Mind the gap: A real-valued distance on combinatorial games

TL;DR

This paper introduces a real-valued distance metric on short combinatorial games, explores its topological properties, and investigates the structure of loopy games and their limits within this metric space.

Contribution

It defines a new distance metric on combinatorial games, analyzes the topological structure of the game space, and characterizes limit points and partitions of loopy games.

Findings

Existence of Cauchy sequences in the game space

Identification of limit points and closure properties

Partitioning of loopy games in the metric space

Abstract

We define a real-valued distance metric $wd$ on the space $\mathcal{C}$ of short combinatorial games in canonical form. We demonstrate the existence of Cauchy sequences informed by sidling sequences, find limit points, and investigate the closure $\overline{\mathcal{C}}$, which is shown to partition the set of loopy games in a non-trivial way. Stoppers, enders, and non-stopper-sided loopy games are explored, as well as the topological properties of $(\mathcal{C},wd)$.