An Investigation of Partizan Misere Games

Abstract

Combinatorial games are played under two different play conventions: normal play, where the last player to move wins, and \mis play, where the last player to move loses. Combinatorial games are also classified into impartial positions and partizan positions, where a position is impartial if both players have the same available moves and partizan otherwise. \Mis play games lack many of the useful calculational and theoretical properties of normal play games. Until Plambeck's indistinguishability quotient and \mis monoid theory were developed in 2004, research on \mis play games had stalled. This thesis investigates partizan combinatorial \mis play games, by taking Plambeck's indistinguishability and \mis monoid theory for impartial positions and extending it to partizan ones, as well as examining the difficulties in constructing a category of \mis play games in a similar manner to Joyal's category of normal play games. This thesis succeeds in finding an infinite set of positions which each have finite \mis monoid, examining conditions on positions for when $* + *$ is equivalent to 0, finding a set of positions which have Tweedledum-Tweedledee type strategy, and the two most important results of this thesis: giving necessary and sufficient conditions on a set of positions $\Upsilon$ such that the \mis monoid of $\Upsilon$ is the same as the \mis monoid of $*$ and giving a construction theorem which builds all positions $\xi$ such that the \mis monoid of $\xi$ is the same as the \mis monoid of $*$.