Some People Never Learn, Rationally : Multidimensional Learning Traps and Smooth Solutions of Dynamic Programs

We give conditions under which the smoothness properties of the value and policy functions of a dynamic program at discount factor zero extend to small positive discount factors. We apply this to a model of Bayesian learning by a decision maker who does not know which of several parameters is true. There is an action that is uninformative insofar as when it is chosen, all parameters give the same distribution over outcomes. We provide conditions under which a policy specifying an action as a function of the current belief results in a positive probability that the sequence of beliefs converges to a belief at which the uninformative action is chosen, so that learning is asymptotically incomplete. Our dynamic programming results imply that when myopically optimal behavior leads with positive probability to asymptotically incomplete learning, such a “learning trap” also exists when the discount factor is positive and sufficiently small