On sequences arising from randomizing subtraction games

TL;DR

This paper investigates the probabilistic sequences from randomized subtraction games, characterizing their recurrence relations and asymptotic behavior, with complete solutions for small sets and partial results for larger sets.

Contribution

It introduces a framework for analyzing randomized subtraction games and characterizes the resulting sequences through linear recurrences, advancing understanding of their long-term behavior.

Findings

Sequences satisfy linear recurrence relations.

Explicit solutions for subtraction sets with fewer than 3 elements.

Partial characterization for arbitrary subtraction sets.

Abstract

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 $S$.