Distributed Nash Equilibrium Seeking Research Articles

Distributed Nash Equilibrium Seeking With Limited Cost Function Knowledge via a Consensus-Based Gradient-Free Method

This article considers a distributed Nash equilibrium seeking problem, where the players only have partial access to other players' actions, such as their neighbors' actions. Thus, the players are supposed to communicate with each other to estimate other players' actions. To solve the problem, a leader-following consensus gradient-free distributed Nash equilibrium seeking algorithm is proposed. This algorithm utilizes only the measurements of the player' local cost function without the knowledge of its explicit expression or the requirement on its smoothness. Hence, the algorithm is gradient-free during the entire updating process. Moreover, the analysis on the convergence of the Nash equilibrium is studied for the algorithm with both diminishing and constant step-sizes, respectively. Specifically, in the case of diminishing step-size, it is shown that the players' actions converge to the Nash equilibrium almost surely, while in the case of fixed step-size, the convergence to the neighborhood of the Nash equilibrium is achieved. The performance of the proposed algorithm is verified through numerical simulations.

Distributed Nash Equilibrium Seeking for a Dynamic Micro-grid Energy Trading Game with Non-quadratic Payoffs

To model energy trading interactions among prosumers, a generic non-quadratic dynamic game model is proposed and solved for a stable Nash equilibrium. The framework is modeled as a non-cooperative, infinite strategy, multiplayer game where the participating players can possess a non-quadratic payoff function. Moreover, the market framework realizes a dynamic mapping from strategy set to payoffs of the participating players. It also carries an embedded notion of reliability and fairness by utilizing the concept of market reputation index. The existence of multiple Nash equilibria is discussed as well as the criteria for stability of Nash equilibrium. It is demonstrated that when an extremum seeking approach is used to model Nash seeking behavior of the players, the system converges to one of the stable Nash equilibriums depending on the initial condition. Simulation results are presented to verify the theoretical results, and a complete day with multiple prosumers is simulated to demonstrate effectiveness of the approach. The results are evaluated using several reliability matrices to show that the proposed framework results in an increased local generation, increased payoffs for prosumers, lower market clearing prices, and higher reliability.

Distributed-Observer-Based Nash Equilibrium Seeking Algorithm for Quadratic Games With Nonlinear Dynamics

In this article, a class of distributed quadratic games is considered over an undirected graph. The issue on the communication topology restriction is introduced and the players’ dynamics are with nonlinear dynamics and unknown time-varying perturbation. A distributed Nash equilibrium seeking algorithm is proposed based on a high-gain observer method, and the convergence is analyzed by the Lyapunov stability theory. It is shown that each player estimates the rival players’ states, and the errors between the players’ states and the Nash equilibrium are ultimately bounded by a small bound. Moreover, the presented algorithm is free of chattering phenomena because it is designed using the hyperbolic tangent function instead of the signum function to dominant the perturbation. The effectiveness of the proposed algorithm is validated via a simulation of the oligopoly game in which five firms produce the same products in a duopoly market structure.

Distributed Nash Equilibrium Seeking for Quadratic Games with Security

Considering a game with quadratic cost functions, this paper presents a distributed algorithm with security, whereby each player updates its strategy variable without using its private data and still achieves the Nash equilibrium. By using the theory of differential inclusions, Lyapunov function and invariance principle, the algorithm is proved to be convergent. Our algorithm can be used when it is required to seek the Nash equilibrium without disclosure of private data.

Distributed Nash Equilibrium Seeking for Aggregative Games via Derivative Feedback

In this paper, we investigate a continuous-time distributed Nash equilibrium seeking algorithm for a class of aggregative games, with application to the real-time pricing demand response. To seek the Nash equilibrium via local communication among neighbors, by combining projected gradient dynamics and consensus tracking dynamics, we propose a novel distributed algorithm for the players. We prove the convergence of the distributed algorithm via a constructed Lyapunov function and the variational inequality technique, and show an illustrative simulation related to the energy consumption control in smart grids.

Distributed averaging integral Nash equilibrium seeking on networks

Continuous-time gradient-based Nash equilibrium seeking algorithms enjoy a passivity property under a suitable monotonicity assumption, which has been exploited to design distributed Nash equilibrium seeking algorithms. We further exploit the passivity property to interconnect the algorithms with distributed nonlinear averaging integral controllers that tune on-line the weights of the communication graph. The main advantage is to guarantee convergence to a Nash equilibrium without requiring a strong coupling condition on the algebraic connectivity of the communication graph over which the players exchange information, nor a global fixed high-gain.

Distributed Nash equilibrium seeking under partial-decision information via the alternating direction method of multipliers

In this paper, we consider the problem of finding a Nash equilibrium in a multi-player game over generally connected networks. This model differs from a conventional setting in that players have partial information on the actions of their opponents and the communication graph is not necessarily the same as the players’ cost dependency graph. We develop a relatively fast algorithm within the framework of inexact-ADMM, based on local information exchange between the players. We prove its convergence to Nash equilibrium for fixed step-sizes and analyse its convergence rate. Numerical simulations illustrate its benefits when compared to a consensus-based gradient type algorithm with diminishing step-sizes.

