A Category for Extensive-Form Games

TL;DR

This paper develops a categorical framework for extensive-form games, defining morphisms and isomorphisms that preserve game structure, and applies it to analyze subgames, equilibria, and subclasses of games.

Contribution

It introduces Gm, a category-theoretic structure for extensive-form games, enabling formal analysis of game properties and subclasses through categorical morphisms.

Findings

Game isomorphisms preserve Nash equilibria and subgame structures.

Selten subgames are characterized as categorical subgames.

Subcategories of action sequence and action-set games are essentially wide.

Abstract

This paper introduces Gm, which is a category for extensive-form games. It also provides some applications. The category's objects are games, which are understood to be sets of nodes which have been endowed with edges, information sets, actions, players, and utility functions. Its arrows are functions from source nodes to target nodes that preserve the additional structure. For instance, a game's information-set collection is newly regarded as a topological basis for the game's decision-node set, and thus a morphism's continuity serves to preserve information sets. Given these definitions, a game monomorphism is characterized by the property of not mapping two source runs (plays) to the same target run. Further, a game isomorphism is characterized as a bijection whose restriction to decision nodes is a homeomorphism, whose induced player transformation is injective, and which strictly preserves the ordinal content of the utility functions. The category is then applied to some game-theoretic concepts beyond the definition of a game. A Selten subgame is characterized as a special kind of categorical subgame, and game isomorphisms are shown to preserve strategy sets, Nash equilibria, Selten subgames, subgame-perfect equilibria, perfect-information, and no-absentmindedness. Further, it is shown that the full subcategory for distinguished-action sequence games is essentially wide in the category of all games, and that the full subcategory of action-set games is essentially wide in the full subcategory for games with no-absentmindedness.