Membership inference attacks (MIAs) expose significant privacy risks by determining whether an individual's data is in a dataset. While differential privacy (DP) mitigates such risks, it faces challenges in general when achieving an optimal balance between privacy and utility, often requiring intractable sensitivity calculations and limiting flexibility in complex compositions. We propose a game-theoretic framework that models privacy protection as a Bayesian game between a defender and an attacker, solved using a general-sum Generative Adversarial Network (general-sum GAN). The Bayes Generative Privacy (BGP) response, based on cross-entropy loss, defines the attacker's optimal strategy, leading to the Bayes-Nash Generative Privacy (BNGP) strategy, which achieves the optimal privacy-utility trade-off tailored to the defender's preferences. The BNGP strategy avoids sensitivity calculations, supports compositions of correlated mechanisms, and is robust to the attacker's heterogeneous preferences over true and false positives. A case study on binary dataset summary statistics demonstrates its superiority over likelihood ratio test (LRT)-based attacks, including the uniformly most powerful LRT. Empirical results confirm BNGP's effectiveness.

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