Guaranteed Scoring Games

TL;DR

This paper introduces the structure of Guaranteed Scoring Games, a class of two-player combinatorial games, and develops methods to analyze their sums, orderings, and inverses, extending existing theories in combinatorial game analysis.

Contribution

It formalizes the structure of Guaranteed Scoring Games as a partially ordered monoid, introduces reduction techniques for unique representatives, and provides algorithms for game comparison.

Findings

GS form a partially ordered monoid with a quotient structure.

Four reduction methods yield unique representatives for each class.

Finite algorithms can compare games via disjunctive sums.

Abstract

The class of Guaranteed Scoring Games (GS) are two-player combinatorial games with the property that Normal-play games (Conway et. al.) are ordered embedded into GS. They include, as subclasses, the scoring games considered by Milnor (1953), Ettinger (1996) and Johnson (2014). We present the structure of GS and the techniques needed to analyze a sum of guaranteed games. Firstly, GS form a partially ordered monoid, via defined Right- and Left-stops over the reals, and with disjunctive sum as the operation. In fact, the structure is a quotient monoid with partially ordered congruence classes. We show that there are four reductions that when applied, in any order, give a unique representative for each congruence class. The monoid is not a group, but in this paper we prove that if a game has an inverse it is obtained by `switching the players'. The order relation between two games is defined by comparing their stops in \textit{any} disjunctive sum. Here, we demonstrate how to compare the games via a finite algorithm instead, extending ideas of Ettinger, and also Siegel (2013).