Generating functional analysis of minority games with inner product strategy definitions

TL;DR

This paper applies generating functional methods to analyze inner product minority games, revealing structural similarities and differences with lookup table versions, and supports findings with numerical simulations.

Contribution

It generalizes the theory of minority games to inner product strategies, highlighting key differences in history processing and phase behavior.

Findings

Inner product MGs share phase diagrams with lookup table MGs, except when histories are sampled linearly.

The role of history frequency distribution is replaced by the eigenvalue spectrum of a covariance matrix.

Theoretical predictions are validated by numerical simulations.

Abstract

We use generating functional methods to solve the so-called inner product versions of the minority game (MG), with fake and/or real market histories, by generalizing the theory developed recently for look-up table MGs with real histories. The phase diagrams of the lookup table and inner product MG versions are generally found to be identical, with the exception of inner product MGs where histories are sampled linearly, which are found to be structurally critical. However, we encounter interesting differences both in the theory (where the role of the history frequency distribution in lookup table MGs is taken over by the eigenvalue spectrum of a history covariance matrix in inner product MGs) and in the static and dynamic phenomenology of the models. Our theoretical predictions are supported by numerical simulations.