Extended Sprague-Grundy theory for locally finite games, and applications to random game-trees

TL;DR

This paper extends Sprague-Grundy theory to locally finite games, demonstrating their equivalence to finite games and analyzing the behavior of randomly generated game-trees using probabilistic models.

Contribution

It generalizes Sprague-Grundy theory to locally finite games and explores the properties of random game-trees via Galton-Watson processes.

Findings

Locally finite games are equivalent to finite games.

Random game-trees exhibit diverse behaviors depending on offspring distributions.

The framework bridges combinatorial game theory and probability.

Abstract

For a collection of papers in memory of Elwyn Berlekamp (1940-2019), John Conway (1937-2020), and Richard Guy (1916-2020). The Sprague-Grundy theory for finite games without cycles was extended to general finite games by Cedric Smith and by Aviezri Fraenkel and coauthors. We observe that the same framework used to classify finite games also covers the case of locally finite games (that is, games where any position has only finitely many options). In particular, any locally finite game is equivalent to some finite game. We then study cases where the directed graph of a game is chosen randomly, and is given by the tree of a Galton-Watson branching process. Natural families of offspring distributions display a surprisingly wide range of behaviour. The setting shows a nice interplay between ideas from combinatorial game theory and ideas from probability.