Impartial Games with Activeness
Kengo Hashimoto
TL;DR
This work extends classical impartial game theory to impartial games with activeness, where each component can be active or inactive and a disjunctive sum ends when all components are inactive. It introduces the extended game space ${\\mathbb Im}^{\\mathcal{B}}$, adapts outcome, addition, and equivalence notions, and develops a comprehensive theory including nimbers with activeness via sequences ${\\bm{\\gamma}}$. Central results comprise the Distinguishing Theorem, Simplification Theorem, and Canonical Form Theorem, along with a treatment of nimbers under activeness and a non-increasing variant ${\\mathbb Im}^{\\mathcal{B}\\downarrow}$. Canonical forms provide unique representatives for each equivalence class, while transitive games are canonical and nimbers with activeness obey familiar mex and addition rules under suitable activeness patterns. The framework broadens impartial game analysis to dynamic end conditions, enabling robust analysis of disjunctive sums where components may deactivate and reactivate, with potential applications in combinatorial game design and algorithmic game reasoning.
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.