{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,18]],"date-time":"2025-12-18T19:00:43Z","timestamp":1766084443506,"version":"build-2065373602"},"reference-count":94,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2011,2,16]],"date-time":"2011-02-16T00:00:00Z","timestamp":1297814400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>Logic and game theory have had a few decades of contacts by now, with the classical results of epistemic game theory as major high-lights. In this paper, we emphasize a recent new perspective toward \u201clogical dynamics\u201d, designing logical systems that focus on the actions that change information, preference, and other driving forces of agency. We show how this dynamic turn works out for games, drawing on some recent advances in the literature. Our key examples are the long-term dynamics of information exchange, as well as the much-discussed issue of extensive game rationality. Our paper also proposes a new broader interpretation of what is happening here. The combination of logic and game theory provides a fine-grained perspective on information and interaction dynamics, and we are witnessing the birth of something new which is not just logic, nor just game theory, but rather a Theory of Play.<\/jats:p>","DOI":"10.3390\/g2010052","type":"journal-article","created":{"date-parts":[[2011,2,17]],"date-time":"2011-02-17T06:08:01Z","timestamp":1297922881000},"page":"52-86","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":["Toward a Theory of Play: A Logical Perspective on Games and Interaction"],"prefix":"10.3390","volume":"2","author":[{"given":"Johan","family":"Van Benthem","sequence":"first","affiliation":[{"name":"ILLC, University of Amsterdam, 1090 GE Amsterdam, The Netherlands"},{"name":"Department of Philosophy, Stanford University, Stanford, CA 94305, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eric","family":"Pacuit","sequence":"additional","affiliation":[{"name":"TiLPS, Tilburg University, Warandelaan 2, 5037 AB Tilburg, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Olivier","family":"Roy","sequence":"additional","affiliation":[{"name":"Faculty of Philosophy, University of Groningen, 9700 AB Groningen, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2011,2,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bacharach, M., Gerard-Varet, L., Mongin, P., and Shin, H. (1997). Epsitemic Logic and the Theory of Games and Decisions, Kluwer Academic Publishers.","DOI":"10.1007\/978-1-4613-1139-3"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Blackburn, P., van Benthem, J., and Wolter, F. (2006). Handbook of Modal Logic, Elsevier.","DOI":"10.1002\/9780470996751.ch27"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1017\/S1357530902000704","article-title":"Modal logic and game theory: Two alternative approaches","volume":"7","author":"Bonanno","year":"2002","journal-title":"Risk Decis. Policy"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1111\/1467-8586.00133","article-title":"Games in dynamic epistemic logic","volume":"53","year":"2001","journal-title":"Bull. Econ. Res."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1007\/s00182-006-0061-2","article-title":"The power of paradox: Some recent developments in interactive epistemology","volume":"35","author":"Brandenburger","year":"2007","journal-title":"Int. J. Game Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1007\/s001820050111","article-title":"Interactive epistemology I: Knowledge","volume":"28","author":"Aumann","year":"1999","journal-title":"Int. J. Game Theory"},{"key":"ref_7","unstructured":"Seligman, J. (, January May). Hybrid logic for analyzing games. Beijing, China."},{"key":"ref_8","unstructured":"Van Benthem, J. (2010). Modal Logic for Open Minds, CSLI Publications."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Blackburn, P., de Rijke, M., and Venema, Y. (2002). Modal Logic, Cambridge University Press.","DOI":"10.1017\/CBO9781107050884"},{"key":"ref_10","unstructured":"Von Wright, G.H. (1963). The Logic of Preference, University of Edinburgh Press."