We propose a categorical framework for processes which interact bidirectionally with both an environment and a 'controller'. Examples include open learners, in which the controller is an optimiser such as gradient descent, and an approach to compositional game theory closely related to open games, in which the controller is a composite of game-theoretic agents. We believe that 'cybernetic' is an appropriate name for the processes that can be described in this framework.

We propose a categorical framework for processes which interact bidirectionally with both an environment and a 'controller'. Examples include open learners, in which the controller is an optimiser such as gradient descent, and an approach to compositional game theory closely related to open games, in which the controller is a composite of game-theoretic agents. We believe that 'cybernetic' is an appropriate name for the processes that can be described in this framework.

We show open games cover extensive form games with both perfect and imperfect information. Doing so forces us to address two current weaknesses in open games: the lack of a notion of player and their agency within open games, and the lack of choice operators. Using the former we construct the latter, and these choice operators subsume previously proposed operators for open games, thereby making progress towards a core, canonical and ergonomic calculus of game operators. Collectively these innovations increase the level of compositionality of open games, and demonstrate their expressiveness. put(,) 222Translating Extensive Form Games to Open Games With Agency • We show that extensive form games can be translated into open games with agency preserving the decision structure and with the same equilibria. Our proof is a significant improvement on the 8-page proof in [7] for perfect information games, while the result for imperfect information games is new. Our translations rely crucially on the new choice operators and reparametrisation operations.Outline of the paper. In Section 2, we introduce extensive form games as an inductive data type.In Section 3, we introduce the new open games with agency, and Section 4 discusses the associated operator calculus. Section 5 contains the translation from the former to the latter. Finally, in Section 6, we draw conclusions and discuss further work.Acknowledgements. The authors would like to thank Scott Cunningham, Clemens Kupke, Glynn Winskel and Radu Mardare for their thoughtful feedback, help and discussions, as well as Jules Hedges, Bruno Gavranović and the rest of MSP group for invaluable input and conversations.

We show open games cover extensive form games with both perfect and imperfect information. Doing so forces us to address two current weaknesses in open games: the lack of a notion of player and their agency within open games, and the lack of choice operators. Using the former we construct the latter, and these choice operators subsume previous proposed operators for open games, thereby making progress towards a core, canonical and ergonomic calculus of game operators. Collectively these innovations increase the level of compositionality of open games, and demonstrate their expressiveness.

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