Stationary states of a spherical Minority Game with ergodicity breaking

TL;DR

This paper analyzes a spherical Minority Game model exhibiting ergodicity breaking, dependence on initial conditions, and phase transitions, using advanced analytical techniques and confirming results with simulations.

Contribution

It introduces a spherical MG with broken ergodicity and initial condition dependence, extending previous models and analyzing effects like market impact correction.

Findings

Identification of a phase with broken ergodicity

Explicit calculation of order parameters including volatility

Confirmation of analytical results through numerical simulations

Abstract

Using generating functional and replica techniques, respectively, we study the dynamics and statics of a spherical Minority Game (MG), which in contrast with a spherical MG previously presented in J.Phys A: Math. Gen. 36 11159 (2003) displays a phase with broken ergodicity and dependence of the macroscopic stationary state on initial conditions. The model thus bears more similarity with the original MG. Still, all order parameters including the volatility can computed in the ergodic phases without making any approximations. We also study the effects of market impact correction on the phase diagram. Finally we discuss a continuous-time version of the model as well as the differences between on-line and batch update rules. Our analytical results are confirmed convincingly by comparison with numerical simulations. In an appendix we extend the analysis of the earlier spherical MG to a model with general time-step, and compare the dynamics and statics of the two spherical models.