String diagrams for game theory

TL;DR

This paper introduces a monoidal category framework with string diagrams for representing and reasoning about game-theoretic concepts, enabling compositional analysis of complex games.

Contribution

It develops a novel categorical and diagrammatic approach to model and analyze games, including a recursive definition of Nash equilibrium based on causal structure.

Findings

Defines a monoidal category of games with diagrammatic language

Provides a recursive Nash equilibrium concept based on causal structure

Uses continuation passing style for reasoning about future consequences

Abstract

This paper presents a monoidal category whose morphisms are games (in the sense of game theory, not game semantics) and an associated diagrammatic language. The two basic operations of a monoidal category, namely categorical composition and tensor product, correspond roughly to sequential and simultaneous composition of games. This leads to a compositional theory in which we can reason about properties of games in terms of corresponding properties of the component parts. In particular, we give a definition of Nash equilibrium which is recursive on the causal structure of the game. The key technical idea in this paper is the use of continuation passing style for reasoning about the future consequences of players' choices, closely based on applications of selection functions in game theory. Additionally, the clean categorical foundation gives many opportunities for generalisation, for example to learning agents.