Bayesian open games

Bolt, Joe; Hedges, Jules; Zahn, Philipp

doi:10.48550/arxiv.1910.03656

Cited by 8 publications

(20 citation statements)

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“…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.…”

Section: Literaturementioning

confidence: 99%

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

Streufert¹

2021

Preprint

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This paper specifies an extensive form as a 5-ary relation (i.e. set of quintuples) which satisfies certain abstract axioms. Each quintuple is understood to list a player, a situation (e.g. information set), a decision node, an action, and a successor node. Accordingly, the axioms are understood to specify abstract relationships between players, situations, nodes, and actions. Such an extensive form is called a "5-form", and a "5-form game" is defined to be a 5-form together with utility functions. The paper's main result is to construct a bijection between (a) those 5-form games with information-set situations and (b) Gm games (Streufert 2021). In this sense, 5-form games equivalently formulate almost all extensive-form games. An application weakens the tree axiom in the presence of the other axioms, which leads to a convenient decomposition of 5-forms.

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Section: Literaturementioning

confidence: 99%

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

Streufert¹

2021

Preprint

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“…The most important of these is the para construction, which yields a category Para(C ) of parametrised morphisms given a category C acted upon by a category of parameters M . We combine this construction with the optics construction [CEG + 20], which is known to model bidirectional information flow in dynamical systems [Mye20] and compositional game theory [BHZ19]. We argue that we can think of optics as being built of coupled parametrised and 'coparametrised' morphisms.…”

Section: Introductionmentioning

confidence: 99%

Towards foundations of categorical cybernetics

Matteo¹,

Gavranović²,

Hedges³

et al. 2021

Preprint

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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.

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“…At its core, CGT involves a completely new representation of games -open games -with operators for constructing larger and more complex games from smaller, simpler (and hence easier to reason about) ones. Despite recent substantial interest and further development [13,10,7,6,4,11,14], open games remain complex structures, e.g. the simplest form of an open game is an 8-tuple consisting of 4 ports, a set of strategies, play and coplay functions and an equilibrium predicate.…”

Section: Introductionmentioning

confidence: 99%

“…the simplest form of an open game is an 8-tuple consisting of 4 ports, a set of strategies, play and coplay functions and an equilibrium predicate. More advanced versions [4] require sophisticated structures such as coends. This causes problems: (i) complex definitions lead to complex and even error prone proofs; (ii) proofs are programs and overly complex proofs turn into overly complex programs; (iii) even worse, this complexity deters experimentation and innovation, for instance the authors of [10] were deterred from trying alternative definitions as the work became prohibitive; and (iv) worse still, this complexity suggests we do not fully understand the mathematical structure of open games.…”

Section: Introductionmentioning

confidence: 99%

“…Category theorists have an answer to such complexity: trade complex definitions in a simple category for simpler definitions in a more complex category. This simplified open games by factoring their definition via the category of lenses, and this more abstract perspective led directly to the discovery of Bayesian Open Games [4] where one naturally swaps out lenses in favour of optics when one starts with a monoidal category. Swapping out a subcomponent for another is a hallmark of compositionality, but, because the definition of open games is not compositional, one has to redevelop all the structure of open games built over optics from scratch.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Compositional Game Theory, Compositionally

Atkey¹,

Gavranović²,

Ghani³

et al. 2021

Electron. Proc. Theor. Comput. Sci.

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We present a new compositional approach to compositional game theory (CGT) based upon Arrows, a concept originally from functional programming, closely related to Tambara modules, and operators to build new Arrows from old. We model equilibria as a module over an Arrow and define an operator to build a new Arrow from such a module over an existing Arrow. We also model strategies as graded Arrows and define an operator which builds a new Arrow by taking the colimit of a graded Arrow. A final operator builds a graded Arrow from a graded bimodule. We use this compositional approach to CGT to show how known and previously unknown variants of open games can be proven to form symmetric monoidal categories.

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Bayesian open games

Cited by 8 publications

References 0 publications

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

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

Towards foundations of categorical cybernetics

Compositional Game Theory, Compositionally

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