Compositional Game Theory, Compositionally

TL;DR

This paper introduces a novel compositional framework for game theory using Arrow concepts from functional programming, enabling modular construction and analysis of complex strategic interactions.

Contribution

It develops a new compositional approach to CGT based on Arrows, bimodules, and graded Arrows, allowing systematic construction of open games and their categorical properties.

Findings

Proves that variants of open games form symmetric monoidal categories

Introduces operators for building new Arrows from bimodules and graded Arrows

Provides a modular, compositional framework for analyzing game-theoretic models

Abstract

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 bimodule over an Arrow and define an operator to build a new Arrow from such a bimodule 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.