On tame, pet, domestic, and miserable impartial games

TL;DR

This paper introduces new subclasses of tame impartial games—pet and domestic—and explores their relationships with miserable games, providing characterizations and demonstrating that many well-known games belong to these classes.

Contribution

It defines and characterizes pet and domestic games, subclasses of tame games, and shows their relation to miserable games, expanding the understanding of impartial game classifications.

Findings

Pet games include all subtraction games.

The game Euclid is miserable and tame.

Many variants of Wythoff and Nim are classified as miserable, pet, or domestic.

Abstract

Playing impartial games under the normal and misere conventions may differ a lot. However, there are also many "exceptions" for which the normal and misere plays are very similar. As early as in 1901 Bouton noticed that this is the case with the game of Nim. In 1976 Conway introduced a large class of such games that he called tame games. Here we introduce a proper subclass, pet games, and a proper superclass, domestic games. For each of these three classes we provide an efficiently verifiable characterization based on the following property. These games are closely related to another important subclass of the tame games introduced in 2007 by the first author and called miserable games. We show that tame, pet, and domestic games turn into miserable games by "slight modifications" of their definitions. We also show that the sum of miserable games is miserable and find several other classes that respect summation. The developed techniques allow us to prove that very many well-known impartial games fall into classes mentioned above. Such examples include all subtraction games, which are pet; game Euclid, which is miserable (and, hence, tame), as well as many versions of the Wythoff game and Nim, which may be miserable, pet, or domestic.