Designing edge-based and node-based fully distributed algorithms for aggregative games with the adaptive technique

This paper focuses on aggregative games with local feasibility decision sets in a partial-decision information scenario. To seek the Nash equilibrium in a fully distributed manner, adaptive algorithms with edge-based and node-based control gains are designed. In the edge-based adaptive algorithm, an auxiliary dynamics is developed with the consensus protocol and adaptively adjusts the edges’ weights. In the node-based adaptive algorithm, fully distributed decision-seeking is achieved by dynamically modifying the player’s weight based on the overall consensus error. By virtue of the designed adaptive parameters, players update decisions without any global information. Utilizing Lyapunov stability theory and the comparison lemma, the proposed algorithms converge exponentially to a small neighborhood of the Nash equilibrium. Furthermore, the proposed adaptive algorithms are extended to the prescribed-time case by combining the prescribed-time gain function and exponential adaptive parameters. Finally, numerical simulations are presented to demonstrate the effectiveness of the proposed algorithms.