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arXiv · 2604.27942

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

Djamel Bouchaffra·Faycal Ykhlef·Mustapha Lebbah·Hanane Azzag
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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.

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Djamel Bouchaffra, Faycal Ykhlef, Mustapha Lebbah, Hanane Azzag. 2026-04-30. A Collective Variational Principle Unifying Bayesian Inference, Game Theory, and Thermodynamics. https://arxiv.org/abs/2604.27942

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