Robust Non-Bayesian Social Learning

We study a canonical setting of learning in networks where initially agents receive conditionally i.i.d. signals about a binary state. The distribution according to which signals are drawn is called an information structure. Agents repeatedly communicate beliefs with their neighbors and update their own belief. We focus on large vanishing-influence networks. Our goal is to design simple non-Bayesian learning heuristics that succeed to robustly learn the correct state for a wide class of initial information structures. We provide an exact characterization of the cases when it is possible. Our main contribution focuses on the positive side of this characterization by introducing the class of virtually additive heuristics. Such a heuristic is characterized by a single function that maps beliefs (elements of [0, 1]) onto the reals which are the virtual beliefs. In the initial period, an agent maps his belief to a virtual belief and in all subsequent periods agent simply sums up all virtual beliefs of his neighbors to obtain his new virtual belief. We show that whenever it is possible to robustly learn the correct state, it is possible to do so with a virtually additive heuristic. Finally, we show that our main positive result can be extended for the case where the agent's initial information is not identically distributed. Moreover, this result remains true even if agents do not share a common prior