1. Introduction

Combinatorial games where the winner is determined by a “score”, rather than who moves last, have been largely ignored by combinatorial game theorists. As far as this author is aware, there have been four previous studies of scoring play combinatorial games, all of which focused on the universe of “well-tempered” scoring games.

There are the works of Milnor [9], Hanner [6], Ettinger [3; 4] and most recently, Johnson [7]. The definition of a scoring game that all of them used is the following.

Definition 1. A scoring game is defined as

The authors would say that a game G, where GL = GR = ∅, is atomic and all other games are not atomic. Using this terminology, they were able to define concepts such as the disjunctive sum, and the “left outcome” and “right outcome”, which are the score at the end of a game under optimal play, when Left and Right move first respectively. Their mathematical definitions are given here.

Definition 2. The disjunctive sum is defined as follows:

Definition 3. The left outcome L(G) and right outcome R(G) are defined as follows:

Effectively they showed that under the disjunctive sum, this class of games forms a nontrivial monoid, and that with certain restrictions, it is equivalent to the set of all small normal play combinatorial games.