Distributed algorithms with linear convergence for aggregative games over time-varying networks

Aggregate games, a special class of non-cooperative games, play an important role in practical applications. In such games, the cost function of each player is influenced not only by their own strategy but also by the aggregate resulting from the strategies of all players, instead of all rival players' direct strategies. This paper proposes the distributed Nash equilibrium (NE) seeking algorithms with linear convergence rates for aggregative games over time-varying (TV) directed and undirected networks. These algorithms employing increasing the amount of information tackle the impact on the convergence rate caused by decreasing connectivity in TV networks, enabling more practical problems to be solved efficiently. Facing TV directed networks, the push-sum concept is introduced to overcome the impact of one-way interaction, while TV undirected networks, as a special case of them, have the property of two-way interaction, so this algorithm design highlights the core of improving convergence rate. Moreover, one limiting condition about the cost function can be derived, under which two algorithms will converge at a linear rate with the proper step-size. Finally, an energy management system is considered to verify the proposed algorithms.