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Each of several agents updates his estimate of the same random variable whenever he makes a new observation or receives the estimate made by another agent. In turn, each agent transmits his estimate to a randomly chosen subset of the other agents. A subset of agents forms a communicating ring if for every pair of ring members, there is a sequence of ring members such that sends his estimate to infinitely often. If each ring member knows that he is a ring member, then the estimates of all the ring members asymptotically agree. However, this common limit can depend upon the order in which estimates are transmitted.