Summary: We introduce the following class of partizan games, called pomax games. Given a partially ordered set whose elements are colored black or white, the players Black and White take turns removing any maximal element of their own color. If there is no such element, the player loses. We prove that pomax games are always integervalued and for colored tree posets and chess-colored Young diagram posets we give a simple formula for the value of the game. However, for pomax games on general posets of height 3 we show that the problem of deciding the winner is PSPACE-complete and for posets of height 2 we prove NP-hardness. Pomax games are just a special case of a larger class of integer-valued games that we call element-removal games, and we pose some open questions regarding element-removal games that are not pomax games.