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In this article, the framework of network aggregative game (NAG) is extended by considering the interaction of players in a dynamic environment whose model is not known to them. The cost function of each player depends on its own strategy, the aggregate strategy of its neighbor players in the network, and also the state of the environment. The state of the environment evolves with a dynamics that is affected by the strategies of players as an input to the model. The players of NAG are modeled as myopic selfish agents who observe the decision value of their neighbors and the state of the environment with some time delay and, then, adjust their decisions using a projected subgradient rule. The main contribution of this article is to provide conditions in which the strategies of myopic agents converge to the unique fixed-point Nash equilibrium point of the proposed dynamic NAG. Furthermore, as a case study, the framework is applied to a wireless network by modeling the service providers as the agents and evolution of service selection by users as the dynamics of the environment. [ABSTRACT FROM AUTHOR]

In this article, we extend the theory of deterministic mean-field/aggregative games to multipopulation games. We consider a set of populations, each managed by a population coordinator (PC), of selfish agents playing a global noncooperative game, whose cost functions are affected by an aggregate term across all agents from all populations. In particular, we impose that the agents cannot exchange information between themselves directly; instead, only a PC can gather information on its own population and exchange local aggregate information with the neighboring PCs. To seek an equilibrium of the resulting (partial-information) game, we propose an iterative algorithm where each PC broadcasts a mean-field signal, namely, an estimate of the overall aggregative term, to its own population only. In turn, we let the local agents react with the best response and the PCs cooperate for estimating the aggregative term. Our main technical contributions are to cast the proposed scheme as a fixed-point iteration with errors, namely, the interconnection of a Krasnoselskij–Mann iteration and a linear consensus protocol, and, under a nonexpansiveness condition, to show convergence towards an $\varepsilon$ -Nash equilibrium, where $\varepsilon$ is inversely proportional to the population size. [ABSTRACT FROM AUTHOR]

This article presents a leader–follower scheme for network aggregative games. The followers and leader are selfish cost minimizing agents. The cost function of each follower is affected by its own strategy, the strategy of leader and the aggregate strategy of its neighbors through a communication graph. Also, the leader's cost function depends on its own strategy and the aggregate strategy of all the followers. The leader infinitely often wakes up, receives the aggregate strategy of the followers, updates its decision value, and broadcasts it to all the followers. Then, the followers apply the updated strategy of the leader into their cost functions. The establishment of information exchange between each neighboring pair of followers, and also, decision updating by a follower at each iteration, that is to say the activeness of the follower in that iteration, are both considered to be drawn from two arbitrary distributions. Moreover, a distributed algorithm based on subgradient method is proposed for updating the strategies of leader and followers. The convergence of the proposed algorithm to the unique Stackelberg equilibrium point of the game is proven in both almost sure and mean square senses. [ABSTRACT FROM AUTHOR]