Multi-player subtraction games

The n-player subtraction game is investigated, assuming that the standard alliance matrix is adopted. Krawec's result is generalized from n = 3 to an arbitrary integer n ź 3 , and the order of subtraction games from k = 2 to an arbitrary integer k ź 2 .The 3-player subtraction games of order 2 are completely analyzed. Our results show that the sequences of game values are always periodic and that the lengths of preperiod and period only depend on the two elements of the subtraction set. We present the explicit representations of preperiods and periods.