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We consider a repeated interaction between a long-run player and a sequence of short-run players, in which the long-run player may either be rational or may be a mechanical type who plays the same (possibly mixed) action in every stage game. We depart from the classical model in assuming that the short-run players make inferences by analogical reasoning, meaning that they correctly identify the average strategy of each type of long-run player, but do not recognize how this play varies across histories. Concentrating on 2 × 2 games, we provide a characterization of equilibrium payoffs, establishing a payoff bound for the rational long-run player that can be strictly larger than the familiar "Stackelberg" bound. We also provide a characterization of equilibrium behavior, showing that play begins with either a reputation-building or a reputation-spending stage (depending on parameters), followed by a reputation-manipulation stage. Copyright 2012, Oxford University Press.