Distributed Nash Equilibrium Seeking for Generalized Convex Games with Shared Constraints

In this paper, we deal with the problem of finding a Nash equilibrium for a generalized convex game. Each player is associated with a convex cost function and multiple shared constraints. Supposing that each player can exchange information with its neighbors via a connected undirected graph, the objective of this paper is to design a Nash equilibrium seeking law such that each agent minimizes its objective function in a distributed way. Consensus and singular perturbation theories are used to prove the stability of the system. A numerical example is given to show the effectiveness of the proposed algorithms.

Distributed Nash Equilibrium Seeking in Multiagent Games Under Switching Communication Topologies.

This paper investigates distributed Nash equilibrium seeking in multiagent games under switching communication topologies. To be specific, the communication topology is supposed to be switching among a set of strongly connected digraphs, which might suffer from occasional loss of communication due to sensor failure, packet loss, etc. The synthesis of the leader-following consensus protocol and the gradient play is exploited to achieve the distributed Nash equilibrium seeking under the switching communication topologies. Switching topology without loss of communication is firstly considered, followed by switching topology subject to missing communication within some time slots. For both situations, nonquadratic and quadratic games are addressed separately. Local convergence results are presented for nonquadratic games and nonlocal convergence results are provided for quadratic games. The theoretical results are verified by numerical examples.

Distributed Nash equilibrium seeking in networked graphical games

This paper considers a gossip approach for finding a Nash equilibrium in networked games on graphs, where a player’s cost function may be affected by the actions of any subset of players. An interference graph illustrates the partially-coupled cost functions, i.e., the asymmetric strategic interaction and information requirements. An algorithm is proposed whereby players make decisions based only on the estimates of their interfering players’ actions. Given the interference graph (not necessarily complete), a communication graph is designed so that players exchange only their required information. When the interference graph is sparse, the algorithm can offer substantial savings in communication and computation. Almost sure convergence to a Nash equilibrium is proved for diminishing step sizes. The effect of the second largest eigenvalue of the expected communication matrix on the convergence rate is quantified.

Distributed Nash Equilibrium Seeking by a Consensus Based Approach

In this paper, Nash equilibrium seeking among a network of players is considered. Different from many existing works on Nash equilibrium seeking in non-cooperative games, the players considered in this paper cannot directly observe the actions of the players who are not their neighbors. Instead, the players are supposed to be capable of communicating with each other via an undirected and connected communication graph. By a synthesis of a leader-following consensus protocol and the gradient play, a distributed Nash equilibrium seeking strategy is proposed for the non-cooperative games. Analytical analysis on the convergence of the players' actions to the Nash equilibrium is conducted via Lyapunov stability analysis. For games with non-quadratic payoffs, where multiple isolated Nash equilibria may coexist in the game, a local convergence result is derived under certain conditions. Then, a stronger condition is provided to derive a non-local convergence result for the non-quadratic games. For quadratic games, it is shown that the proposed seeking strategy enables the players' actions to converge to the Nash equilibrium globally under the given conditions. Numerical examples are provided to verify the effectiveness of the proposed seeking strategy.

Distributed Nash equilibrium seeking for aggregative games with coupled constraints

In this paper, we study a distributed continuous-time design for aggregative games with coupled constraints in order to seek the generalized Nash equilibrium by a group of agents via simple local information exchange. To solve the problem, we propose a distributed algorithm based on projected dynamics and non-smooth tracking dynamics, even for the case when the interaction topology of the multi-agent network is time-varying. Moreover, we prove the convergence of the non-smooth algorithm for the distributed game by taking advantage of its special structure and also combining the techniques of the variational inequality and Lyapunov function.

Stability Criterion for Nonsmooth Systems and Nash Equilibrium Seeking via Projected Gradient Dynamics

We derive a novel stability criterion for nonsmooth dynamical systems by virtue of a new set-valued Lie derivative of nonsmooth Lyapunov functions. This set-valued Lie derivative requires no computation of generalized gradients. Instead, it only calculates the directional derivatives. Moreover, our criterion allows for Lyapunov function candidates to be locally Lipschitz continuous and not necessarily regular. These merits strengthen the existing stability criterion and simplify its checking process. As an application, we establish the stability of projected gradient dynamics for distributed Nash equilibrium seeking under very mild conditions, compared to the existing ones.

Distributed Nash equilibrium seeking: A gossip-based algorithm

This paper presents an asynchronous gossip-based algorithm for finding a Nash equilibrium (NE) of a game in a distributed multi-player network. The algorithm is designed in such a way that players make decisions based on estimates of the other players’ actions obtained from local neighbors. Using a set of standard assumptions on the cost functions and communication graph, the paper proves almost sure convergence to a NE for diminishing step sizes. For constant step sizes an error bound on expected distance from a NE is established. The effectiveness of the proposed algorithm is demonstrated via simulation for both diminishing and constant step sizes.