A Modal Logic of Epistemic Games

We propose some variants of a multi-modal of joint action, preference and knowledge that support reasoning about epistemic games in strategic form. The first part of the paper deals with games with complete information. We first provide syntactic proofs of some well-known theorems in the area of interactive epistemology that specify some sufficient epistemic conditions of equilibrium notions such as Nash equilibrium and Iterated Deletion of Strictly Dominated Strategies (IDSDS). Then, we present a variant of the logic extended with dynamic operators of Dynamic Epistemic Logic (DEL). We show that it allows to express the notion IDSDS in a more compact way. The second part of the paper deals with games with weaker forms of complete information. We first discuss several assumptions on different aspects of perfect information about the game structure (e.g., the assumption that a player has perfect knowledge about the players’ strategy sets or about the preference orderings over strategy profiles), and show that every assumption is expressed by a corresponding logical axiom of our logic. Then we provide a proof of Harsanyi’s claim that all uncertainty about the structure of a game can be reduced to uncertainty about payoffs. Sound and complete axiomatizations of the logics are given, as well as some complexity results for the satisfiability problem.

[1]  Robert J. Aumann,et al.  Interactive epistemology I: Knowledge , 1999, Int. J. Game Theory.

[2]  Joseph Y. Halpern,et al.  A Guide to Completeness and Complexity for Modal Logics of Knowledge and Belief , 1992, Artif. Intell..

[3]  B. Douglas Bernheim,et al.  Axiomatic Characterizations of Rational Choice in Strategic Environme nts , 1986 .

[4]  Carsten Lutz,et al.  Complexity and succinctness of public announcement logic , 2006, AAMAS '06.

[5]  Oliver J. Board.  Knowledge, Beliefs, and Game-Theoretic Solution Concepts , 2002 .

[6]  Fenrong Liu,et al.  Changing for the better : preference dynamics and agent diversity , 2008 .

[7]  Johan van Benthem,et al.  Dynamic logic of preference upgrade , 2007, J. Appl. Non Class. Logics.

[8]  H. Gintis.  The Bounds of Reason: Game Theory and the Unification of the Behavioral Sciences , 2014 .

[9]  Henrik Sahlqvist.  Completeness and Correspondence in the First and Second Order Semantics for Modal Logic , 1975 .

[10]  Johan van Benthem,et al.  Everything Else Being Equal: A Modal Logic for Ceteris Paribus Preferences , 2009, J. Philos. Log..

[11]  G. Bonanno.  A Syntactic Approach to Rationality in Games with Ordinal Payoffs , 2008 .

[12]  Ernst Fehr,et al.  Theories of Fairness and Reciprocity , 2001 .

[13]  Johan van Benthem,et al.  Rational Dynamics and Epistemic Logic in Games , 2007, IGTR.

[14]  Wiebe van der Hoek,et al.  Modal logic for games and information , 2007, Handbook of Modal Logic.

[15]  Emiliano Lorini,et al.  A Dynamic Logic of Agency II: Deterministic $${\mathcal{DLA}}$$ , Coalition Logic, and Game Theory , 2010, J. Log. Lang. Inf..

[16]  Lawrence S. Moss,et al.  Logics for Epistemic Programs , 2004, Synthese.

[17]  Leandro Chaves Rêgo,et al.  Generalized solution concepts in games with possibly unaware players , 2007, TARK '07.

[18]  Emiliano Lorini,et al.  A Dynamic Logic of Agency II: Deterministic VCA, Coalition Logic, and Game Theory , 2010 .

[19]  Adam Brandenburger,et al.  Knowledge and Equilibrium in Games , 1992 .

[20]  David Pearce.  Rationalizable Strategic Behavior and the Problem of Perfection , 1984 .

[21]  H. Gaifman,et al.  Symbolic Logic , 1881, Nature.

[22]  R. Aumann,et al.  Epistemic Conditions for Nash Equilibrium , 1995 .

[23]  J. Horty.  Agency and Deontic Logic , 2001 .

[24]  Wojciech Jamroga,et al.  A logic for strategic reasoning , 2005, AAMAS '05.

[25]  N. Belnap,et al.  Facing the Future: Agents and Choices in Our Indeterminist World , 2001 .

[26]  Adam Brandenburger,et al.  Rationalizability and Correlated Equilibria , 1987 .

[27]  J. Harsanyi.  Games with Incomplete Information Played by 'Bayesian' Players, Part III. The Basic Probability Distribution of the Game , 1968 .

[28]  Michael P. Wellman,et al.  Proceedings of the fifth international joint conference on Autonomous agents and multiagent systems , 2006, AAMAS 2006.

[29]  Emiliano Lorini,et al.  A Logical Account of Social Rationality in Strategic Games , 2010 .

[30]  B. D. Bruin,et al.  Explaining games : on the logic of game theoretic explanations , 2004 .

[31]  Joseph Y. Halpern.  Reasoning about uncertainty , 2003 .

[32]  Robert Stalnaker.  On the evaluation of solution concepts , 1994 .

[33]  Ariel Rubinstein,et al.  A Course in Game Theory , 1995 .

[34]  Walter J. Savitch,et al.  Relationships Between Nondeterministic and Deterministic Tape Complexities , 1970, J. Comput. Syst. Sci..

[35]  Pierpaolo Battigalli,et al.  Recent results on belief, knowledge and the epistemic foundations of game theory , 1999 .

[36]  Hong Hu,et al.  An epistemic analysis of the Harsanyi transformation , 2002, Int. J. Game Theory.

[37]  N. Rescher.  The Logic of Preference , 1968 .

[38]  Richard Spencer-Smith,et al.  Modal Logic , 2007 .

[39]  Ronald Fagin,et al.  Reasoning about knowledge and probability , 1988, JACM.

[40]  Ernst Fehr,et al.  Advances in Economics and Econometrics: Theories of Fairness and Reciprocity: Evidence and Economic Applications , 2003 .

[41]  U. Dulleck,et al.  μ-σ Games , 2012, Games.

[42]  Marc Pauly,et al.  A Modal Logic for Coalitional Power in Games , 2002, J. Log. Comput..

[43]  T. Tan,et al.  The Bayesian foundations of solution concepts of games , 1988 .

[44]  O. Roy.  Thinking before acting : intentions, logic, rational choice , 2008 .

[45]  Vaughan R. Pratt,et al.  A Near-Optimal Method for Reasoning about Action , 1980, J. Comput. Syst. Sci..

[46]  Jelle Gerbrandy,et al.  Dynamic epistemic logic , 1998 .

[47]  Salil P. Vadhan,et al.  Computational Complexity , 2005, Encyclopedia of Cryptography and Security.

[48]  Emiliano Lorini,et al.  A Dynamic Logic of Agency I: STIT, Capabilities and Powers , 2010, J. Log. Lang. Inf..

[49]  Marc Pauly,et al.  Logic for social software , 2000 .

[50]  Jelle Gerbrandy,et al.  Reasoning about Information Change , 1997, J. Log. Lang. Inf..

[51]  I-Peng Lin,et al.  The Complexity of Propositional Modal Theories and the Complexity of Consistency of Propositional Modal Theories , 1994, LFCS.