Three Views of Common Knowledge

This paper investigates the relationships between three different views of common knowledge: the iterate approach, the fixed point approach, and the shared environment approach. We show that no two of these approaches are equivalent, contrary to accepted wisdom. We argue that the fixed point is the best conceptual analysis of the pretheoretic notion, but that the shared environment approach has its own role to play in understanding how common knowledge is used. We also discuss the assumptions under which various versions of the iterate approach are equivalent to the fixed point approach. We find that, for common knowledge, these assumptions are false, but that for simply having information, the assumptions are not so implausible, at least in the case of finite situations. "The research reported here was partially supported by an award from the System Development Foundation.: The results discussed in this paper appeared in somewhat different form in the unpublished working paper: "Modeling shared understanding," CSLI Working Paper, August 1985. Special thanks go to Peter Aczel for discussions based on that paper.