A Collective Variational Principle Unifying Bayesian Inference, Game Theory, and Thermodynamics

2026-04-30

AI Summary

This study addresses the absence of a unified theoretical framework for explaining the emergence of collective intelligence in decentralized multi-agent systems. The authors propose a game-theoretic free energy principle, establishing the first variational framework that unifies Bayesian inference, game theory, and thermodynamics. By embedding local free energy minimization within stochastic games under constraints of bounded rationality and partial observability, they demonstrate that stationary points of collective free energy correspond to approximate Nash equilibria of the induced game and provide a Gibbs-distribution-based variational representation for cooperative games. The work introduces a novel free-energy formulation of the Harsanyi dividend to capture irreducible synergistic effects and reveals a non-monotonic relationship between perceptual precision and agent influence. The theory is validated across neural, biological, and artificial multi-agent systems, thereby unifying foundational principles of inference, thermodynamics, and game-theoretic equilibrium.

Abstract

Collective intelligence emerges across biological, physical, and artificial systems without central coordination, yet a unifying principle governing such behaviour remains elusive. The Free Energy Principle explains how individual agents adapt through variational inference, while game theory formalises strategic interactions. Here we introduce the Game-Theoretic Free Energy Principle, a unified framework showing that multi-agent systems performing local free-energy minimisation implicitly implement a stochastic game. We prove that, under bounded rationality and local information constraints, stationary points of collective free energy correspond to approximate Nash equilibria of an induced game. Conversely, a broad class of cooperative games admits a variational representation in which equilibria arise as Gibbs distributions over coalitions, establishing a bridge between Bayesian inference and strategic interaction. To characterise higher-order effects, we introduce a free-energy formulation of the Harsanyi dividend, isolating irreducible multi-agent synergy. This yields a predictive theory of cooperation, including a falsifiable non-monotonic relationship between sensory precision and agent influence. We validate this prediction across neural, biological, and artificial multi-agent systems. These results identify a common variational principle underlying inference, thermodynamics, and game-theoretic equilibrium.

Problem

• collective intelligence
• unifying principle
• Bayesian inference
• game theory
• thermodynamics

Innovation

• Free Energy Principle
• Variational Inference
• Stochastic Games
• Nash Equilibrium
• Harsanyi Dividend