Transfinite game values in infinite chess

TL;DR

This paper explores the complex transfinite game values in infinite chess, establishing bounds, constructing specific positions with various ordinal values, and demonstrating the vast diversity of game outcomes in infinite and three-dimensional chess.

Contribution

It introduces new bounds on the omega one of chess, constructs specific infinite positions with transfinite values, and shows that all countable ordinals can be realized in infinite 3D chess.

Findings

Infinite chess has transfinite game values up to omega one.

Specific positions with values omega, omega^2, omega^2*k, and omega^3 are constructed.

Every countable ordinal can be realized as a game value in infinite 3D chess.

Abstract

We investigate the transfinite game values arising in infinite chess, providing both upper and lower bounds on the supremum of these values---the omega one of chess---with two senses depending on whether one considers only finite positions or also positions with infinitely many pieces. For lower bounds, we present specific infinite positions with transfinite game values of omega, omega^2, omega^2 times k, and omega^3. By embedding trees into chess, we show that there is a computable infinite chess position that is a win for white if the players are required to play according to a deterministic computable strategy, but which is a draw without that restriction. Finally, we prove that every countable ordinal arises as the game value of a position in infinite three-dimensional chess, and consequently the omega one of infinite three-dimensional chess is as large as it can be, namely, true omega one.