  We study analytically and by computer simulations a complex system of
adaptive agents with finite memory. Borrowing the framework of the Minority
Game and using the replica formalism we show the existence of an equilibrium
phase transition as a function of the ratio between the memory $\lambda$ and
the learning rates $\Gamma$ of the agents. We show that, starting from a random
configuration, a dynamic phase transition also exists, which prevents the
system from reaching any Nash equilibria. Furthermore, in a non-stationary
environment, we show by numerical simulations that agents with infinite memory
play worst than others with less memory and that the dynamic transition
naturally arises independently from the initial conditions.
