We consider an infinite collection of agents who make decisions,
sequentially, about an unknown underlying binary state of the world. Each
agent, prior to making a decision, receives an independent private signal whose
distribution depends on the state of the world. Moreover, each agent also
observes the decisions of its last K immediate predecessors. We study
conditions under which the agent decisions converge to the correct value of the
underlying state. We focus on the case where the private signals have bounded
information content and investigate whether learning is possible, that is,
whether there exist decision rules for the different agents that result in the
convergence of their sequence of individual decisions to the correct state of
the world. We first consider learning in the almost sure sense and show that it
is impossible, for any value of K. We then explore the possibility of
convergence in probability of the decisions to the correct state. Here, a
distinction arises: if K equals 1, learning in probability is impossible under
any decision rule, while for K greater or equal to 2, we design a decision rule
that achieves it. We finally consider a new model, involving forward looking
strategic agents, each of which maximizes the discounted sum (over all agents)
of the probabilities of a correct decision. (The case, studied in previous
literature, of myopic agents who maximize the probability of their own decision
being correct is an extreme special case.) We show that for any value of K, for
any equilibrium of the associated Bayesian game, and under the assumption that
each private signal has bounded information content, learning in probability
fails to obtain
