Learning to coordinate in a complex and nonstationary world.

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 agents from reaching optimal coordination. Furthermore, in a nonstationary environment, we show by numerical simulations that the phase transition becomes discontinuous.