We show that for many classes of symmetric two-player games, the simple decision rule "imitate-if-better" can hardly be beaten by any strategy. We provide necessary and sufficient conditions for imitation to be unbeatable in the sense that there is no strategy that can exploit imitation as a money pump. In particular, imitation is subject to a money pump if and only if the relative payoff function of the game is of the rock-scissors-paper variety. We also show that a sufficient condition for imitation not being subject to a money pump is that the relative payoff game is a generalized ordinal potential game or a quasiconcave game. Our results apply to many interesting examples of symmetric games including 2 × 2 games, Cournot duopoly, price competition, public goods games, common pool resource games, and minimum effort coordination games.

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We use a large-scale internet experiment to explore how subjects learn to play against computers that are programmed to follow one of a number of standard learning algorithms. The learning theories are (unbeknown to subjects) a best response process, fictitious play, imitation, reinforcement learning, and a trial & error process. We explore how subjects' performances depend on their opponents' learning algorithm. Furthermore, we test whether subjects try to influence those algorithms to their advantage in a forward-looking way (strategic teaching). We find that strategic teaching occurs frequently and that all learning algorithms are subject to exploitation with the notable exception of imitation.Financial support by the DFG through SFB/TR 15 and SFB 504 is gratefully acknowledged. We thank two anonymous referees, David Cooper, Drew Fudenberg, Tim Grebe, Aaron Lowen, and seminar participants in Edinburgh, Heidelberg, Mannheim, Vienna, Tsukuba, the University of Arizona, and at the ESA Meetings 2005 in Tucson for helpful comments.

We observe that a symmetric two-player zero-sum game has a pure strategy equilibrium if and only if it is not a generalized rock-paper-scissors matrix. Moreover, we show that every finite symmetric quasiconcave two-player zero-sum game has a pure equilibrium. Further sufficient conditions for existence are provided. Our findings extend to general two-player zero-sum games using the symmetrization of zero-sum games due to von Neumann. We point out that the class of symmetric twoplayer zero-sum games coincides with the class of relative payoff games associated with symmetric two-player games. This allows us to derive results on the existence of finite population evolutionary stable strategies.

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