Membership inference attacks (MIAs) pose significant privacy risks by determining whether individual data is in a dataset. While differential privacy (DP) mitigates these risks, it has limitations including limited resolution in expressing privacy-utility tradeoffs and intractable sensitivity calculations for tight guarantees. We propose a game-theoretic framework modeling privacy protection as a Bayesian game between defender and attacker, where privacy loss corresponds to the attacker's membership inference ability. To address strategic complexity, we represent the defender's mixed strategy as a neural network generator mapping private datasets to public representations (e.g., noisy statistics) and the attacker's strategy as a discriminator making membership claims. This \textit{general-sum Generative Adversarial Network} trains iteratively through alternating updates, yielding \textit{Bayes-Nash Generative Privacy (BNGP)} strategies. BNGP avoids worst-case privacy proofs such as sensitivity calculations, supports correlated mechanism compositions, handles heterogeneous attacker preferences. Empirical studies on sensitive dataset summary statistics show our approach significantly outperforms state-of-the-art methods by generating stronger attacks and achieving better privacy-utility tradeoffs.