The *-operator and invariant subtraction games

An invariant subtraction game is a 2-player impartial game defined by a set of invariant moves (k-tuples of non-negative integers) M. Given a position (another k-tuple) x = (x1, . . . , xk), each option is of the form (x1 − m1, . . . , xk − mk), where m = (m1, . . . ,mk) ∈ M (and where xi − mi ≥ 0, for all i). Two players alternate in moving and the player who moves last wins. The set of non-zero P-positions of the game M defines the moves in the dual game M. For example, in the game of (2-pile Nim) a move consists in removing the same positive number of tokens from both piles. Our main results concern a double application of ⋆, the operation M → (M). We establish a fundamental ‘convergence’ result for this operation. Then, we give necessary and sufficient conditions for the relation M = (M) to hold. We show that it holds for example with M = k-pile Nim.