Continuum time limit and stationary states of the minority game - art. no.056138

We discuss in detail the derivation of stochastic differential equations fo r the continuum time limit of the minority game. We show that all propertie s of the minority game can be understood by a careful theoretical analysis of such equations. In particular, (i) we confirm that the stationary state properties are given by the ground state configurations of a disordered (so ft) spin system. (ii) we derive the full stationary state distribution, (ii i) we characterize the dependence on initial conditions in the symmetric ph ase, and (iv) we clarify the behavior of the system as a function of the le arning rate. This leaves us with a complete and coherent picture of the col lective behavior of the minority game. Strikingly we find that the temperat urelike parameter, which is introduced in the choice behavior of individual agents turns out to play the role, at the collective level, of the inverse of a thermodynamic temperature.