The gameplay of strategic board games such as chess, Go and Hex is often characterized by combinatorial, relational structures -- capturing distinct interactions and non-local patterns -- and not just images. Nonetheless, most common self-play reinforcement learning (RL) approaches simply approximate policy and value functions using convolutional neural networks (CNN). A key feature of CNNs is their relational inductive bias towards locality and translational invariance. In contrast, graph neural networks (GNN) can encode more complicated and distinct relational structures. Hence, we investigate the crucial question: Can GNNs, with their ability to encode complex connections, replace CNNs in self-play reinforcement learning? To this end, we do a comparison with Hex -- an abstract yet strategically rich board game -- serving as our experimental platform. Our findings reveal that GNNs excel at dealing with long range dependency situations in game states and are less prone to overfitting, but also showing a reduced proficiency in discerning local patterns. This suggests a potential paradigm shift, signaling the use of game-specific structures to reshape self-play reinforcement learning.