Tame and Wild Scoring Play Games

TL;DR

This paper proves mathematically that scoring play combinatorial game theory encompasses all types of combinatorial games, including normal and misère play, establishing it as a complete analytical framework.

Contribution

It demonstrates that scoring play combinatorial game theory fully covers all combinatorial games, unifying various game types under a single theoretical framework.

Findings

Scoring play theory includes normal and misère play as subsets.

It provides a complete analysis method for all combinatorial games.

The proof establishes the universality of scoring play theory.

Abstract

In this paper, we will be proving mathematically that scoring play combinatorial game theory covers all combinatorial games. That is, there is a sub-set of scoring play games that are identical to the set of normal play games, and a different sub-set that is identical to the set of mis\`ere play games. This proves conclusively, that scoring play combinatorial game theory is a complete theory for combinatorial games, and that every combinatorial game, regardless of the rule-set, can be analysed using scoring play combinatorial game theory.