},{"key":"ref_11","unstructured":"Zalta, E.N. (2009). The Stanford Encyclopedia of Philosophy, Stanford University. [Spring 2009 ed.]."},{"key":"ref_12","unstructured":"Liu, F. (2008). Changing for the Better: Preference Dynamics and Agent Diversity. [Ph.D Thesis, Institute for Logic, Language and Computation (DS-2008-02), University of Amsterdam]."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/s10992-008-9085-3","article-title":"Everything Else Being Equal: A Modal Logic for Ceteris Paribus Preferences","volume":"38","author":"Girard","year":"2009","journal-title":"J. Phil. Logic"},{"key":"ref_14","unstructured":"Girard, P. (2008). Modal Logic for Belief and Preference Change. [Ph.D Thesis, Stanford University]."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/0022-0000(89)90039-1","article-title":"The complexity of reasoning about knowledge and time","volume":"38","author":"Halpern","year":"1989","journal-title":"J. Comput. Syst. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Blackburn, P., van Benthem, J., and Wolter, F. (2006). Handbook of Modal Logic, Elsevier.","DOI":"10.1002\/9780470996751.ch27"},{"key":"ref_17","unstructured":"Gabbay, D., Kurucz, A., Wolter, F., and Zakharyaschev, M. (2003). Many-Dimensional Modal Logics: Theory and Applications, Elsevier."},{"key":"ref_18","unstructured":"Governatori, G., Hodkinson, I., and Venema, Y. (2006, January 25\u201328). The tree of knowledge in action: Towards a common perspective. Queensland, Australia."},{"key":"ref_19","unstructured":"Van Benthem, J. (1996). Exploring Logical Dynamics, CSLI Press."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/1911154","article-title":"Correlated equilibrium as an expression of bayesian rationality","volume":"55","author":"Aumann","year":"1987","journal-title":"Econometrica"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.jet.2007.09.012","article-title":"Intrinsic correlations in games","volume":"141","author":"Brandenburger","year":"2008","journal-title":"J. Econ. Theor."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1142\/S0219198907001254","article-title":"Rational dynamics and epistemic logic in games","volume":"9","year":"2007","journal-title":"Int. J. Game Theory Rev."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"De Bruin, B. (2010). Explaining Games: The Epsitemic Programme in Game Theory, Springer.","DOI":"10.1007\/978-1-4020-9906-9"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Bonanno, G., van der Hoek, W., and Wooldridge, M. (2008). Logic and the Foundations of Game and Decision Theory, Amsterdam University Press.","DOI":"10.5117\/9789089640260"},{"key":"ref_25","unstructured":"Artemov, S., Barringer, H., d'Avila Garcez, A., Lamb, L., and Woods, J. (2005). We Will Show Them! Essays in Honour of Dov Gabbay, College Publications."},{"key":"ref_26","unstructured":"He, X., Horty, J., and Pacuit, E. Epistemic games in modal logic: Joint actions, knowledge and preferences all together. Chongqing, China."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Harel, D., Kozen, D., and Tiuryn, J. (2000). Dynamic Logic, The MIT Press.","DOI":"10.7551\/mitpress\/2516.001.0001"},{"key":"ref_28","first-page":"281","article-title":"A modal characterization of Nash Equilibrium","volume":"57","author":"Harrenstein","year":"2003","journal-title":"Fund. Inform."},{"key":"ref_29","unstructured":"Van Benthem, J. (2008). Technical report, PP-2008-03, Institute for Logic, Language and Computation."},{"key":"ref_30","unstructured":"Lagerlund, H., Lindstr\u00f6m, S., and Sliwinski, R. (2006). Modality Matters: Twenty-Five Essays in Honour of Krister Segerberg, Uppsala University Press."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Ebbinghaus, H.D., and Flum, J. (1995). Finite Model Theory, Springer.","DOI":"10.1007\/3-540-28788-4"},{"key":"ref_32","unstructured":"Gheerbrant, A. (2010). Fixed-Point Logics on Trees. [Ph.D Thesis, Institute for Logic, Language and Information, University of Amsterdam]."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Van Benthem, J. (2010). Logical Dynamics of Information and Interaction, Cambridge University Press.","DOI":"10.1017\/CBO9780511974533"},{"key":"ref_34","unstructured":"Van Benthem, J. Rational Animals: What is \u2018KRA\u2019?. Malaga, Spain."