Synthetic data generation overcomes limitations of real-world machine learning. Traditional methods are valuable for augmenting costly datasets but only optimize one criterion: realism. In this paper, we tackle the problem of generating synthetic data that optimize multiple criteria. This goal is necessary when real data are replaced by synthetic for privacy preservation. We introduce HydraGAN, a new approach to synthetic data generation that introduces multiple generator and discriminator agents into the system. The multi-agent GAN optimizes the goal of privacy-preservation as well as data realism. To facilitate multi-agent training, we adapt game-theoretic principles to offer equilibrium guarantees. We observe that HydraGAN outperforms baseline methods for three datasets for multiple criteria of maximizing data realism, maximizing model accuracy, and minimizing re-identification risk.