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) , each option is of the form (x1−m1,…,xk−mk), where , 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, as is the case for example with M=k-pile Nim.