Common Knowledge and Consensus with Noisy Communication

Abstract

Parikh and Krasucki (1990, JET 52) have suggested in an informal manner that a consensus does not require common knowledge. Weyers (1992, CORE DP 9228) has proved that their model does not permit such a conclusion, and that a more general one has to be constructed. Heifetz (1996, JET 70) has given an example with three agents, inspired by computer science works, which illustrates the intuition of the first authors, i.e. where a consensus is obtained without common knowledge of it. We propose a general setting of noisy communication to confirm this result. We show that for any non public and noisy communication, no event can become common knowledge if it was not at the beginning, but that under some assumptions a consensus and arbitrary high levels of interactive knowledge are achievable. A minimal example is given, with two agents and two states. Nevertheless, for public and noisy communication, some results on common knowledge are obtained, depending on the richness of available language. We apply our results to describe some conditions that ensure or prevent epistemic conditions for Nash equilibrium. In general, non public and noisy communication is not sufficient for the conjectures to form, during time, a Nash equilibrium, even if the game and mutual rationality are mutually known. However, with only two agents, or with a noisy and public communication protocol, sufficient conditions are given for the conjectures to form a Nash equilibrium in a finite number of communication periods.