Best reply structure and equilibrium convergence in generic games
Game theory often assumes rational players that play equilibrium strategies. But when the players have to learn their strategies by playing the game repeatedly, how often do the strategies converge? We analyze generic two player games using a standard learning algorithm, and also study replicator dynamics, which is closely related. We show that the frequency with which strategies converge to a fixed point can be understood by analyzing the best reply structure of the payoff matrix. A Boolean transformation of the payoff matrix, replacing all best replies by one and all other entries by zero, provides a reasonable approximation of the asymptotic strategic dynamics. We analyze the generic structure of randomly generated payoff matrices using combinatorial methods to compute the frequency of cycles of different lengths under the microcanonical ensemble. For a game with $N$ possible moves the frequency of cycles and non-convergence increases with $N$, becoming dominant when $N > 10$. This is especially the case when the interactions are competitive.
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