Differentially Private Distributed Nash Equilibrium Seeking for Aggregative Games With Linear Convergence

The problem of differentially private distributed Nash equilibrium seeking for aggregative games with unknown nonlinear players under unbalanced directed graphs is investigated in this article. We utilize an auxiliary variable to eliminate the influence of the unbalancedness caused by the topological structure. The consensus protocol is employed to estimate the aggregate function. To protect players' sensitive information, Laplace noise is added on shared messages among players to perturb the raw information. By combining with the heavy-ball method, Nesterov gradient descent strategy and the consensus protocol, a momentum-based auxiliary system is designed to generate the Nash equilibrium signals with differential privacy guarantee with the help of the fixed gradient step size. A weakening factor is adopted to ensure the almost sure convergence of the designed auxiliary system in the presence of noise. Then, we analyze the linear convergence of the auxiliary system to the Nash equilibrium by using the linear systems inequalities. The rigorous differential privacy with a finite cumulative privacy budget without requiring a tradeoff between the convergence accuracy and differential privacy is also proven. In addition, an adaptive fuzzy method is used to deal with the unknown nonlinear dynamics, and the controller is designed to drive the actions of the players to the neighborhood of Nash equilibrium with the aid of auxiliary system. Finally, we present a numerical example to present the effectiveness of the proposed algorithm.