Abstract
This chapter re-visits Johan van Benthem’s proposal to study the logic of “best actions” in games. After introducing the main ideas behind this proposal, this chapter makes three general arguments. First, we argue that the logic of best action has a natural deontic rider. Second, that this deontic perspective on the logic of best action opens the door to fruitful contributions from deontic logic to the normative foundation of solution concepts in game theory. Third, we argue that the deontic logic of solution concepts in games takes a specific form, which we call “obligation as weakest permission”. We present some salient features of that logic, and conclude with remarks about how to apply it to specific understandings of best actions in games.
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Notes
- 1.
The relation \(S\) in this formula is the sub-relation of \(move\) that will correspond to sub-game perfect equilibrium. It is a one-step relation in the tree, with its usual reflexive-transitive closure \(S^*\). The other predicates used in this formula are meant to identify which player is to choose at a given node (\(turn_i\)), whether a node is terminal (\(end\)) and to make payoff comparisons.
- 2.
- 3.
The same holds mutatis mutandis for games in strategic form.
- 4.
Cf. [26] for a general presentation.
- 5.
That is not choosing weakly dominated strategies or, equivalently, maximizing expected utility under cautious beliefs. Cf. [33] for this equivalence.
- 6.
A normal modal operator is one that satisfies the \(K\) axiom, i.e. that distributes over material implication, and the rule of necessitation. See [14] for details.
- 7.
We are here taking \(P\) as a “box”, i.e. not as dual of \(O\), which is in line with the interpretation we want to give to the two notions. But if one insists on them being dual, then \(P\top \) comes out as a theorem whenever one assumes that obligations are always consistent.
References
Apt KR, Zvesper JA (2010) The role of monotonicity in the epistemic analysis of strategic games. Games 1(4):381–394
Baltag A, Smets S, Zvesper JA (2009) Keep hoping for rationality: a solution to the backward induction paradox. Synthese 169(2):301–333
Battigalli P, Siniscalchi M (2002) Strong belief and forward induction reasoning. J Econ Theory 106(2):356–391
Belnap ND, Perloff M, Xu M (2001) Facing the future: agents and choices in our indeterminist world. Oxford University Press, Oxford
van Benthem J (2002) Extensive games as process models. J Log Lang Inf 11(3):289–313
van Benthem J (2007) Rational dynamics and epistemic logic in games. Int Game Theory Rev, vol. 9
van Benthem J (2011) Exploring a theory of play. In: Proceedings of the 13th conference on theoretical aspects of rationality and knowledge. ACM, pp 12–16.
van Benthem J (2011) Logic in a social setting. Episteme 8(03):227–247
van Benthem J (2011) Logical dynamics of information and interaction. Cambridge University Press, Cambridge
van Benthem J, Gheerbrant A (2010) Game solution, epistemic dynamics and fixed-point logics. Fundamenta Informaticae 100(1):19–41
van Benthem J, Girard P, Roy O (2009) Everything else being equal: a modal logic for ceteris paribus preferences. J Philos Log 38(1):83–125
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
van Benthem J, Roy O, van Otterloo S (2006) Preference logic, conditionals, and solution concepts in games. In Lagerlund H, Lindstrom S, Sliwinski R (eds) Modality matters: twenty-five essays in honour of Krister Segerberg. Uppsala Philosophical Studies, vol 53. Uppsala University Press, Uppsala.
Blackburn P, De Rijke M, Venema Y (2002) Modal logic, vol 53. Cambridge University Press, Cambridge
Braham M, van Hees M (2012) From wall street to guantánamo bay: the right and the good, practically speaking. http://www.rug.nl/staff/martin.van.hees/Ordering.pdf
de Bruin B (2010) Explaining games: the epistemic programme in game theory. Synthese library, Springer, Dordrecht
Camerer CF (2003) Behavioral game theory: experiments in strategic interaction. Princeton University Press, Princeton
Fagin R, Halpern JY, Moses Y, Vardi MY (1995) Reasoning about knowledge, vol 4. MIT Press Cambridge, Massachusetts
Gheerbrant AP (2010) Fixed-point logics on trees. Institute for Logic, Language and Computation, Amsterdam
Ghosh S (2008) Strategies made explicit in dynamic game logic. Logic and the foundations of game and decision theory.
Grossi D, Lorini E, Schwarzentruber F (2013) Ceteris paribus structure in logics of game forms. In: Proceedings of the 14th conference on theoretical aspects of rationality and knowledge. ACM.
van der Hoek W, Pauly M (2006) Modal logic for games and information. Handbook of modal logic, vol 3. Elsevier, Amsterdam, pp 1077–1148.
Horty JF (2001) Agency and deontic logic. Oxford University Press, Oxford
Horty JF (2011) Perspectival act utilitarianism. In: Girard P, Roy O, Marion M (eds) Dynamic formal epistemology. Springer, Berlin, pp 197–221
Horty JF (2012) Reasons as defaults. OUP, Oxford
Joyce JM (2004) Bayesianism. In: Mele AR, Rawling P (eds) The Oxford handbook of rationality. Oxford University Press, Oxford
Kooi B, Tamminga A (2008) Moral conflicts between groups of agents. J Philos Log 37(1):1–21
Lorini E, Schwarzentruber F (2010) A modal logic of epistemic games. Games 1(4):478–526
McNamara P (2010) Deontic logic. In: Zalta EN (ed) The Stanford encyclopedia of philosophy. Fall 2010 edition.
Olde Loohuis L (2009) Obligations in a responsible world. Logic, rationality, and interaction. Springer, Heidelberg, pp 251–262
Pacuit E, Roy O (2011) A dynamic analysis of interactive rationality. Logic, rationality, and interaction, pp 244–257.
Pacuit E, Roy O (forthcoming) Epistemic foundations of game theory. In: Zalta E (ed) Stanford encyclopedia of philosophy
Pearce DG (1984) Rationalizable strategic behavior and the problem of perfection. Econometrica: J Econometric Soc 52:1029–1050.
Ramanujam R, Simon S (2008) Dynamic logic on games with structured strategies. In: Proceedings of the conference on principles of knowledge representation and reasoning, pp 49–58.
Roy O, Anglberger A, Gratzl N (2012) The logic of obligation as weakest permission. In: Broersen J, Agotnes T, Elgensem D (eds) Deontic logic in computer science. Springer, Heidelberg, pp 139–150
Roy O, Pacuit E (2011) Interactive rationality and the dynamics of reasons. Manuscript.
Samuelson L (1992) Dominated strategies and common knowledge. Games Econ Behav 4(2):284–313
Tamminga A (2013) Deontic logic for strategic games. Erkenntnis 78:183–200
Acknowledgments
This research has been supported by the Alexander von Humboldt Foundation and by a LMU Research Fellowship as part of the LMU Academic Career Program and the Excellence Initiative.
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Roy, O., Anglberger, A.J.J., Gratzl, N. (2014). The Logic of Best Actions from a Deontic Perspective. 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_24
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