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 $s\in S$ to be removed at equal probability at any given position and we study the sequences $(a_n^S)_{n\in\mathbb{N}}$ 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 $n\rightarrow\infty$ 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 $S$.