Abstract
In the context of strategic games, we provide an axiomatic proof of the statement
- (Imp):
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Common knowledge of rationality implies that the players will choose only strategies that survive the iterated elimination of strictly dominated strategies.
Rationality here means playing only strategies one believes to be best responses. This involves looking at two formal languages. One, \(\mathcal{L}_O\), is first-order, and is used to formalise optimality conditions, like avoiding strictly dominated strategies, or playing a best response. The other, \(\mathcal{L}_\nu\), is a modal fixpoint language with expressions for optimality, rationality and belief. Fixpoints are used to form expressions for common belief and for iterated elimination of non-optimal strategies.
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References
Apt, K.R.: Relative strength of strategy elimination procedures. Economics Bulletin 3(21), 1–9 (2007), http://economicsbulletin.vanderbilt.edu/Abstract.asp?PaperID=EB-07C70015
Apt, K.R.: The many faces of rationalizability. Berkeley Electronic Journal of Theoretical Economics 7(1), 38 pages (2007)
Aumann, R.J., Brandenburger, A.: Epistemic conditions for nash equilibrium. Econometrica 63(5), 1161–1180 (1995)
Battigalli, P., Bonanno, G.: Recent results on belief, knowledge and the epistemic foundations of game theory. Research in Economics 53, 149–225 (1999)
Benthem, J.v.: Rational dynamics and epistemic logic in games. International Game Theory Review 9(1), 13–45 (2007) (Erratum reprint, 9(2), 377–409)
Bernheim, B.D.: Rationalizable strategic behavior. Econometrica 52, 1007–1028 (1984)
Bruin, B.d.: Explaining Games: On the logic of game theoretic explanations. PhD thesis, ILLC, Amsterdam (2004)
Dawar, A., Grädel, E., Kreutzer, S.: Inflationary fixed points in modal logics. ACM Transactions on Computational Logic, TOCL 5(2), 282–315 (2004)
Fagin, R., Halpern, J.Y., Vardi, M., Moses, Y.: Reasoning about knowledge. MIT Press, Cambridge (1995)
Fine, K.: Propositional quantifiers in modal logic. Theoria 36, 336–346 (1970)
Kozen, D.: Results on the propositional mu-calculus. Theoretical Computer Science 27(3), 333–354 (1983)
Lipman, B.L.: A note on the implications of common knowledge of rationality. Games and Economic Behaviour 6, 114–129 (1994)
Osborne, M.J., Rubinstein, A.: A Course in Game Theory. MIT Press, Cambridge (1994)
Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics 5, 285–309 (1955)
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Zvesper, J.A., Apt, K.R. (2010). Proof-Theoretic Analysis of Rationality for Strategic Games with Arbitrary Strategy Sets. In: Dix, J., Leite, J., Governatori, G., Jamroga, W. (eds) Computational Logic in Multi-Agent Systems. CLIMA 2010. Lecture Notes in Computer Science(), vol 6245. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14977-1_15
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DOI: https://doi.org/10.1007/978-3-642-14977-1_15
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