Interactive and common knowledge of information partitions

This paper addresses the question whether interactive knowledge and common knowledge of information partitions are additional assumptions in the state-space model of knowledge. Robert Aumann and others have already discussed this issue, but in a way that appears unsatisfactory. This paper provides a thorough answer to the question in four steps. First, it makes clear the methodological nature of the puzzles surrounding interactive and common knowledge of information partitions. Second, it points out two properties – labeled as Substitutivity and Immediacy – that knowledge holds in the state-space model, and that have received little attention in the literature. Third, based on the previous two steps, the paper demonstrates that interactive and common knowledge of information partitions are not additional assumptions of the state-space model of knowledge; this is the main contribution of the work. Finally, the Appendix of the paper offers a critical examination of Aumann’s discussion of the issue.

[1]  Robert J. Aumann,et al.  Chapter 43 Incomplete information , 2002 .

[2]  Dov Samet,et al.  Coherent beliefs are not always types , 1999 .

[4]  Philippe Mongin,et al.  Strong Completeness Theorems for Weak Logics of Common Belief , 2003, J. Philos. Log..

[5]  J. Mill.  A System of Logic , 1843 .

[6]  Aviad Heifetz.  How canonical is the canonical model? A comment on Aumann's interactive epistemology , 1999, Int. J. Game Theory.

[7]  Eddie Dekel,et al.  Hierarchies of Beliefs and Common Knowledge , 1993 .

[8]  Edward L. Keenan,et al.  The Intensional Logic , 1985 .

[9]  Martin Meier,et al.  On the nonexistence of universal information structures , 2005, J. Econ. Theory.

[10]  John Corcoran,et al.  Church, Alonzo , 2003 .

[11]  John Geanakoplos,et al.  Common Knowledge , 1992, TARK.

[12]  M. Bacharach.  Some extensions of a claim of Aumann in an axiomatic model of knowledge , 1985 .

[13]  E. Zalta,et al.  Intensional Logic and the Metaphysics of Intentionality. , 1991 .

[14]  G. Frege.  Über Sinn und Bedeutung , 1892 .

[15]  Alonzo Church,et al.  A Formulation of the Logic of Sense and Denotation , 1952 .

[16]  Jing Li.  Information Structures with Unawareness , 2008 .

[17]  S. Zamir,et al.  Formulation of Bayesian analysis for games with incomplete information , 1985 .

[18]  Dov Samet,et al.  'Knowing Whether', 'Knowing That' and the Cardinality of State Spaces , 1996 .

[19]  Myron Louis Gochnauer,et al.  The Analysis of Knowledge , 1941, Nature.

[20]  R. Montague.  On the Nature of Certain Philosophical Entities , 1969 .

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

[22]  Daniel Gallin,et al.  Intensional and Higher-Order Modal Logic , 1975 .

[23]  Robert J. Aumann,et al.  Musings on Information and Knowledge , 2005 .

[24]  Territoire Urbain,et al.  Convention , 1955, Hidden Nature.

[25]  Mark Perlman,et al.  The Rational Foundations of Economic Behaviour , 1996 .

[26]  G. Bealer.  Intensional Entities , 2022 .

[27]  Richard Montague.  Pragmatics and intensional logic , 1970 .

[28]  Dov Samet,et al.  Knowledge Spaces with Arbitrarily High Rank , 1998 .

[29]  Ronald Fagin,et al.  The hierarchical approach to modeling knowledge and common knowledge , 1999, Int. J. Game Theory.

[30]  Burkhard C. Schipper,et al.  Interactive Unawareness , 2005, J. Econ. Theory.

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

[32]  Gösta Grahne.  Incomplete Information , 2009, Encyclopedia of Database Systems.

[33]  S. Morris.  COWLES FOUNDATION FOR RESEARCH IN ECONOMICS , 2001 .

[34]  R. Aumann.  Correlated Equilibrium as an Expression of Bayesian Rationality Author ( s ) , 1987 .

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

[36]  Rudolf Carnap,et al.  Meaning and Necessity , 1947 .

[37]  Robert Sugden,et al.  COMMON KNOWLEDGE, SALIENCE AND CONVENTION: A RECONSTRUCTION OF DAVID LEWIS' GAME THEORY , 2003, Economics and Philosophy.

[38]  Itzhak Gilboa,et al.  Information and Meta Information , 1988, TARK.

[39]  Philippe Mongin,et al.  A Non-Minimal But Very Weak Axiomatization of Common Belief , 1994, Artif. Intell..

[40]  A. Heifetz,et al.  Hierarchies of knowledge: an unbounded stairway☆ , 1999 .

[41]  Nancy L. Stokey,et al.  Information, Trade, and Common Knowledge , 1982 .

[42]  Ken Binmore,et al.  Common Knowledge and Game Theory , 1988 .

[43]  Martin Meier,et al.  Hierarchies of beliefs for compact possibility models , 2005 .

[44]  Barton L. Lipman,et al.  Standard state-space models preclude unawareness , 1998 .

[45]  L. Samuelson.  Modeling knowledge in economic analysis , 2004 .

[46]  R. Aumann.  Agreeing to disagree. , 1976, Nature cell biology.

[47]  David M. Kreps,et al.  Rationality and knowledge in game theory , 1997 .

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

[49]  Ronald Fagin,et al.  Reasoning about knowledge , 1995 .

[50]  Matthias Steup,et al.  The Analysis of Knowledge , 2001 .

[51]  Adam Brandenburger,et al.  The power of paradox: some recent developments in interactive epistemology , 2007, Int. J. Game Theory.

[52]  D. Monderer,et al.  Approximating common knowledge with common beliefs , 1989 .