Diffusion and localization of relative strategy scores in the Minority Game

TL;DR

This paper models the distribution of relative strategy scores in the Minority Game's asymmetric phase, revealing how scores evolve as a random walk and matching simulations without complex minimization.

Contribution

It introduces a simple, flexible statistical model for strategy score dynamics in the Minority Game, applicable to various payoff functions.

Findings

Distribution of scores is either static or diffusive, depending on agent behavior.

Model predictions align quantitatively with simulations.

Framework is adaptable to different payoff structures.

Abstract

We study the equilibrium distribution of relative strategy scores of agents in the asymmetric phase ($\alpha\equiv P/N\gtrsim 1$) of the basic Minority Game using sign-payoff, with $N$ agents holding two strategies over $P$ histories. We formulate a statistical model that makes use of the gauge freedom with respect to the ordering of an agent's strategies to quantify the correlation between the attendance and the distribution of strategies. The relative score $x\in\mathbb{Z}$ of the two strategies of an agent is described in terms of a one dimensional random walk with asymmetric jump probabilities, leading either to a static and asymmetric exponential distribution centered at $x=0$ for fickle agents or to diffusion with a positive or negative drift for frozen agents. In terms of scaled coordinates $x/\sqrt{N}$ and $t/N$ the distributions are uniquely given by $\alpha$ and in quantitative agreement with direct simulations of the game. As the model avoids the reformulation in terms of a constrained minimization problem it can be used for arbitrary payoff functions with little calculational effort and provides a transparent and simple formulation of the dynamics of the basic Minority Game in the asymmetric phase.