Inertia in social learning from a summary statistic

We model normal-quadratic social learning with agents who observe a summary statistic over past actions, rather than complete action histories. Because an agent with a summary statistic cannot correct for the fact that earlier actions influenced later ones, even a small presence of old actions in the statistic can introduce very persistent errors. Depending on how fast these old actions fade from view, social learning can either be as fast as if agents' private information were pooled (rate n  ) or it can slow to a crawl (rate ln⁡nln⁡n). Consistent with Vives (1993), the fastest possible rate of learning falls to rate n(1/3)n(1/3) if actions are also observed with noise, but may be much slower. Increasing the sample size of the summary statistic does not lead to faster asymptotic learning and may reduce short run welfare.