Replica symmetry breaking in the minority game

We extend and complete recent work concerning the analytic solution of the minority game. Nash equilibria (NE) of the game have been found to be related to the ground states of a disordered hamiltonian with replica symmetry breaking (RSB), signalling the presence of a large number of them. Here we study the number of NE both analytically and numerically. We then analyze the stability of the recently-obtained replica-symmetric (RS) solution and, in the region where it becomes unstable, derive the solution within one-step RSB approximation. We are finally able to draw a detailed phase diagram of the model.