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/j.entcs.2006.05.034","article-title":"Conditional Doxastic Models: A Qualitative Approach to Dynamic Belief Revision","volume":"165","author":"Mints","year":"2006","journal-title":"Electronic Notes in Theoretical Computer"},{"key":"ref_36","unstructured":"Zalta, E.N. (2008). The Stanford Encyclopedia of Philosophy, Stanford University. [Fall 2008 ed.]."},{"key":"ref_37","unstructured":"Emrich, M.L., Pfeifer, M.S., Hadzikadic, M., and Ras, Z. Logics of public communications. Charlotte, NC, USA."},{"key":"ref_38","unstructured":"Gerbrandy, J. (1999). Bisimulations on Planet Kripke. [Ph.D Thesis, Institute for Logic, Language and Computation, University of Amsterdam]."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Fagin, R., Halpern, J., Moses, Y., and Vardi, M. (1995). Reasoning about Knowledge, The MIT Press.","DOI":"10.7551\/mitpress\/5803.001.0001"},{"key":"ref_40","unstructured":"Zalta, E.N. (2009). The Stanford Encyclopedia of Philosophy, Stanford University. [Spring 2009 ed.]."},{"key":"ref_41","unstructured":"Lewis, D. (1969). Convention, Harvard University Press."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1017\/S0266267103001123","article-title":"Common knowledge, salience and convention: A reconstruction of David Lewis' game theory","volume":"19","author":"Cubitt","year":"2003","journal-title":"Econ. Phil."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s11225-006-8301-9","article-title":"Modal frame correspondences and fixed-points","volume":"83","year":"2006","journal-title":"Studia Logica"},{"key":"ref_44","unstructured":"Gilboa, I. The Logic of Common Knowledge, Public Announcements and Private Suspicions. Evanston, IL, USA."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1620","DOI":"10.1016\/j.ic.2006.04.006","article-title":"Logics of communication and change","volume":"204","author":"Kooi","year":"2006","journal-title":"Inform. Comput."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"510","DOI":"10.2307\/2274239","article-title":"On the logic of theory change: Partial meet contraction and revision functions","volume":"50","author":"Makinson","year":"1985","journal-title":"J. Symbolic Logic"},{"key":"ref_47","first-page":"129","article-title":"Dynamic logic for belief revision","volume":"14","year":"2004","journal-title":"J. Appl Non Classical Logics"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s11229-005-1349-7","article-title":"Prolegomena to dynamic logic for belief revision","volume":"147","year":"2005","journal-title":"Synthese: Knowl. Ration. Action"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.geb.2003.10.006","article-title":"Dynamic interactive epistemology","volume":"49","author":"Board","year":"2004","journal-title":"Games Econ. Behav."},{"key":"ref_50","unstructured":"Boutilier, C. (1992). Conditional Logics for Default Reasoning and Belief Revision. [Ph.D Thesis, University of Toronto]."},{"key":"ref_51","unstructured":"Doyle, J., Sandewall, E., and Torasso, E. (, January 24\u201327). Knowledge, Certainty, Belief and Conditionalisation. Bonn, Germany."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Shoham, Y., and Leyton-Brown, K. (2008). Multiagent Systems, Cambridge University Press.","DOI":"10.1017\/CBO9780511811654"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s10670-008-9147-5","article-title":"Iterated belief revision","volume":"70","author":"Stalnaker","year":"2009","journal-title":"Erkenntnis"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1006\/jeth.2001.2942","article-title":"Strong belief and forward induction reasoning","volume":"105","author":"Battigalli","year":"2002","journal-title":"J. Econ. Theory"},{"key":"ref_55","unstructured":"Baltag, A., and Smets, S. ESSLLI 2009 course: Dynamic logics for interactive belief revision, Bordeaux, France, 2009. Available online: http:\/\/alexandru.tiddlyspot.com\/ (accessed on 15 February 2011)."