States And Microstates In A Mean-Field Approach To The Minority Game And Its Generalizations

We developed a mean-field approach to the Minority Game, that allows us to reproduce the behavior of the model in the phase in which a so-called "dynamics of period two" is present. This dynamics describes the situation where the model is controlled by the presence of crowds of agents participating in the game. Our approach is based on the hypothesis that we can introduce states representative of the system, in such a form that averages over time can be replaced by averages over those states. The main idea is to work with virtual agents, rather than working with the actual set of agents of a particular game. Virtual agents are built from all the possible pairs of the strategies available in the model (the Full Strategy Space FSS). Moreover, we define an ensemble of microstates and, thereafter, states compatible with the specifications of the game. In this work we explain in detail how to introduce these elements, and how to actually calculate the ensemble of states and microstates. We have developed one generalization of the Minority Game as an attempt to make the model more realistic, by introducing interactions among the agents. We also discuss and explain the adequate ensemble of states and microstates for that generalization.