Prediction, Optimization, and Learning in Repeated Games

Consider a two-player discounted repeated game in which each player optimizes with respect to prior beliefs about his opponent's repeated game strategy. One would like to argue that if beliefs are cautious then players will learn as the game unfolds to predict the continuation path of play. If this conjecture were true then a convergence result due to Kalai and Lehrer would imply that the continuation path would asymptotically resemble the path of a Nash equilibrium. One would thus have constructed a theory which predicts Nash equilibrium as the necessary long-run consequence of optimization by cautious players. This paper points out that there is an obstacle to such a result in the form of a potential conflict between prediction and optimization.