  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.
