Cyclic impartial games with carry-on moves

TL;DR

This paper introduces a new theoretical framework for analyzing cyclic impartial games with carry-on moves, extending existing theories to handle infinite plays and entailing moves in such games.

Contribution

It combines the Smith-Frankel-Perl and Larsson-Nowakowski-Santos theories to analyze cyclic impartial games with carry-on moves, a novel class of games.

Findings

The generalized theory successfully analyzes cyclic impartial games with carry-on moves.

Application to sc green-lime hackenbush demonstrates the theory's effectiveness.

Provides insights into infinite plays and entailing move structures.

Abstract

In an impartial combinatorial game, both players have the same options in the game and all its subpositions. The classical Sprague-Grundy Theory was developed for short impartial games, where players have a finite number of options, there are no special moves, and an infinite run is not possible. Subsequently, many generalizations have been proposed, particularly the Smith-Frankel-Perl Theory devised for games where the infinite run is possible, and the Larsson-Nowakowski-Santos Theory able to deal with entailing moves that disrupt the logic of the disjunctive sum. This work presents a generalization that combines these two theories, suitable for analyzing cyclic impartial games with carry-on moves, which are particular cases of entailing moves where the entailed player has no freedom of choice in their response. This generalization is illustrated with sc green-lime hackenbush, a game inspired by the classic green hackenbush.