Restricted Universes of Partizan Misere Games

Abstract

This thesis considers three restricted universes of partizan combinatorial games and finds new results for misere play using the recently-introduced theory of indistinguishability quotients. The universes are defined by imposing three different conditions on game play: alternating, dicot (all-small), and dead-ending. General results are proved for each main universe, which in turn facilitate detailed analysis of specific subuniverses. In this way, misere monoids are constructed for alternating ends, for pairs of day-2 dicots, and for normal-play numbers, as well as for sets of positions that occur in variations of nim, hackenbush, and kayles, which fall into the alternating, dicot, and dead-ending universes, respectively. Special attention is given to equivalency to zero in misere play. With a new sufficiency condition for the invertibility of games in a restricted universe, the thesis succeeds in demonstrating the invertibility (modulo specific universes) of all alternating ends, all but previous-win alternating non-ends, all but one day-2 dicot, over one thousand day-3 dicots, hackenbush ‘sprigs’, dead ends, normal-play numbers, and partizan kayles positions. Connections are drawn between the three universes, including the recurrence of monoids isomorphic to the group of integers under addition, and the similarities of universe-specific outcome determinants. Among the suggestions for future research is the further investigation of a natural and promising subset of dead-ending games called placement games.