Optimal Dimensionality Reduction of Complex Dynamics: The Chess Game as Diffusion on a Free Energy Landscape

Abstract

Dimensionality reduction is ubiquitous in analysis of complex dynamics. The conventional dimensionality reduction techniques, however, focus on reproducing the underlying configuration space, rather than the dynamics itself. The constructed low-dimensional space does not provide complete and accurate description of the dynamics. Here I describe how to perform dimensionality reduction while preserving the essential properties of the dynamics. The approach is illustrated by analyzing the chess game - the archetype of complex dynamics. A variable that provides complete and accurate description of chess dynamics is constructed. Winning probability is predicted by describing the game as a random walk on the free energy landscape associated with the variable. The approach suggests a possible way of obtaining a simple yet accurate description of many important complex phenomena. The analysis of the chess game shows that the approach can quantitatively describe the dynamics of processes where human decision-making plays a central role, e.g., financial and social dynamics.