Cumulative subtraction games

TL;DR

This paper analyzes a variant of subtraction games called Cumulative Subtraction, proving that the game outcomes become eventually periodic with a period related to the largest move, and provides explicit strategies for simple cases.

Contribution

It proves the eventual periodicity of Cumulative Subtraction game outcomes with a quadratic bound and generalizes the result to reward-based variants.

Findings

Outcome in optimal play is eventually periodic with period 2s.

Explicit optimal strategies are provided for two-action cases.

Periodicity bound is quadratic in the size of the largest action.

Abstract

We study zero-sum games, a variant of the classical combinatorial Subtraction games (studied for example in the monumental work "Winning Ways", by Berlekamp, Conway and Guy), called Cumulative Subtraction (CS). Two players alternate in moving, and get points for taking pebbles out of a joint pile. We prove that the outcome in optimal play (game value) of a CS with a finite number of possible actions is eventually periodic, with period $2s$, where $s$ is the size of the largest available action. This settles a conjecture by Stewart in his Ph.D. thesis (2011). Specifically, we find a quadratic bound, in the size of $s$, on when the outcome function must have become periodic. In case of two possible actions, we give an explicit description of optimal play. We generalize the periodicity result to games with a so-called reward function, where at each stage of game, the change of `score' does not necessarily equal the number of pebbles you collect.