The combinatorial game theory of well-tempered scoring games

Abstract

The general behavior of combinatorial scoring games is not well-understood. In this paper, we focus on a special class of “well-tempered” scoring games. By analogy with Grossman and Siegel’s notion of even- and odd-tempered normal play games, we declare a dicot scoring game to be even-tempered if all its options are odd-tempered, and odd-tempered if it is not atomic and all its options are even-tempered. Games of either sort are called well-tempered. These show up naturally when analyzing one of the “knot games” introduced by Henrich et al. We consider disjunctive sums of well-tempered scoring games, and develop a theory for them analogous to the standard theory of disjunctive sums of normal play partizan games. We isolate a special class of inversive well-tempered games which behave like normal play partizan games or like the “Milnor games” considered by Milnor and Hanner. In particular, inversive games (modulo equivalence) form a partially ordered abelian group, and there is an effective description of the partial order. Moreover, the full monoid of well-tempered scoring games (modulo equivalence) admits a complete description in terms of the group of inversive games. We also describe several examples of well-tempered scoring games and provide dictionaries listing the values of some small positions in two of these games.