Polynomial rate convergence to an invariant measure for the continuum time limit of the Minority Game

TL;DR

This paper proves polynomial rate convergence to an invariant measure for a continuum time version of the Minority Game, focusing on cases with asymmetric initial conditions and finite choices, providing bounds on convergence time.

Contribution

It establishes conditions under which the continuum time Minority Game converges to an invariant measure and provides bounds on the convergence rate for large populations.

Findings

Polynomial rate of convergence to invariant measure.

Upper bounds on waiting time for stationarity.

Convergence results for asymmetric initial conditions.

Abstract

In this paper we show that the continuum time version of the Minority Game satisfies the criteria for the application of a theorem on the existence of an invariant measure. We consider the special case of a game with "sufficiently" asymmetric initial condition where the number of possible choices for each individual is S=2 and $\Gamma<+\infty$. An upper bound for the asymptotic behavior, as the number of agents grows to infinity, of the waiting time for reaching the stationary state is then obtained.