Partizan octal games: Partizan subtraction games
An impartial subtraction game \(S=\{s_ 1,...,s_ k\}\) is a two-player game where each player in turn chooses a number \(m\) in \(S\) and removes \(m\) tokens from a pile, the size of which is known. The first player unable to move loses. The game becomes partizan if instead of one set \(S\) there are two sets \(S_ L\) and \(S_ R\), and Left must choose a number from \(S_ L\) and Right from \(S_ R\). \(S_ L\) dominates \(S_ R\) if, for all sufficiently large piles, Left wins both as a first player and as a second player. The game is periodic if there are integers \(a\) and \(c\) such that for all \(n\geq a\), the win possibilities for each player are the same from a pile with \(n+c\) token as from a pile of \(n\); purely periodic if \(a=0\). It is shown that every partizan subtraction game is periodic, and various theorems involving dominance and periodicity are established.
scientific article; zbMATH DE number 4083405