[Paper Review] Compositional Game Theory with Mixed Strategies: Probabilistic Open Games Using a Distributive Law

Neil Ghani, Clemens Kupke+2 more|Strathprints: The University of Strathclyde institutional repository (University of Strathclyde)|Sep 15, 2020

Game Theory and Voting Systems4 citations

TL;DR

This paper extends compositional game theory to mixed strategies using the discrete probability distribution monad, constructing a symmetric monoidal category of probabilistic open games. It introduces a relational Kleisli lifting to model mixed strategies without requiring probabilistic play functions, and proves that the resulting framework supports parallel and sequential composition, functors, and adjunctions between pure and probabilistic games.

ABSTRACT

We extend the open games framework for compositional game theory to encompass also mixed strategies, making essential use of the discrete probability distribution monad. We show that the resulting games form a symmetric monoidal category, which can be used to compose probabilistic games in parallel and sequentially. We also consider morphisms between games, and show that intuitive constructions give rise to functors and adjunctions between pure and probabilistic open games.

Motivation & Objective

• To extend compositional game theory to include mixed strategies, which are essential for ensuring Nash equilibrium existence in finite games.
• To address the limitation of pure open games, which may lack equilibria, by incorporating probabilistic strategy profiles via the discrete distribution monad.
• To develop a categorical framework that supports both sequential and parallel composition of probabilistic games using symmetric mononoidal structure.
• To define morphisms between games that preserve game-theoretic structure and support functors and adjunctions between pure and probabilistic game categories.
• To establish a foundation for modeling correlated equilibria and other effectful game-theoretic phenomena using monadic structures beyond the distribution monad.

Proposed method

• Uses the discrete probability distribution monad $\mathcal{D}$ on $\mathsf{Set}$ to model mixed strategies, with $\mathcal{D}(X)$ representing the set of finite-support probability distributions over a set $X$.
• Introduces a relational Kleisli lifting to transform predicates with non-probabilistic parameters into predicates with mixed parameters, enabling non-trivial handling of probabilistic strategy profiles.
• Defines probabilistic open games as tuples $(\Sigma_{\mathcal{G}}, P_{\mathcal{G}}, C_{\mathcal{G}}, E_{\mathcal{G}})$ where strategy sets are standard, but play and coutility functions are lifted to handle distributions via the monad structure.
• Endows outcome sets $R$ and $S$ with $\mathcal{D}$-algebra structure to support expected utility computation, ensuring compatibility with probabilistic reasoning.
• Constructs a determinisation functor $\Delta^\prime$ mapping probabilistic games to pure games over distribution spaces, using the double strength of $\mathcal{D}$.
• Defines a fully faithful embedding $\Theta$ from pure to probabilistic games and proves it has a right adjoint $\Psi$, establishing a categorical adjunction between the two categories.

Experimental results

Research questions

• RQ1 How can compositional game theory be extended to include mixed strategies while preserving the categorical structure of open games?
• RQ2 What is the correct categorical lifting mechanism to model mixed strategies without requiring probabilistic play functions?
• RQ3 Can the resulting framework of probabilistic open games form a symmetric monoidal category closed under sequential and parallel composition?
• RQ4 How do functors and adjunctions between pure and probabilistic games arise, and what is their game-theoretic significance?
• RQ5 Can this framework be generalized to model other effectful game-theoretic phenomena, such as correlated equilibria or quitting games?

Key findings

• The category $\mathsf{Game}_{\mathsf{Prob}}$ of probabilistic open games forms a symmetric monoidal category under parallel and sequential composition.
• The relational Kleisli lifting enables non-trivial handling of mixed strategies by transforming predicates over pure strategies into predicates over probability distributions.
• The determinisation functor $\Delta^\prime$ maps probabilistic games to pure games over distribution spaces, with $\Sigma_{\Delta^\prime(\mathcal{G})} = \mathcal{D}(\Sigma_{\mathcal{G}})$, and extends to a functor $\Delta^\prime: \mathsf{Game}_{\mathsf{Prob}} \to \mathsf{Game}_{\mathsf{Pure}}^\prime$.
• The embedding functor $\Theta: \mathsf{Game}_{\mathsf{Pure}}^\prime \to \mathsf{Game}_{\mathsf{Prob}}$ has a right adjoint $\Psi$, establishing a categorical duality between pure and probabilistic games.
• The framework successfully models core game-theoretic examples such as Matching Pennies and the Market Entry Game using mixed strategy equilibria.
• The results are general enough that they do not rely on specific properties of $\mathcal{D}$, suggesting applicability to other monads such as the exceptions monad for quitting games.