Continuum time limit and stationary states of the minority game.

We discuss in detail the derivation of stochastic differential equations for the continuum time limit of the minority game. We show that all properties 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 (soft) spin system, (ii) we derive the full stationary state distribution, (iii) we characterize the dependence on initial conditions in the symmetric phase, and (iv) we clarify the behavior of the system as a function of the learning rate. This leaves us with a complete and coherent picture of the collective behavior of the minority game. Strikingly we find that the temperaturelike 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.

Relevance of memory in minority games

By considering diffusion on De Bruijn graphs, we study in detail the dynamics of the histories in the minority game, a model of competition between adaptative agents. Such graphs describe the structure of the temporal evolution of M bit strings, each node standing for a given string, i.e., a history in the minority game. We show that the frequency of visit of each history is not given by 1/2(M) in the limit of large M when the transition probabilities are biased. Consequently, all quantities of the model do significantly depend on whether the histories are real or uniformly and randomly sampled. We expose a self-consistent theory of the case of real histories, which turns out to be in very good agreement with numerical simulations.

Statistical mechanics of systems with heterogeneous agents: minority games

We study analytically a simple game theoretical model of heterogeneous interacting agents. We show that the stationary state of the system is described 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.

Phase transition and symmetry breaking in the minority game.

We show that the minority game, a model of interacting heterogeneous agents, can be described as a spin system and displays a phase transition between a symmetric phase and a symmetry broken phase where the game's outcome is predictable. As a result a "spontaneous magnetization" arises in the spin formalism.