Binary dicots, a core of dicot games

TL;DR

This paper explores the relationships between different classes of combinatorial games under misère play, revealing that certain equivalences and comparisons are consistent across these classes, especially between binary dicot and dicot games.

Contribution

It establishes that comparison and equivalence relations in misère combinatorial games are often consistent across binary dicot, dicot, and impartial game classes.

Findings

Comparison modulo binary dicot games often matches comparison modulo dicot games.

Equivalence modulo dicot games and impartial games coincide when restricted to impartial games.

Connections between various sets of misère combinatorial games are clarified.

Abstract

We study combinatorial games under mis\`ere convention. Several sets of games have been considered earlier to better understand the behaviour of mis\`ere games. We here connect several of these sets. In particular, we prove that comparison modulo binary dicot games is often the same as comparison modulo dicot games, and that equivalence modulo dicot games and modulo impartial games are the same when they are restricted to impartial games.