  We study a three-player variation of the impartial avoidance game introduced
by Anderson and Harary. Three players take turns selecting
previously-unselected elements of a finite group. The losing player is the one
who selects an element that causes the set of jointly-selected elements to be a
generating set for the group, with the previous player winning and the
remaining player coming in second place. We describe the winning strategy for
these games on cyclic, dihedral, and nilpotent groups.
