Multimodal learning can complete the picture of information extraction by uncovering key dependencies between data sources. However, current systems fail to fully leverage multiple modalities for optimal performance. This has been attributed to modality competition, where modalities strive for training resources, leaving some underoptimized. We show that current balancing methods struggle to train multimodal models that surpass even simple baselines, such as ensembles. This raises the question: how can we ensure that all modalities in multimodal training are sufficiently trained, and that learning from new modalities consistently improves performance? This paper proposes the Multimodal Competition Regularizer (MCR), a new loss component inspired by mutual information (MI) decomposition designed to prevent the adverse effects of competition in multimodal training. Our key contributions are: 1) Introducing game-theoretic principles in multimodal learning, where each modality acts as a player competing to maximize its influence on the final outcome, enabling automatic balancing of the MI terms. 2) Refining lower and upper bounds for each MI term to enhance the extraction of task-relevant unique and shared information across modalities. 3) Suggesting latent space permutations for conditional MI estimation, significantly improving computational efficiency. MCR outperforms all previously suggested training strategies and is the first to consistently improve multimodal learning beyond the ensemble baseline, clearly demonstrating that combining modalities leads to significant performance gains on both synthetic and large real-world datasets.