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

TL;DR

This paper introduces a unifying variational principle linking Bayesian inference, game theory, and thermodynamics to explain collective intelligence across systems.

Contribution

It presents the Game-Theoretic Free Energy Principle, connecting free-energy minimization with strategic game equilibria in multi-agent systems.

Findings

Stationary points of collective free energy approximate Nash equilibria.

A variational representation of cooperative games with Gibbs distributions.

Prediction of a non-monotonic relationship between sensory precision and influence.

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.