Distributed Nash equilibrium computation in aggregative games: An event-triggered algorithm

doi:10.1016/j.ins.2019.03.047

Information Sciences

Volume 489, July 2019, Pages 289-302

https://doi.org/10.1016/j.ins.2019.03.047 Get rights and content

Abstract

This paper is concerned with the problem of distributed Nash equilibrium computation in aggregative games. Note that the traditional computation algorithms are designed based on time-scheduled communication strategy, which may lead to high communication consumption of the whole network. To reduce the consumption, this paper proposes a novel distributed algorithm with an event-triggered mechanism, where the communication between any two agents is only carried out when an edge-based event condition is triggered. In the convergence analysis of the proposed algorithm, an important event-related error variable is firstly defined. Then, based on a zero-sum property of this event-related error, two key relations on the agents’ estimates in the proposed algorithm are provided. Further, by using these relations, it is proven that the agents’ estimates can achieve a Nash equilibrium under a proper event-triggering condition. Finally, examples on the demand response of power systems are presented to verify the theoretical findings.

Introduction

The problems of networked optimization and control have received much attention due to the rapid developments of sensor networks and interconnected systems [8], [18], [24], [39], [40]. In particular, Nash equilibrium seeking in network games is an important issue since many real applications involve such a setup, including cloud computation, power allocation, mobile ad-hoc networks, and so on [1], [5], [7], [29], [34]. In the Nash equilibrium seeking problem, each player (which is also called agent) in the network aims to minimize its own cost function by selfishly taking a strategy in response to other players’ strategies [32]. Due to the topological limitation of communication network, each player can only exchange the information with its neighboring players. Thus, algorithms for the Nash equilibrium computation are often designed in a distributed manner.

In general, distributed subgradient method (see, e.g., [25], [26], [31], [38], [41], [42]) is an important technique to solve Nash equilibrium seeking problems. For instance, by combining the gradient play and a consensus scheme, a distributed Nash equilibrium seeking algorithm is developed in [45]. Further, the reference [16] considers a noncooperative game with coupled constraints. In this case, an interesting distributed gradient-based algorithm is designed to search a generalized Nash equilibrium. On the other hand, the problem of distributed Nash equilibrium seeking in the presence of communication delay and dynamic network topologies is addressed in [48]. In [32], a gossip-based subgradient algorithm is designed to solve the Nash equilibrium computation problems. The reference [44] investigates the distributed Nash equilibrium computation under switching communication topologies. Moreover, a subgradient-based algorithm is developed in [23] to compute Nash equilibrium in two-network zero-sum games, and the reference [46] further studies the Nash equilibrium computation in N-coalition noncooperative games. In addition, an inexact-ADMM method is used in [33] to find a Nash equilibrium in a multi-player game.

More recently, a number of efforts have focused on aggregative games due to their extensive applications in Nash–Cournot games [11], demand response of power systems [43], and congestion control of communication networks [2]. Especially, some interesting distributed algorithms are developed for Nash equilibrium seeking in aggregative games. For example, in [12], both synchronous and asynchronous discrete-time algorithms are provided to solve the Nash equilibrium seeking problem in aggregative games. In [43], the demand response scheme of power systems is modeled by an aggregative game, and a Nash equilibrium seeking strategy is proposed to control the user’s energy consumption to the desired profile. Moreover, the references [21] and [28] consider the aggregative games with coupled constraints, and provide efficient distributed algorithms to seek Nash equilibrium in such games.

Note that in the aforementioned distributed algorithms, each player needs to communicate with its neighbors at every iteration, which may lead to severe communication congestion and high energy consumption. Therefore, reducing the communication frequency of network is a significant and practical task. Recently, the event-triggered strategy is recognized as an effective method to reduce the communication frequency. In the event-triggered strategy, the communication between any two agents is only carried out when a pre-designed event condition is triggered. Based on this idea, many efficient networked control approaches are proposed, such as [3], [6], [10], [19], [20], [35], [37], [47]. In particular, there are some interesting results on the event-triggered distributed optimization. For example, a continuous-time event-based algorithm is developed in [13] to solve the distributed convex optimization problems. In [15] and [22], both the quantized and event-triggered techniques are introduced to design the distributed optimization algorithms. On the other hand, by using an edge-based event-triggered scheme, the discrete-time projected subgradient algorithms are studied in [14] and [17]. Note that in the above optimization algorithms [13], [14], [15], [17], [22], all the agents focus on cooperatively minimizing a sum of local cost functions. This objective is essentially different from that of the Nash equilibrium computation algorithms, in which each agent focuses on selfishly minimizing its own cost function. Taking into account the broad applications of aggregative games and the importance of reducing communication frequency, it is necessary to study the event-triggered Nash equilibrium seeking problem in aggregative games. This reason motivates our current work.

In this paper, the aggregative game over multi-agent network is further studied. Compared with the existing results, the main contributions and differences of this paper contain the following aspects:

• The existing Nash equilibrium computation methods [12], [21], [28], [43] in aggregative games are developed under time-scheduled communication. By comparison, this paper introduces an edge-based event-triggered mechanism to the communication network. Furthermore, by combining the gradient descent method and the average tracking technique, a novel event-triggered distributed algorithm (Algorithm 1) is designed to compute Nash equilibrium in aggregative games.

• Due to the introduction of the event-triggered mechanism, the convergence analysis of the proposed algorithm is more complicated than those of the time-scheduled algorithms [12], [21], [28], [43]. Especially, the agents’ estimates in the proposed algorithm are affected by an additional event-related error. To study the behaviors of agents’ estimates, an important zero-sum property on the event-related error is applied. Then, by combining with a proper event-triggering condition, it is proven that the agents’ estimates can converge to a Nash equilibrium of the considered aggregative game.

