Minority games, evolving capitals and replicator dynamics

TL;DR

This paper analyzes a simplified Minority Game with evolving capitals, revealing phase transitions and ergodicity breaking, and connects it to replicator dynamics, providing insights into market-like behaviors and evolutionary game theory.

Contribution

It introduces a simplified MG model with dynamic capitals, computes its stationary states, and links it to replicator dynamics, highlighting new phase transition phenomena.

Findings

Presence of ergodicity breaking phase transition

Divergence of total capital in the majority game

Different phase transition in the replicator dynamics model

Abstract

We discuss a simple version of the Minority Game (MG) in which agents hold only one strategy each, but in which their capitals evolve dynamically according to their success and in which the total trading volume varies in time accordingly. This feature is known to be crucial for MGs to reproduce stylised facts of real market data. The stationary states and phase diagram of the model can be computed, and we show that the ergodicity breaking phase transition common for MGs, and marked by a divergence of the integrated response is present also in this simplified model. An analogous majority game turns out to be relatively void of interesting features, and the total capital is found to diverge in time. Introducing a restraining force leads to a model akin to replicator dynamics of evolutionary game theory, and we demonstrate that here a different type of phase transition is observed. Finally we briefly discuss the relation of this model with one strategy per player to more sophisticated Minority Games with dynamical capitals and several trading strategies per agent.