# Compositional game theory

arXiv: https://arxiv.org/abs/1603.04641 · HTML: https://mlsanity.com/paper/1603.04641/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Mar 2016.

## Built on by (most cited first, 13 of 13)

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- [Active Inference in String Diagrams: A Categorical Account of Predictive Processing and Free Energy](https://mlsanity.com/paper/2308.00861/) (2023)
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- [Space-time tradeoffs of lenses and optics via higher category theory](https://mlsanity.com/paper/2209.09351/) (2022)
- [Cross-Entropy Games for Language Models: From Implicit Knowledge to General Capability Measures](https://mlsanity.com/paper/2506.06832/) (2025)
- [Cognitive Training for Language Models: Towards General Capabilities via Cross-Entropy Games](https://mlsanity.com/paper/2603.22479/) (2026)
- [AutoBayes: A Compositional Framework for Generalized Variational Inference](https://mlsanity.com/paper/2503.18608/) (2025)
- [Game-Theoretic Gradient Control for Robust Neural Network Training](https://mlsanity.com/paper/2507.19143/) (2025)

## Abstract (arXiv)

We introduce open games as a compositional foundation of economic game theory. A compositional approach potentially allows methods of game theory and theoretical computer science to be applied to large-scale economic models for which standard economic tools are not practical. An open game represents a game played relative to an arbitrary environment and to this end we introduce the concept of coutility, which is the utility generated by an open game and returned to its environment. Open games are the morphisms of a symmetric monoidal category and can therefore be composed by categorical composition into sequential move games and by monoidal products into simultaneous move games. Open games can be represented by string diagrams which provide an intuitive but formal visualisation of the information flows. We show that a variety of games can be faithfully represented as open games in the sense of having the same Nash equilibria and off-equilibrium best responses.