Accordingly to Duchet (1987), the first paper of convexity on general graphs, in english, is the 1981 paper “Convexity in graphs”. One of its authors, Frank Harary, introduced in 1984 the first graph convexity games, focused on the geodesic convexity, which were investigated in a sequence of five papers that ended in 2003. In this paper, we continue this research line, extend these games to other graph convexities, and obtain winning strategies and complexity results. Among them, we obtain winning strategies for general convex geometries in graphs. We also obtain the first PSPACE-hardness results on convexity games, by proving that the normal play and the misère play of the hull game on the geodesic and the monophonic convexities are PSPACE-complete.