• Besides, it is worth mentioning that there are several effective continuous-time algorithms for Nash equilibrium seeking, where the agents’ updates are performed according to differential equations, such as [16], [45], [46]. Different from these methods, the proposed algorithm is designed in a discrete-time manner, where the agents’ updates are performed based on iterative computation. Consequently, the convergence analysis in our work is different from those of [16], [45], [46].

Paper organization: The preliminaries are provided in Section 2. Section 3 shows the main objective of Nash equilibrium seeking in aggregative games. In Section 4, a novel event-triggered algorithm for Nash equilibrium computation is provided, and the convergence analysis of the proposed algorithm is given. Further, the simulation examples are shown in Section 5. Finally, the conclusions and future works are presented in Section 6.

Section snippets

Notation

$N$ and $R$ denote the nonnegative integer set and the real set, respectively. For a vector $v \in R^{n},$ ‖v‖ denotes the Euclidian norm. For any matrix $W \in R^{n \times m},$ W^T represents its transpose, [W]_ij denotes the entry in the jth column and ith row of W. A matrix W is said to be row stochastic if [W]_ij ≥ 0, ∀i, j and $\sum_{j} {[W]}_{i j} = 1$ . If both W and W^T are row stochastic, then W is said to be doubly stochastic.

Graph theory

An undirected graph $G = (V, E)$ consists of a set $V = {1, \dots, N}$ of nodes, and a set $E \subseteq {(i, j) | i, j \in V}$ of edges. Denote

Problem formulation

Consider a group of N players (agents) arranged on an undirected graph $G = (V, E),$ each player i is characterized by a cost function $C_{i} (x_{i}, x_{- i}) : X \to R,$ where $x_{i} \in X_{i}$ is the ith player’s decision from a local strategy set $X_{i},$ $x_{- i} ≜ {[x_{1}, \dots, x_{i - 1}, x_{i + 1}, \dots, x_{N}]}^{T}$ denotes all the players’ decisions other than x_i, and $X ≜ \prod_{i = 1}^{N} X_{i}$ is the Cartesian product of $X_{i}$ for all $i \in V$ . In this paper, the cost function is specified as $C_{i} (x_{i}, x_{- i}) = J_{i} (x_{i}, \bar{x})$ with a function $J_{i} : X_{i} \times \bar{X} \to R,$ where $\bar{x} ≜ \sum_{i = 1}^{N} x_{i}$ denotes the aggregate of all

Distributed event-triggered algorithm for Nash equilibrium computation

In this section, a novel distributed event-triggered algorithm is provided to compute Nash equilibrium of the game (2). Moreover, the convergence analysis of the proposed algorithm is conducted.

Simulation example

In this section, simulation examples on demand response management of power systems [43] are provided. We consider a network of 5 electricity users, where the underlying graph is a cycle graph, as shown in Fig. 1. Let x_i ∈ [l_i, h_i] be the energy consumption of the user i, and $J_{i} (x_{i}, \bar{x})$ be the corresponding cost function, which is described by $\begin{matrix} J_{i} (x_{i}, \bar{x}) = ν_{i} {(x_{i} - μ_{i})}^{2} + P (\bar{x}) x_{i}, \end{matrix}$

where $\bar{x} = \sum_{i = 1}^{N} x_{i}$ is the aggregate energy consumption, ν_i is a thermal coefficient, μ_i is a nominal energy consumption, and $P (\bar{x}$

Conclusion

In this paper, the Nash equilibrium seeking problem in aggregative games is considered. To reduce the communication frequency of the network, a novel distributed computation algorithm with edge-based event-triggered strategy is developed. In the technical part, it is proven that the agents’ estimates generated by the proposed algorithm can achieve a Nash equilibrium of the considered game. Finally, simulation results show the effectiveness of our method. The future works consist of the

Acknowledgments

This work was supported in part by the Funds of the National Natural Science Foundation of China (Grant Nos. 61621004 and 61420106016), the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant No. 2018ZCX03 and the Fundamental Research Funds for the Central Universities (Grant No. N180406001).

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Distributed Nash equilibrium computation in aggregative games: An event-triggered algorithm

Abstract

Introduction

Section snippets

Notation

Graph theory

Problem formulation

Distributed event-triggered algorithm for Nash equilibrium computation

Simulation example

Conclusion

Acknowledgments

Automatica

Automatica

Automatica

IEEE Trans. Ind. Electron.

IEEE Trans. Control Netw. Syst.

Automatica

Automatica

IEEE Trans. Autom. Control

Inf. Sci.

IEEE Trans. Cybern.

IET Control Appl.

IEEE Trans. Cybern.

Generalized Nash equilibria for the service provisioning problem in multi-cloud systems

IEEE Trans. Serv. Comput.

Dynamic incentives for congestion control

IEEE Trans. Autom. Control

Distributed event-triggered control for multi-agent systems

IEEE Trans. Autom. Control

Finite-Dimensional Variational Inequalities and Complementarity Problems

Nash equilibrium seeking in noncooperative games

IEEE Trans. Autom. Control

Event-based set-membership leader-following consensus of networked multi-agent systems subject to limited communication resources and unknown-but-bounded noise

IEEE Trans. Ind. Electron.

Distributed economic dispatch for smart grids with random wind power

IEEE Trans. Smart Grid

Periodic event-triggered control for linear systems

IEEE Trans. Autom. Control

Distributed computation of equilibria in monotone Nash games via iterative regularization techniques

SIAM J. Optim.

Distributed algorithms for aggregative games on graphs

Oper. Res.

Distributed subgradient method with edge-based event-triggered communication

IEEE Trans. Autom. Control

Event-triggered communication and data rate constraint for distributed optimization of multiagent systems

IEEE Trans. Syst. Man Cybern. Syst.