Combinatorial two-player games have recently been applied to knot theory. Examples of this include the Knotting-Unknotting Game and the Region Unknotting Game, both of which are played on knot shadows. These are turn-based games played by two players, where each player has a separate goal to achieve in order to win the game. In this paper, we introduce the Linking-Unlinking Game which is played on two-component link shadows. We then present winning strategies for the Linking-Unlinking Game played on all shadows of two-component rational tangle closures and played on a large family of general two-component link shadows.