Optimal Distribution of Players into Two Alliances in Mis\`{e}re Subtraction Game

Abstract

In combinatorial game theory, multi-player games expands on the research of two-person games and are widely applicable to computer science and business. The given paper studies the mis\`{e}re version of the subtraction game (Nim with one pile of counters) for two alliances extending the results of Kelly in \cite{2,3}; Liu and Zhao in \cite{5}, Liu and Wang in \cite{6} and Suetsugu in \cite{7}. The main result of the current paper determines the optimal distribution of the players among two alliances that minimizes $N$, the smallest number of counters such that the larger alliance has a winning strategy for all the games with more than $N$ counters.