Ordinal Sums with Substitution of Impartial Games

Abstract

A combinatorial game is a two-player game without hidden information or chance elements. The disjunctive sum $G + H$ of games $G$ and $H$ is the game in which $G$ and $H$ are played in parallel, and a player makes a move on exactly one of $G$ and $H$ in a turn. The ordinal sum $G \colon H$ is similar to the disjunctive sum, but once the left game $G$ is played, the right game $H$ is discarded and can no longer be played. It is known that the outcome of a mixture of disjunctive sums and ordinal sums, such as $(G_1 \colon G_2) + ((G_3 + G_4) \colon G_5)$, is determined by the variation sets, the set of Grundy numbers of all options, of the components in the normal-play. In this paper, we propose a generalization of an ordinal sum, called an ordinal sum with substitution $G \colon_{\widehat{H}} H$, which is the game made by combining $G$, $H$, and $\widehat{H}$ in the following way: the games $G$ and $H$ are played in parallel; a player makes a move on exactly one of $G$ and $H$ in a turn; each time the left game $G$ is played, the right game $H$ is replaced with $\widehat{H}$. We investigate their fundamental properties and prove a simple formula for the variation sets of ordinal sums with substitution. Apply the formula, we give an explicit expression of the Grundy number of a chain of ordinal sums with substitution consisting of nimbers. We also provide an example illustrating the generalization of ordinal sums with substitution to poset structures.