A tractable evolutionary model for the Minority Game with asymmetric payoffs

We set up a simple behavioral model for a large population of agents who are repeatedly playing the Minority Game and whose interaction is modeled by means of the so-called replicator dynamics. This allows us to specify the dynamics of the aggregate variables, the number of agents choosing each side, in terms of a low-dimensional dynamical system that gives qualitatively the same results of the existing computational approaches. As an extension we introduce asymmetric payoffs, i.e., we analyze the case where the minority and majority payoffs are side dependent. In this case the fluctuations out of the equilibrium are qualitatively different. In particular, contrary to the previous case, they are associated with a difference in the average payoff gained by each side. Furthermore, a parameter region exists where the dynamics does not converge to any isolated periodic attractor.