Being realistic about common knowledge: a Lewisian approach

Abstract

Defined and formalized several decades ago, widely used in philosophy and game theory, the concept of common knowledge is still considered as problematic, although not always for the right reasons. I suggest that the epistemic status of a group of human agents in a state of common knowledge has not been thoroughly analyzed. In particular, every existing account of common knowledge, whether formal or not, is either too strong to fit cognitively limited individuals, or too weak to adequately describe their state. I provide a realistic definition of common knowledge, based on a formalization of David Lewis' seminal account and show that it is formally equivalent to probabilistic common belief. This leads to a philosophical analysis of common knowledge which answers several common criticisms and sheds light on its nature.

Community-Based Reasoning in Games: Salience, Rule-Following, and Counterfactuals

“…2 More precisely, the meaning of environmental cues for a given person arguably depends on the context in which she is embedded and, for strategic interactions, on the identity of the other persons with whom she is interacting. In this sense, salience can be said to be community-based [4]: the degree of salience of a strategy, an outcome or more generally of an event 3 is a function of the community in which the interaction is taking place. Consider for instance the market panic game depicted by Figure 3 ([4,10]): 2 Of course, as culture is itself shaped by our evolutionary history, in particular through the genetically and cognitively-based learning mechanisms that have been selected for, salience is surely ultimately natural.…”

“…The contrary would imply that speaking of culture is meaningless for anyone endorsing naturalism and materialism, which is surely not the case. 3 As it will appear in the next section, playing a given strategy or implementing a given outcome can be ultimately formalized as an event in a set-theoretic framework. The looseness of the statement in the text is thus unproblematic.…”

“…Following a recent literature reflecting on social conventions and the role of reasoning and common belief (e.g., [2][3][4][5]), this paper develops a game-theoretic and epistemic account of a peculiar mode of practical reasoning that sustains focal points but also more general forms of rule-following behavior which I call community-based reasoning (henceforth, CBR). The CBR notion emphasizes the fact that the working of focal points and rules depends on the ability for each player to infer correctly others' behavior on the basis of a mutually believed state of affairs.…”

Common Reasoning in Games: A Lewisian Analysis of Common Knowledge of Rationality

“…141-143). Others have recognised a distinction between reason to believe and knowledge, while still using a set-theoretic framework in which 'i has reason to believe that p' is an event (Sillari, 2005;Paternotte, 2010). Such frameworks are not conducive to the representation of processes of reasoning.…”

“…Cubitt and Sugden (2003) is one of several recent analyses of Lewis's argument. It contrasts that approach with that of Aumann. Other formalisations of Lewis's argument, based on different understandings of what is central to it, are offered by Vanderschraaf (1998), Sillari (2005), Gintis (2009) and Paternotte (2010). We discuss them further in Section 4.…”