Distributed Nash Equilibrium Seeking for Aggregative Games With Directed Communication Graphs

One key factor affecting the distributed Nash equilibrium (NE) seeking in aggregative games is the unbalanced communication structure for multiple players. Although some results on seeking NE over undirected or weight-balanced graphs were established, how to address the distributed NE seeking problem over general directed communication graphs is still an outstanding challenge. This paper addresses the NE seeking problem for a class of aggregative games with general directed communication graphs. To achieve this objective, two new kinds of distributed discrete-time NE seeking algorithms are developed for aggregative games over fixed digraphs and time-varying digraphs, respectively. In particular, motivated by the heavy-ball method in optimization studies, a momentum term is introduced to the update law of the players' actions and it is numerically verified that this momentum term accelerates the convergence of the proposed algorithms. For both strongly connected fixed graph and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$B$ </tex-math></inline-formula> -strongly connected time-varying graph, it is theoretically proved that the actions of players will converge to the NE of aggregative games for the case of decreasing step-size implemented by the proposed NE seeking algorithms if the cost functions and the aggregation of players satisfy some certain conditions. Finally, the developed NE seeking algorithms are applied to the energy consumption control of plug-in hybrid electric vehicles (PHEVs), which demonstrates the effectiveness of the theoretical results.