The Combinatorial Game Theory of Well-Tempered Scoring Games

TL;DR

This paper develops a theoretical framework for well-tempered scoring games, analyzing their algebraic structure and classifying Boolean-valued instances, extending combinatorial game theory concepts.

Contribution

It introduces a novel class of scoring games, establishes their algebraic properties, and classifies Boolean-valued cases, advancing the mathematical understanding of such games.

Findings

The monoid of well-tempered scoring games is cancellative but not a group.

The structure of these games relates to normal-play partizan games.

There are exactly seventy Boolean-valued well-tempered scoring games up to equivalence.

Abstract

We consider the class of "well-tempered" integer-valued scoring games, which have the property that the parity of the length of the game is independent of the line of play. We consider disjunctive sums of these games, and develop a theory for them analogous to the standard theory of disjunctive sums of normal-play partizan games. We show that the monoid of well-tempered scoring games modulo indistinguishability is cancellative but not a group, and we describe its structure in terms of the group of normal-play partizan games. We also classify Boolean-valued well-tempered scoring games, showing that there are exactly seventy, up to equivalence.