Online Distributed Learning for Aggregative Games With Feedback Delays

This article proposes a new aggregative game (AG) model with feedback delays. The strategies of players are selected from given strategy sets and subject to global nonlinear inequality constraints. Both cost functions and constrained functions of players are time varying, which reflects the changing nature of environments. At each time, each player only has access to its strategy set information, and the information of its current cost function and current constrained function is unknown. Due to feedback delays, the feedback information of corresponding cost functions and constrained functions is not revealed to players immediately after the selection of strategies. It would take a period of time for players to observe their feedback information. To address such an AG problem, a distributed learning algorithm is proposed with the local information from their neighbors and the delayed feedback information from environments, and it is applicable to time-varying weighted digraphs. We find that the two metrics of the algorithm grow sublinearly with respect to the learning time. A simulation example is given to illustrate the performance of the proposed algorithm.