Impartial games with entailing moves

TL;DR

This paper extends combinatorial game theory to include games with entailing moves, introducing new axioms and algebraic structures to analyze such games like Nimstring and Top Entails.

Contribution

It axiomatizes normal play for games with entailing moves using new terminating games and extends Sprague-Grundy theory to analyze affine impartial games.

Findings

Extended Sprague-Grundy theory for affine impartial games.

Provided solutions for Nimstring and Top Entails.

Introduced new axioms using games ∞ and ¯∞ for normal play.

Abstract

Combinatorial Game Theory has also been called `additive game theory', whenever the analysis involves sums of independent game components. Such {\em disjunctive sums} invoke comparison between games, which allows abstract values to be assigned to them. However, there are rulesets with {\em entailing moves} that break the alternating play axiom and/or restrict the other player's options within the disjunctive sum components. These situations are exemplified in the literature by a ruleset such as {\sc nimstring}, a normal play variation of the classical children's game {\sc dots \& boxes}, and {\sc top~entails}, an elegant ruleset introduced in the classical work Winning Ways, by Berlekamp Conway and Guy. Such rulesets fall outside the scope of the established normal play theory. Here, we axiomatize normal play via two new terminating games, $\infty$ (Left wins) and $\overline\infty$ (Right wins), and a more general theory is achieved. We define {\em affine impartial}, which extends classical impartial games, and we analyze their algebra by extending the established Sprague-Grundy theory, with an accompanying minimum excluded rule. Solutions of {\sc nimstring} and {\sc top~entails} are given to illustrate the theory.