A foundational assumption in economics is that people are rational: they choose optimal plans of action given their predictions about future states of the world. In games of strategy this means that each player's strategy should be optimal given his or her prediction of the opponents' strategies. We demonstrate that there is an inherent tension between rationality and prediction when players are uncertain about their opponents' payoff functions. Specifically, there are games in which it is impossible for perfectly rational players to learn to predict the future behavior of their opponents (even approximately) no matter what learning rule they use. The reason is that in trying to predict the next-period behavior of an opponent, a rational player must take an action this period that the opponent can observe. This observation may cause the opponent to alter his next-period behavior, thus invalidating the first player's prediction. The resulting feedback loop has the property that, a positive fraction of the time, the predicted probability of some action next period differs substantially from the actual probability with which the action is going to occur. We conclude that there are strategic situations in which it is impossible in principle for perfectly rational agents to learn to predict the future behavior of other perfectly rational agents based solely on their observed actions.
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