Common knowledge without differentiating counterfactuals

In this paper, I present results on communication and common knowledge in a group with a learning mechanism which is less demanding than the one most commonly used in the existing literature. More specifically, I consider that people distinguish between reality and counterfactuals but not between different counterfactual possibilities. This learning process is qualitatively similar to the two person learning process used by Geanakoplos and Polemarchakis. First I present two results on group communication, one static and one dynamic. The static result presents a sufficient condition for common knowledge based on groupwise comparisons and the dynamic result states that common knowledge will be reached in finite time. This is the problem studied by Cave. The static result is independent of Cave's work. In the dynamic result, I weaken the requirements on the learning process. Second, I examine the case of pairwise communication in a group of n individuals. In a. static result I state a sufficient condition for common knowledge of the value of a function, in which only n pairwise comparisons are involved. Each of these comparisons is based on the distinction between reality and counterfactuals as was described above. Finally, I discuss the difficulty of using the above learning mechanism for pairwise communication.