The Game Value of Sequential Compounds of Integers and Stars

TL;DR

This paper analyzes the game values of sequential compounds involving integers and stars, providing a comprehensive understanding of their properties in combinatorial game theory.

Contribution

It introduces the game values of sequential compounds of integers and stars, expanding the theoretical framework for analyzing such game structures.

Findings

Game values of sequential compounds of integers and stars are characterized.

Properties of sequential compounds are systematically analyzed.

Results facilitate understanding of complex game compositions.

Abstract

A combinatorial game is a two-player game without hidden information or chance elements. One of the major approaches to analyzing games in combinatorial game theory is to break down a given game position into a disjunctive sum of multiple sub-positions, then evaluate the game value of each component of the sum, and finally integrate these game values to find which player has a winning strategy in the whole position. Accordingly, finding the game value of a given position is a major topic in combinatorial game theory. The sequential compound proposed by Stromquist and Ullman is a combinatorial game consisting of two combinatorial games. In the sequential compound of games $G$ and $H$, the players make moves on $G$ until $G$ is over, and then they play on $H$. In this paper, we investigate the general properties of sequential compounds. As the main result, we give the game values of sequential compounds of a finite number of integers and stars, which are basic and typical games in combinatorial game theory.