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Cumulative subtraction games
arXiv - CS - Discrete Mathematics Pub Date : 2018-05-23 , DOI: arxiv-1805.09368
Gal Cohensius, Urban Larsson, Reshef Meir, David Wahlstedt

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.

中文翻译:

累积减法游戏

我们研究零和游戏,它是经典组合减法游戏的变体(例如在 Berlekamp、Conway 和 Guy 的巨著“Winning Ways”中进行了研究),称为累积减法 (CS)。两名玩家交替移动,并从联合堆中取出鹅卵石获得积分。我们证明了具有有限数量的可能动作的 CS 的最佳游戏结果(游戏价值)最终是周期性的,周期为 $2s$,其中 $s$ 是最大可用动作的大小。这解决了斯图尔特在他的博士论文中的一个猜想。论文(2011)。具体来说,我们找到了一个二次边界,大小为 $s$,关于结果函数何时必须变为周期性。在两个可能的动作的情况下,我们给出了最佳游戏的明确描述。我们将周期性结果推广到具有所谓奖励函数的游戏,
更新日期:2020-02-14
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