Transfinite game values in infinite 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 chessdenoted by ω Ch 1 in the context of finite positions only and by ω Ch ∼ 1 in the context of all positions, including those with infinitely many pieces. For lower bounds to ω Ch ∼ 1 , we present specific positions with transfinite game values of ω, ω 2 , ω 2 • k and ω 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 threedimensional chess is as large as it can be, namely, ω