},{"key":"ref_56","first-page":"13","article-title":"A qualitative theory of dynamic interactive belief revision","volume":"3","author":"Bonanno","year":"2008","journal-title":"Logic and the Foundation of Game and Decision Theory (LOFT7)"},{"key":"ref_57","first-page":"359","article-title":"Shifting priorities: Simple representations for 27 iterated theory change operators","volume":"53","author":"Lagerlund","year":"2006","journal-title":"Modality Matters: Twenty-Five Essays in Honour of Krister Segerberg"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"129","DOI":"10.3166\/jancl.17.129-155","article-title":"Dynamic logic for belief revision","volume":"17","year":"2007","journal-title":"J. Appl Non-Classical Logics"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1007\/s10992-008-9099-x","article-title":"Merging frameworks for interaction","volume":"38","author":"Gerbrandy","year":"2009","journal-title":"J. Phil. Logic"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1236","DOI":"10.1214\/aos\/1176343654","article-title":"Agreeing to disagree","volume":"4","author":"Aumann","year":"1976","journal-title":"Ann. Stat."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1016\/0899-8256(89)90017-1","article-title":"Approximating common knowledge with common beliefs","volume":"1","author":"Monderer","year":"1989","journal-title":"Games Econ. Behav."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.geb.2008.09.032","article-title":"Agreeing to disagree: The non-probabilistic case","volume":"69","author":"Samet","year":"2010","journal-title":"Games Econ. Behav."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1016\/0022-0531(82)90099-0","article-title":"We can't disagree forever","volume":"28","author":"Geanakoplos","year":"1982","journal-title":"J. Econ. Theory"},{"key":"ref_64","unstructured":"Bonanno, G., and Nehring, K. (1997, January 1\u20132). Agreeing to Disagree: A Survey. Firenze, Italy."},{"key":"ref_65","unstructured":"Heifetz, A. Agreement theorems in dynamic-epistemic logic. Stanford, CA, USA."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/0022-0531(85)90035-3","article-title":"Some extensions of a claim of Aumann in an axiomatic model of knowledge","volume":"37","author":"Bacharach","year":"1985","journal-title":"J. Econ. Theory"},{"key":"ref_67","unstructured":"Baldoni, M. (2009, January 7\u201310). Talking Your Way into Agreement: Belief Merge by Persuasive Communication. Turin, Italy."},{"key":"ref_68","unstructured":"Beklemishev, L., Goranko, V., and Shehtman, V. Moorean phenomena in epistemic logic. Moscow, Russia."},{"key":"ref_69","unstructured":"Chatzidakis, Z., Koepke, P., and Pohlers, W. (2006, January 3\u201311). One is a lonely number: On the logic of communication. Munster, Germany."},{"key":"ref_70","unstructured":"Roelofsen, F. (2005). Exploring Logical Perspectives on Distributed Information and Its Dynamics. [Master Thesis, University of Amsterdam]. ILLC (LDC 2005-05)."},{"key":"ref_71","unstructured":"Heifetz, A. Group Belief Dynamics under Iterated Revision: Fixed Points and Cycles of Joint Upgrades. Stanford, CA, USA."},{"key":"ref_72","unstructured":"D\u00e9gremont, C. (2010). The Temporal Mind. Observations on the Logic of Belief Change in Interactive Systems. [Ph.D Thesis, Institute for Logic, Language and Computation (DS-2010-03), University of Amsterdam]."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF00302338","article-title":"Truth and Paradox","volume":"11","author":"Gupta","year":"1982","journal-title":"J. Phil. Logic"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/BF00302339","article-title":"Notes on naive semantics","volume":"11","author":"Herzberger","year":"1982","journal-title":"J. Phil. Logic"},{"key":"ref_75","first-page":"149","article-title":"Semantics and the Liar Paradox","volume":"11","author":"Gabbay","year":"2004","journal-title":"Handbook of Philosophical Logic"},{"key":"ref_76","doi-asserted-by":"crossref","unstructured":"Kelly, K. (1996). The Logic of Reliable Inquiry, Oxford University Press.","DOI":"10.1093\/oso\/9780195091953.001.0001"},{"key":"ref_77","doi-asserted-by":"crossref","unstructured":"Osherson, D., Stob, M., and Weinstein, S. (1986). Systems that Learn, The MIT Press.","DOI":"10.7551\/mitpress\/6609.001.0001"},{"key":"ref_78","unstructured":"Zalta, E.N. (2008). The Stanford Encyclopedia of Philosophy, Standford University. [Winter 2008 ed.]