Continuous time dynamics of the Thermal Minority Game

We study the continuous time dynamics of the Thermal Minority Game. We find that the dynamical equations of the model reduce to a set of stochastic differential equations for an interacting disordered system with non-trivial random diffusion. This is the simplest microscopic description which accounts for all the features of the system. Within this framework, we study the phase structure of the model and find that its macroscopic properties strongly depend on the initial conditions.