Symbolic Dynamics for Discrete Adaptive Games

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We use symbolic dynamics to study discrete adaptive games, such as the minority game and the El Farol Bar problem. We show that no such game can have deterministic chaos. We put upper bounds on the statistical complexity and period of these games; the former is at most linear in the number agents and the size of their memories. We extend our results to cases where the players have infinite-duration memory (they are still non-chaotic) and to cases where there is ``noise'' in the play (leaving the complexity unchanged or even reduced). We conclude with a mechanism that can reconcile our findings with the phenomenology, and reflections on the merits of simple models of mutual adaptation.