."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/S0019-9958(67)91165-5","article-title":"Language identification in the limit","volume":"10","author":"Gold","year":"1967","journal-title":"Inform. Control."},{"key":"ref_80","unstructured":"He, X., Horty, J., and Pacuit, E. (2009, January 6-11). Can Doxastic Agents Learn? On the Temporal Structure of Learning. Chongqing, China."},{"key":"ref_81","unstructured":"Gierasimczuk, N. (2011). Knowing Ones Limits: Logical Analysis of Inductive Inference. [Ph.D Thesis, Institute for Logic, Language and Computation, University of Amsterdam]."},{"key":"ref_82","doi-asserted-by":"crossref","unstructured":"Hendricks, V. (2005). Mainstream and Formal Epistemology, Cambridge University Press.","DOI":"10.1017\/CBO9780511616150"},{"key":"ref_83","unstructured":"Baltag, A., Gierasimczuk, N., and Smets, S. (2010). Truth-tracking by belief revision. (Unpublished)."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.2307\/2951492","article-title":"Rational learning leads to Nash Equilibrium","volume":"61","author":"Kalai","year":"1993","journal-title":"Econometrica"},{"key":"ref_85","first-page":"1","article-title":"Game solution, epistemic dynamics and fixed-point logics","volume":"100","author":"Gheerbrant","year":"2010","journal-title":"Fund. Inform."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/s11229-009-9559-z","article-title":"Keep \u2018hoping\u2019 for rationality: A solution to the backward induction paradox","volume":"169","author":"Baltag","year":"2009","journal-title":"Synthese"},{"key":"ref_87","unstructured":"Zvesper, J. (2010). Playing with Information. [Ph.D Thesis, Institute for Logic, Language and Computation, University of Amsterdam]."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/BF00184816","article-title":"Self-refuting theories of strategic interaction: A paradox of common knowledge","volume":"30","author":"Bicchieri","year":"1989","journal-title":"Erkenntnis"},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1016\/S0899-8256(05)80015-6","article-title":"Backward induction and common knowledge of rationality","volume":"8","author":"Aumann","year":"1995","journal-title":"Games Econ. Behav."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1006\/game.1996.0104","article-title":"Hypothetical knowledge and games with perfect information","volume":"17","author":"Samet","year":"1996","journal-title":"Games Econ. Behav."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1017\/S0266267100004132","article-title":"Knowledge, belief and counterfactual reasoning in games","volume":"12","author":"Stalnaker","year":"1996","journal-title":"Econ. Philos."},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1006\/game.2000.0838","article-title":"Substantive rationality and backward induction","volume":"37","author":"Stalnaker","year":"2001","journal-title":"Games Econ. Behav."},{"key":"ref_93","unstructured":"Vel\u00e1zquez-Quesada, F.R. (2011). Small Steps in the Dynamics of Information. [Ph.D Thesis, Institute for Logic, Language and Computation, University of Amsterdam]."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s11229-007-9168-7","article-title":"Logics of public communications","volume":"158","author":"Plaza","year":"2007","journal-title":"Synthese: Knowl. Ration. Action"}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/2\/1\/52\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:55:15Z","timestamp":1760219715000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/2\/1\/52"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,2,16]]},"references-count":94,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,3]]}},"alternative-id":["g2010052"],"URL":"https:\/\/doi.org\/10.3390\/g2010052","relation":{},"ISSN":["2073-4336"],"issn-type":[{"type":"electronic","value":"2073-4336"}],"subject":[],"published":{"date-parts":[[2011,2,16]]}}}