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 relat ed to the ground states of a disordered Hamiltonian with replica symmetry b reaking (RSB), signalling the presence of a large number of NE. Here we stu dy the number of NE both analytically and numerically. We then analyse the stability of the recently obtained replica-symmetric solution and, in the r egion where it becomes unstable, derive the solution within one-step RSB ap proximation. We are finally able to draw a detailed phase diagram of the mo del.