Differentially Private Distributed Algorithms for Aggregative Games Over Directed Graphs With Linear Convergence

This article studies privacy-preserving distributed Nash equilibrium (NE) seeking for aggregative games over directed graphs, where agents' cost functions contain sensitive information. A novel differentially private algorithm using decaying Laplace noise is developed to address two key issues: 1) how to design a distributed algorithm over directed graphs that achieves linear convergence while satisfying differential privacy requirements and 2) how to characterize the tradeoff between convergence accuracy and the privacy budget. First, sufficient conditions for linear convergence are established through the appropriate design of constant step sizes and convex combination parameters. Second, the differential privacy properties of the algorithm are analyzed without assuming bounded gradients, and a quantitative relationship between convergence accuracy and privacy budget is characterized. Furthermore, under additional restrictions on adjacent functions, the cumulative privacy budget admits an explicit expression and remains finite over an unbounded horizon, while the proposed algorithm is proven to converge to the exact NE. Finally, the effectiveness of the proposed algorithm is validated through a Nash-Cournot game and comparative simulations, which demonstrate its superior convergence performance compared to existing methods.