Impartial Games with Activeness

TL;DR

This paper introduces a new framework for impartial combinatorial games where each game has an active or inactive status that can change, affecting the end condition of the disjunctive sum.

Contribution

It formalizes the concept of impartial games with activeness, extending traditional theory to include dynamic status changes and new end conditions.

Findings

Defined the concept of activeness in impartial games

Analyzed properties and implications of active/inactive statuses

Established foundational results for the new framework

Abstract

A combinatorial game is a two-player game without hidden information or chance elements. The main object of combinatorial game theory is to obtain the outcome, which player has a winning strategy, of a given combinatorial game. Positions of many well-known combinatorial games are naturally decomposed into a disjunctive sum of multiple components and can be analyzed independently for each component. Therefore, the study of disjunctive sums is a major topic in combinatorial game theory. Combinatorial games in which both players have the same set of possible moves for every position are called impartial games. In the normal-play convention, it is known that the outcome of a disjunctive sum of impartial games can be obtained by computing the Grundy number of each term. The theory of impartial games is generalized in various forms. This paper proposes another generalization of impartial games to a new framework, impartial games with activeness: each game is assigned a status of either ``active'' or ``inactive''; the status may change by moves; a disjunctive sum of games ends immediately, not only when no further moves can be made, but also when all terms become inactive. We formally introduce impartial games with activeness and investigate their fundamental properties.