The Combinatorial Game Theory of Well-Tempered Scoring Games

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.