Statistical mechanics of systems with heterogeneous agents: Minority games

We study analytically a simple game theoretical model of heterogeneous inte racting agents. We show that the stationary state of the system is describe d by the ground state of a disordered spin model which is exactly solvable within the simple replica symmetric ansatz. Such a stationary state differs from the Nash equilibrium where each agent maximizes her own utility. The latter turns out to be characterized by a replica symmetry broken structure . Numerical results fully agree with our analytical findings.