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

arXiv: https://arxiv.org/abs/2604.27942 · HTML: https://mlsanity.com/paper/2604.27942/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 39 references that are in the map.
- First posted on arXiv 30 Apr 2026.

## Builds on (most cited first, 4 of 4)

- [NeuroGame Transformer: Gibbs-Inspired Attention Driven by Game Theory and Statistical Physics](https://mlsanity.com/paper/2603.18761/) (2026)
- [Q-Learning-Driven Adaptive Rewiring for Cooperative Control in Heterogeneous Networks](https://mlsanity.com/paper/2509.01057/) (2025)
- [Neural Computation Without Slots: Steps Towards Biologically Plausible Memory and Attention in Natural and Artificial Intelligence](https://mlsanity.com/paper/2511.04593/) (2025)
- [Sequential Causal Normal Form Games: Theory, Computation, and Strategic Signaling](https://mlsanity.com/paper/2511.06934/) (2025)

## Built on by (most cited first, 1 of 1)

- [A Game Theoretic Free Energy Analysis of Higher Order Synergy in Attention Heads of Large Language Models](https://mlsanity.com/paper/2605.09515/) (2026)

## Abstract (arXiv)

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.