Common Learning on a Continuum State Space

Abstract

In Bayesian games the strategic outcome of a game might be sensitive to the higher order beliefs of the players. Robust strategic behavior around a perfect information type requires high enough common belief in the true state of the world. We ask whether players in a private observations framework can attain common p-belief in the true state, for p arbitrarily close to one, assuming that the state space is a continuum, the signal space is finite, and the observed process is independently and identically distributed over time. We show that, under some smoothness and convexity assumptions, if the observations are sufficiently uncorrelated, common learning fails with arbitrarily high probability. This includes the case of independent observations. The failure obtains even if any neighborhood of the true state is asymptotically assigned probability one by each agent. We also show that the failure is an almost certain if for each agent the conditional signal distributions make an open set on the set of all probability distributions on the signal set. In this case common learning fails unless there is perfect correlation. This provides justification for the presence of higher order uncertainty in Bayesian games.