Leader–Follower Network Aggregative Game With Stochastic Agents’ Communication and Activeness

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.

I. Introduction

Distributed optimization over networks has attracted widespread attention of researchers in recent years [1]. As a typical framework, each agent in a network aims to minimize a social or an individual cost function while it communicates with some other agents through the network. In case that each agent is modeled as a selfish player who aims to minimize its own cost and also, the agent's cost is affected by decision variables of its neighbors through the network topology, the problem can be studied as a noncooperative network game [2]. If the effect of decision variables of rivals on the agent's cost function appears as an aggregative term (e.g., summation or weighted sum), the network game is known as network aggregative game (NAG) [3]. Many applications can be studied via this framework including, power system [4], opinion dynamics [5], communication system [6], provision of public goods [7], and criminal networks [8].