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.
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