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  • Open Access

Complex dynamics in the Sherrington-Kirkpatrick game

Desmond Chan*

Tobias Galla†

  • Department of Informatics, King's College, London WC2B 4BG, United Kingdom

  • Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC, Edifici Instituts Universitaris de Recerca, Campus Universitat de les Illes Balears, E-07122 Palma, Mallorca, Spain

  • *Contact author: desmond.chan@kcl.ac.uk
  • †Contact author: tobias.galla@ifisc.uib-csic.es
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Phys. Rev. E 114, 044325 – Published 28 October, 2026

DOI: https://doi.org/10.1103/b1qh-k8pn

Abstract

We study the outcome of adaptive learning among many players engaged in sets of two-strategy, two-player games. We are interested in typical games and therefore generate the payoff matrices randomly at the outset. These payoff matrices then remain fixed throughout the learning process. This provides a game-theoretic foundation for the Sherrington-Kirkpatrick (SK) game recently introduced by Garnier-Brun et al. [Phys. Rev. X 14, 021039 (2024)]. The original model studied by these authors corresponds to the special case with no bias toward either strategy. Here, we determine the stability of learning in SK games with general random biases and show 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 may choose to abstain. For this extended model, we again determine the stability of learning. Our analysis confirms that complex situations involving many players are often unlearnable, even if each player chooses between only two actions. The rate at which players forget past payoffs and the competitiveness of the game emerge as key parameters determining whether learning converges to a unique fixed point, whether multiple fixed points arise, or whether the dynamics remain persistently volatile.

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Phys. Rev. E 114, 044325– Published 28 October, 2026
  • Received 2 July 2026
  • Accepted 7 October 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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