A Game on a Finite Projective Plane

See "Tic-Tac-Toe on a Finite Plane", Maureen T. Carroll and Steven T. Dougherty,
Mathematics Magazine, Vol. 77, No. 4 (Oct., 2004), pp. 260-274. (Preprint here: http://academic.scranton.edu/faculty/carrollm1/tictac.pdf) The second player can force a draw on the 3-by-3 and 4-by-4 projective planes.