Replica symmetry breaking in the minority game

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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.