Spin Glass Mapping of the Parallel Minority Game

TL;DR

This paper maps the parallel minority game to an Ising spin glass model, revealing the complex optimization landscape and inherent frustration in agents' choices.

Contribution

It establishes an exact equivalence between the PMG's variance minimization and the ground state of a mean-field spin glass, offering a novel theoretical perspective.

Findings

Mapping shows the PMG as a quadratic Hamiltonian with quenched disorder.

Reveals frustration and sub-optimal frozen states in the game.

Connects game dynamics to well-studied spin glass theory.

Abstract

The parallel minority game (PMG) extends the classical minority game to many choices, with each agent restricted to two predetermined alternatives. In this condition, minimizing the population variance across all choices is a complex combinatorial optimization problem. We show that this minimization is exactly equivalent to finding the ground state of an Ising spin glass in the mean-field limit, i.e., the Sherrington-Kirkpatrick model. By encoding the agent choices as spin variables, the variance becomes a quadratic Hamiltonian with quenched random couplings $J_{ij}$ and random fields $h_i$. This mapping reveals inherent frustration and connects the PMG to the well developed theory of spin glasses, providing a new perspective on the frozen, sub-optimal configurations observed in stochastic strategies.