全支撑与截断支撑的对抗减法游戏
Partizan Subtraction with Full and Truncated Support
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中文总结 AI 辅助
本文研究全支撑与截断支撑的对抗减法游戏，明确移动集更丰富的玩家除有限情况外总能获胜，发现中间平衡截断水平会产生非平凡周期性及无限多P、N-位置。
中文摘要 AI 辅助
我们研究全支撑（Full Support, FS）这一对抗减法游戏，其中玩家可从石堆中移除数量不超过特定界限的石子，左右玩家的界限通常不同。除有限个石堆大小外，移动集更丰富的玩家总能获胜。我们通过确定该游戏的一般标准型和原子权值验证了这一优势。为恢复公平性，我们引入截断支撑（Truncated Support, TS），本质是从下方修剪较大的减法集。若截断程度浅，超过特定石堆大小时不公平性仍存在，某一玩家持续主导；若截断程度深，则另一玩家开始主导。有趣的是，中间存在一个额外的平衡截断水平，此时会出现非平凡周期性，且存在无限多个P-位置和N-位置。我们还探究了轻度截断场景下的原子权值。
英文摘要
We investigate {\sc Full Support (FS)}, a {\sc Partizan Subtraction} game in which the players can remove any number of pebbles from the heap up to certain bounds that are typically different for Left and Right. The player with the richer move set always wins for all but finitely many heap sizes. We confirm this advantage by finding the general canonical form and the atomic weights of this game. To restore fairness (and peace), we introduce {\sc Truncated Support (TS)}, which essentially trims the larger subtraction set from below. If the truncation is shallow, the unfairness persists above a certain heap size, and one player continues ruling. If the truncation is deep, another player starts ruling. Interestingly, there is one more balanced truncation level in the middle, for which a non-trivial periodicity emerges, and where it has infinitely many $\mathcal P$ and $\mathcal N$-positions. We also explore the atomic weights for the lightly trimmed scenarios.
