Limits to Rational Learning

A long-standing open question raised in the seminal paper of Kalai and Lehrer (1993) is whether or not the play of a repeated game, in the rational learning model introduced there, must eventually resemble play of exact equilibria, and not just play of approximate equilibria as demonstrated there.  This paper shows that play may remain distant - in fact, mutually singular - from the play of any equilibrium of the repeated game.  We further show that the same inaccessibility holds in Bayesian games, where the play of a Bayesian equilibrium may continue to remain distant from the play of any equilibrium of the true game.