Complex dynamics in the Sherrington-Kirkpatrick game

2026-07-02

AI Summary

This study investigates adaptive learning dynamics in multi-player stochastic two-strategy games, with a focus on how random biases and voluntary participation (abstention) mechanisms influence convergence and stability. Building upon the Sherrington–Kirkpatrick game model, the authors introduce an extension incorporating general random biases and a grand-canonical formulation of abstention. By integrating tools from statistical physics, random matrix theory, and dynamical systems analysis, they develop an analytical framework for high-dimensional learning dynamics. Their findings reveal that the interplay between memory decay rate and game competitiveness governs convergence behavior; random biases substantially alter the nature of stable states, while abstention modulates learning stability. Notably, even in two-strategy settings, most multi-player games generically exhibit persistent oscillations rather than converging to a unique equilibrium, highlighting an intrinsic unlearnability inherent in such systems.

Abstract

We study the outcome of adaptive learning of a large number of players engaging in sets of two-strategy two-player games. We are interested in typical games, and generate the payoff matrices at random at the beginning. The payoff matrices then remain fixed during the learning process. This provides a game theoretic foundation for the Sherrington-Kirkpatrick (SK) game, recently introduced by Garnier-Brun, Benzaquen and Bouchaud. The original model by these authors is a special case, with no bias towards any strategy. We here determine stability of learning for SK games with general random bias, and find that the nature of the stable state is affected by random fields. We also introduce a grand-canonical version of the SK game, in which players can choose to abstain. We determine the stability of learning for this game. Our analysis confirms that complex situations involving many players are frequently unlearnable, even if each player only chooses between two different actions. The rate with which players lose memory of past payoffs and the competitiveness of the game emerge as key parameters determining whether learning converges to a unique fixed point, whether there are many fixed points, or if the dynamics remains persistently volatile.

Problem

• Sherrington-Kirkpatrick game
• adaptive learning
• random payoff matrices
• learning stability
• multi-player games

Innovation

• Sherrington-Kirkpatrick game
• adaptive learning
• random bias
• grand-canonical formulation
• learning stability