Translating Extensive Form Games to Open Games with Agency

Abstract

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.

Infinite Horizon Extensive Form Games, Coalgebraically

“…In future work, we plan to exploit techniques from coalgebra and automata theory to use our framework to solve various infinite-horizon games, i.e., to compute the set of equilibria for some given notion of equilibrium -Nash, sub game perfect equilibria, etc. We also hope to extend our recent translation of finite extensive form games into the framework of open games [9] to infinite games; since open games are wellsuited for software implementation, this might point to another approach for computing the equilibria of infinite extensive form games in practice.…”

Towards Foundations of Categorical Cybernetics

“…We hereby present the first categorical formalization of this neural network architecture and interpret its game-theoretic behaviour in terms of opposing reparametrisations: gradient descent and gradient ascent. 4 A GAN is composed of two networks: a generator (g) and a discriminator (d) network in Para(Optic(Smooth)), obtained as the image under the functor Para(R) (Equation 1). The discriminator (an element of Para(Smooth)(R x , R)) is tasked with assigning an R-valued cost to each point of a particular "image space" (denoted with R x ) as a measure of how much it looks like it belongs to a particular predefined dataset, where low cost means realistic-looking samples.…”

“…This results in a category in which we can do 'compositional game theory', which refines open games [12] by adding an explicit account of agents. More detail on this perspective can be found in [4].…”

“…Although this is only a very simple example, separating an open game into these 'horizontal' and 'vertical' parts provides a very flexible language for describing a wide variety of game-theoretic situations, such as those described in [4].…”

“…However, abstractly the same construction works if selection functions are replaced with any suitable lax monoidal pseudofunctor on a category of optics. Further applications of this framework are given in a companion paper [4].…”