Fragility of Asymptotic Agreement under Bayesian Learning

Under the assumption that individuals know the conditional distributions of signals given the payoff-relevant parameters, existing results conclude that, as individuals observe infinitely many signals, their beliefs about the parameters will eventually merge. We first show that these results are fragile when individuals are uncertain about the signal distributions: given any such model, a vanishingly small individual uncertainty about the signal distributions can lead to a substantial (non-vanishing) amount of differences between the asymptotic beliefs. We then characterize the conditions under which a small amount of uncertainty leads only to a small amount of asymptotic disagreement. According to our characterization, this is the case if the uncertainty about the signal distributions is generated by a family with rapidly-varying tails (such as the normal or the exponential distributions). However, when this family has regularly-varying tails (such as the Pareto, the log-normal, and the t-distributions), a small amount of uncertainty leads to a substantial amount of asymptotic disagreement