Annotations

Lewis (1969) is the classic pioneering study of common knowledge and its potential applications to conventions and game theory. As Lewis acknowledges, parts of his work are foreshadowed in Hume (1740) and Schelling (1960).

Aumann (1976) gives the first mathematically rigorous formulation of common knowledge using set theory. Schiffer (1972) uses the formal vocabulary of epistemic logic (Hintikka 1962) to state his definition of common knowledge. Schiffer’s general approach is to augment a system of sentential logic with a set of knowledge operators corresponding to a set of agents, and then to define common knowledge as a hierarchy of propositions in the augmented system. Bacharach (1992), Bicchieri (1993) and Fagin, et al. (1995) adopt this approach, and develop logical theories of common knowledge which include soundness and completeness theorems. Fagin, et al. show that the syntactic and set-theoretic approaches to developing common knowledge are logically equivalent.

Aumann (1995) gives a recent defense of the classical view of backwards induction in games of imperfect information. For criticisms of the classical view, see Binmore (1987), Reny (1992), Bicchieri (1989) and especially Bicchieri (1993). Brandenburger (1992) surveys the known results connecting mutual and common knowledge to solution concepts in game theory. For more in-depth survey articles on common knowledge and its applications to game theory, see Binmore and Brandenburger (1989), Geanakoplos (1994) and Dekel and Gul (1997). For her alternate account of common knowledge along with an account of conventions which opposes Lewis’ account, see Gilbert (1989).

Monderer and Samet (1989) remains one of the best resources for the study of common p-belief.

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