Federated Learning (FL) has emerged as a prominent paradigm for privacy-preserving distributed machine learning, yet two fundamental challenges hinder its large-scale adoption. First, gradient inversion attacks can reconstruct sensitive training data from uploaded model updates, so privacy risk persists even when raw data remain local. Second, without adequate monetary compensation, rational clients have little incentive to contribute high-quality gradients, limiting participation at scale. To address these challenges, a privacy trading market is developed in which clients sell their differential privacy budgets as a commodity and receive explicit economic compensation for privacy sacrifice. This market is formalized as a Privacy Auction Game (PAG), and the existence of a Bayesian Nash Equilibrium is established under dominant-strategy incentive compatibility (DSIC), individual rationality (IR), and budget feasibility. To overcome the NP-hard, high-dimensional Nash Equilibrium computation at scale, \textit{MFG-RegretNet} is introduced as a deep-learning-based auction mechanism that combines mean-field game (MFG) approximation with differentiable mechanism design. The MFG reduction lowers per-round computational complexity from $\mathcal{O}(N^2 \log N)$ to $\mathcal{O}(N)$ while incurring only an $\mathcal{O}(N^{-1/2})$ equilibrium approximation gap. Extensive experiments on MNIST and CIFAR-10 demonstrate that MFG-RegretNet outperforms state-of-the-art baselines in incentive compatibility, auction revenue, and social welfare, while maintaining competitive downstream FL model accuracy.

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