The Role of Absolute Continuity in "Merging of Opinions" and "Rational Learning"

Two agents with different priors watch a sequence unfold over time, updating their priors about the future course of the sequence with each new observation. Blackwell and Dubins (1962) show that the agents' opinions about the future will converge if their priors over the sequence space are absolutely continuous: i.e., if they agree on what events are possible. From this Kalai and Lehrer (1993) conclude that the players in a repeated game will eventually agree about the future course of play and thus that "rational learning leads to Nash equilibrium." We provide an alternative proof on convergence that clarifies the role of absolute continuity and in doing so casts doubt on the relevance of the result. From the existence of continued disagreement we construct a sequence of mutually favorable, uncorrelated "bets." By a law of large numbers, both agents are sure that they win these bets on average over the long run and this disagreement over what is possible violates absolute continuity