  A subset of the vertex set of a graph is geodetically convex if it contains
every vertex on any shortest path between two elements of the set. The convex
hull of a set of vertices is the smallest convex set containing the set. We
study two games where two players take turns selecting previously-unselected
vertices of a graph until the convex hull of the jointly-selected vertices
becomes too large. The last player to move is the winner. The achievement game
ends when the convex hull contains every vertex. In the avoidance game, the
convex hull is not allowed to contain every vertex. We determine the nim-number
of these games for several graph families.
