Distributed Strategy Design for Free-In and Free-Out Aggregative Games

This article proposes a new aggregative game model over free-in and free-out networks, in which each player can freely join and leave the network at its timing and aims to minimize its total cost during its active period. To enable the players to make self-beneficial decisions in such a dynamic environment, it is assumed that the players can locally store and exchange the historical action information. Based on the stored information, a distributed strategy is established, in which each player updates its action by a dual averaging method. In order to prevent excessive storage requirements, a storage mechanism is designed so that only the information generated from a certain historical time horizon is retained. The performance of the proposed strategy is evaluated by the static regret, which quantifies player's loss between the actual cost and the cost with the fixed best response during its active period. It is shown that the upper bound of each player's static regret grows sublinearly under local diminishing step sizes, indicating that the proposed strategy performs as well as choosing the best stationary action in hindsight for the long-term players. Finally, a simulation study on energy consumption games is given to verify the effectiveness of the developed methods.