The Sprague-Grundy theory of normal-play impartial combinatorial games generalizes to misere play via a local construction known as the indistinguishability quotient.
Suppose is a set of impartial combinatorial games that is closed in both of the following senses:
(1) Additive closure: If and are games in , then their disjunctive sum is also in .
(2) Hereditary closure: If is a game in and is an option of , then is also in .
Next, define on the indistinguishability congruence ≈ that relates two games and if for every choice of a game in , the two positions and have the same outcome (i.e., are either both first-player wins in best play of , or alternatively are both second-player wins).
One easily checks that ≈ is indeed a congruence on the set of all disjunctive position sums in , and that this is true regardless of whether the game is played in normal or misere play. The totality of all the congruence classes form the indistinguishability quotient.
If is played as a normal-play (last-playing winning) impartial game, then the congruence classes of are in one-to-one correspondence with the nim values that occur in the play of the game (themselves determined by the Sprague-Grundy theorem).
In misere play, the congruence classes form a Monoid#Commutative monoid, instead, and it has become known as a misere quotient.
- Plambeck, Thane E. (2005-07-19). "Taming the Wild in Impartial Combinatorial Games" (PDF). Integers: Electronic Journal of Combinatorial Number Theory (PDF) 5 (G05). Retrieved 2010-12-21.