Universality of Infinite Chess

TL;DR

This paper demonstrates that infinite chess on an infinite board with infinitely many pieces can simulate any open Gale-Stewart game, establishing its maximal strategic complexity and connecting it to the hierarchy of countable ordinals.

Contribution

It proves the universality of infinite chess in representing all open Gale-Stewart games and their game values, including all countable ordinals, with a computable construction.

Findings

Infinite chess can simulate all open Gale-Stewart games.

All countable ordinals can be realized as game values in infinite chess.

A simplified construction uses only kings and pawns to realize all countable ordinals.

Abstract

We prove that chess played on the infinite chessboard $\mathbb{Z}^2$ with infinitely many pieces is as powerful as it could possibly be, by showing that every open Gale-Stewart game with draws is strategically equivalent to some infinite chess position and vice versa. As our construction is computable and open Gale-Stewart games are well understood, this allows us to resolve many open questions about the complexity of infinite chess with infinitely many pieces. In particular, all countable ordinals arise as the game value of some such chess position. We also give an alternate construction that realizes all countable ordinals as game values, with the pleasing property that it consists only of the king pair and pawns.