On sequences arising from randomizing subtraction games

In this article, we study the behavior of a broad family of real sequences derived from randomized one-pile subtraction games. For any subtraction set S, we allow any valid number of chips s∈ S to be removed at equal probability at any given position and we study the sequences (aₙS)ₙ representing the probability of winning the game from a position with n chips. We characterize these sequences in terms of linear recurrence relations and examine their behavior as n→∞ for all finite S. We fully solve the cases for subtraction sets of fewer than 3 elements and partially complete the general case for arbitrary S.