Let two opponents, Oh and Ex, play the following game on the complete bipartite graph Kn,n. Oh colors one of the edges green and Ex colors a different edge red, and so on. The goal of each player is to be the first one to construct in his own color a predetermined bipartite graph M with no isolated points. The minimum n for which Oh can win on Kn,n regardless of the moves made by Ex is called the bipartite achievement number of M. In the corresponding misere game, the first player (if any) who forms M is the loser. The minimum n for which Ex can force Oh to make a monochromatic M is called the bipartite avoidance number of M. Bipartite achievement and avoidance numbers of some small graph are presented as well as their bipartite ramsey numbers.