Communication, timing, and common learning

We study the effect of stochastically delayed communication on common knowledge acquisition (common learning). If messages do not report dispatch times, communication prevents common learning under general conditions even if common knowledge is acquired without communication. If messages report dispatch times, communication can destroy common learning under more restrictive conditions. The failure of common learning in the two cases is based on different infection arguments. Communication can destroy common learning even if it ends in finite time, or if agents communicate all of their information. We also identify conditions under which common learning is preserved in the presence of communication.