Peg solitaire is a game in which pegs are placed in every hole but one and the player jumps over pegs along rows or columns to remove them. Usually, the goal of the player is to leave only one peg. In a 2011 paper, this game is generalized to graphs. When the game is played between two players it is called duotaire. In this paper, we consider two variations of peg duotaire on graphs. In the first variation, the last player to remove a peg wins. Inspired by the work of Slater, we also investigate a variation in which one player tries to maximize the number of pegs at the end of the game while their opponent seeks to minimize this number. For both variations, we give explicit strategies for several families of graphs. Finally, we give a number of open problems as possible avenues for future research.