits to, more than strategies.

  6. 6.

    Compare the not wholly unrelated case of epistemology and informational action, where natural language has a rich and telling repertoire of common expressions such as ‘know’, ‘suspect’, ‘learn’, ‘note’, ‘discover’, ‘tell’ that we use with a certain amount of stability and even sophistication when engaging in actions of our own, or reporting and reflecting on actions by others. For more on this theme, cf. [10].

  7. 7.

    As noted before, intuitively, a plan restricts my choices in helpful ways, but it need not fix my behavior uniquely: cf. [20] on the conceptual importance of this ‘slack’.

  8. 8.

    Coalgebraic strategies [54] are typically top-down objects that can be used by making an observation of their head after which an infinite tail of the strategy remains available. This never-ending feature is very different from the bottom-up behavior of terminating programs highlighted in PDL.

  9. 9.

    [11] explores a follow-up to this concrete style of proof analysis for strategic reasoning in infinite games with simultaneous moves.

  10. 10.

    Much further background, including game constructions associated with a strategy calculus in our sense, is found in [9]. That book also discusses how strategies can change our view of logic itself when we move from logic of games to logic as games, reading formulas as complex game expressions.

  11. 11.

    One might seek the robustness already in the standard game-theoretic notion of a strategy, that has to work under every eventuality. One can turn all relevant forms of change into moves in a ‘supergame’, asking for one strategy there. But the latter ‘pre-encoding’ seems far removed from our ordinary understanding of agency.

  12. 12.

    Compare the nice example of repairing programs discussed in [38]. We know very little by way of relevant systematic results in logic. Thus, I am not even aware of model-theoretic preservation theorem under submodels or model extensions for such a simple logic as PDL with programs. However, re-planning in multi-agent systems has been investigated in computer science, cf. [25, 26].

  13. 13.

    Such natural extensions with explicit epistemic features do not seem to exist yet for other logical formats for strategies, such as linear game semantics.

  14. 14.

    This issue plays in the area of epistemic planning (cf. [3]), where different kinds of knowledge or beliefs become important: about where we are in following some current plan, but also beliefs about how we expect the process to develop over time.

  15. 15.

    Similar issues arise in analyzing what it means to understand a formal proof, and useful intuitions might be drawn from our experience with mathematical practice.

  16. 16.

    Some game-theoretic sources for Forward Induction are [19, 44, 45].

  17. 17.

    In cognitive reality, zooming out and hiding procedural detail mirror processes of automation turning explicit skills into unconscious routines in the brain, cf. [21].

  18. 18.

    For relevant issues, notions, and results, see the entry on combining logics in the Stanford Encyclopedia of Philosophy [22].

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Acknowledgments

I would like to thank my co-editors Sujata Ghosh and Rineke Verbrugge, as well as the very helpful anonymous reviewers of this volume. I also received valuable feedback from audiences for talks on the theme of designing an explicit logic of strategies.

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  1. Institute for Logic, Language and Computation (ILLC), University of Amsterdam, Amsterdam, The Netherlands

    Johan van Benthem

  2. Department of Philosophy, Stanford University, Stanford, USA

    Johan van Benthem

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  1. Inst for Logic Language & Computation, University of Amsterdam, Amsterdam, Noord-Holland, The Netherlands

    Johan van Benthem

  2. Indian Statistical Institute, Chennai, Tamil Nadu, India

    Sujata Ghosh

  3. Faculty of Math & Natural Sci, University of Groningen, Groningen, Groningen, The Netherlands

    Rineke Verbrugge

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van Benthem, J. (2015). Logic of Strategies: What and How?. In: van Benthem, J., Ghosh, S., Verbrugge, R. (eds) Models of Strategic Reasoning. Lecture Notes in Computer Science(), vol 8972. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48540-8_10

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