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

• URL: http://arxiv.org/abs/2604.27942v1
• Date: Thu, 30 Apr 2026 14:42:23 GMT
• Title: A Collective Variational Principle Unifying Bayesian Inference, Game Theory, and Thermodynamics
• Abstract summary: We introduce the Game-Theoretic Free Energy Principle, showing that multi-agent systems implicitly implement a game.<n>We show that stationary points of collective free energy correspond to approximate Nash equilibria of an induced game.<n>This yields a predictive theory of cooperation, including a falsifiable non-monotonic relationship between sensory precision and agent influence.
• Score: 1.9365949510386082
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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