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 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.

Interpreting Dynamical Systems as Bayesian Reasoners

“…These include approaches to Bayesian inference [37] and game theory [10]. The idea that agency is related to parametrisation has also arisen in these contexts [11,38,12]. These works focus on the notion of a lens and its generalisations.…”

“…This is a key component of work on the Free Energy Principle (FEP) [19,16,34] and also of [39], although our approach differs from these in that our model is not derived from the dynamics of the true environment. The notion that agency is closely related to parametrisation is also central to recent approaches to agency based on category theory [11,38,12].…”

Categorical Foundations of Gradient-Based Learning

“…loss ϵ + dense P P A (10) This morphism is constructed essentially as below, where apply update(α, f ) represents the 'vertical stacking' of α atop f : a p p l y u p d a t e ( b a s i c u p d a t e , d e n s e ) >> l o s s >> l e a r n i n g r a t e ( ϵ ) Now, given the parametric lens of (10), one can construct a morphism step : B ×P ×A → P which is simply the put map of the lens. Training the model then consists of iterating the step function over dataset examples (x, y) ∈ A×B to optimise some initial choice of parameters θ 0 ∈ P , by letting θ i+1 = step(y i , θ i , x i ).…”

“…A general survey of category theoretic approaches to machine learning, covering many of the above papers, can be found in [43]. Lastly, the concept of parametric lenses has started appearing in recent formulations of categorical game theory and cybernetics [9,10]. The work of [9] generalises the study of parametric lenses into parametric optics and connects it to game thereotic concepts such as Nash equilibria.…”

Specifying a Game-Theoretic Extensive Form as an Abstract 5-ary Relation

“…The application to decomposition is related to the composable open games of Capucci, Ghani, Ledent, and Nordvall Forsberg 2021, Bolt, Hedges, and Zahn 2019, and Ghani, Kupke, Lambert, and Nordvall Forsberg 2018, 2020. The mathematics there is very different from the mathematics here, and more is said there about utility.…”