In this paper, we extend the study of graph labelings to combinatorial game theory by introducing a new competitive optimization game based on cordial labelings of graphs. In this game, two players (Admirable and Impish) take turns labeling vertices of a graph with 0s and 1s, with edge labels determined by the sum (mod 2) of their incident vertices. Admirable and Impish aim to minimize and maximize, respectively, the difference between edges labeled 0 and 1. We define the “game cordiality number” as the resulting difference when both players play optimally and prove bounds for this number on paths (graphs with vertices connected in a straight line) and trees (connected acyclic graphs).