From Agreement to Asymptotic Learning

Since Aumann's Agreement Theorem (1976), the study of the exchange of information between Bayesian agents has resulted in broad theoretical insight into the phenomenon of agreement and the dynamics that lead to it. However, while questions regarding convergence to common posterior belief/action are now well understood, the correctness of posterior beliefs/actions is understood only in special cases, where either agents act only once, or where the update of beliefs is not fully Bayesian. We present the first results showing "asymptotic learning" for agents who are fully Bayesian, and act or communicate until they converge to the same posterior belief/action. We consider a group of Bayesian agents who have to choose between two alternative actions. They are initially given informative, i.i.d. private signals, and proceed to communicate with their peers. We consider two cases: in the first they learn each others' beliefs regarding the optimal action, while in the second only their chosen actions are revealed. For large groups of agents the combined private signals contain enough information to identify the optimal action with probability approaching one. It is natural to ask whether the agents learn, i.e., whether this information reaches the agents or not. We show that asymptotic learning occurs for a wide variety of models, so that as the size of the group grows the agents learn the correct action with probability that approaches one. Regardless of the communication model, we show that if n agents always have common knowledge of posterior beliefs then they know the correct action with probability 1-O(1/n). When private beliefs are unbounded then common knowledge of the optimal action is a sufficient condition for asymptotic learning. For a natural communication model introduced by Gale and Kariv (2003) we show that asymptotic learning occurs even when private beliefs are bounded.