Silverman Games on Disjoint Discrete Sets

Abstract

Previous work by Heuer on Silverman games on odd versus even positive integers with threshold $T > 1$ and penalty $\nu > 0$, is extended to games on more general strategy sets $S_1 $ and $S_2 $ of positive numbers. Much of this paper is devoted to games where $S_1 $ and $S_2 $ are disjoint and discrete. Results include complete classification and solutions for $T\leqq T^* $ (where $T^*$ depends on $S_1$ and $S_2$), complete classification and solutions for $T > T^* $ and $\nu \geqq 1$, and partial classification with solutions for $T > T^ * $ and $\nu < 1$.