Partizan Subtraction with Full and Truncated Support

Abstract

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.