Ordinal Sums with Substitution of Impartial Games

TL;DR

This paper introduces a generalized ordinal sum for combinatorial games, analyzes its properties, and derives explicit formulas for Grundy numbers, extending the concept to poset structures.

Contribution

It proposes a new ordinal sum with substitution, providing fundamental properties, formulas for variation sets, and explicit Grundy number expressions, expanding the theory of combinatorial game sums.

Findings

Derived a simple formula for variation sets of ordinal sums with substitution.

Explicitly expressed Grundy numbers for chains of such sums involving nimbers.

Extended the concept to poset structures with illustrative examples.

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.