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.
