Nash Equilibrium Seeking with Non-doubly Stochastic Communication Weight Matrix

A distributed Nash equilibrium seeking algorithm is presented for networked games. We assume an incomplete information available to each player about the other players’ actions. The players communicate over a strongly connected digraph to send/receive the estimates of the other players’ actions to/from the other local players according to a gossip communication protocol. Due to asymmetric information exchange between the players, a non-doubly (row) stochastic weight matrix is defined. We show that, due to the non-doubly stochastic property, there is no exact convergence. Then, we present an almost sure convergence proof of the algorithm to a Nash equilibrium of the game. Moreover, we extend the algorithm for graphical games in which all players’ cost functions are only dependent on the local neighboring players over an interference digraph. We design an assumption on the communication digraph such that the players are able to update all the estimates of the players who interfere with their cost functions. It is shown that the communication digraph needs to be a superset of a transitive reduction of the interference digraph. Finally, we verify the efficacy of the algorithm via a simulation on a social media behavioral case.

One key factor affecting the distributed Nash equilibrium (NE) seeking in aggregative games is the unbalanced communication structure for multiple players. Although some results on seeking NE over undirected or weight-balanced graphs were established, how to address the distributed NE seeking problem over general directed communication graphs is still an outstanding challenge. This paper addresses the NE seeking problem for a class of aggregative games with general directed communication graphs. To achieve this objective, two new kinds of distributed discrete-time NE seeking algorithms are developed for aggregative games over fixed digraphs and time-varying digraphs, respectively. In particular, motivated by the heavy-ball method in optimization studies, a momentum term is introduced to the update law of the players' actions and it is numerically verified that this momentum term accelerates the convergence of the proposed algorithms. For both strongly connected fixed graph and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$B$ </tex-math></inline-formula> -strongly connected time-varying graph, it is theoretically proved that the actions of players will converge to the NE of aggregative games for the case of decreasing step-size implemented by the proposed NE seeking algorithms if the cost functions and the aggregation of players satisfy some certain conditions. Finally, the developed NE seeking algorithms are applied to the energy consumption control of plug-in hybrid electric vehicles (PHEVs), which demonstrates the effectiveness of the theoretical results.

In this paper, we study a Nash equilibrium (NE) seeking problem in a multi-player non-cooperative game over a directed communication graph. Specifically, the players' costs are functions of all players' actions, but only part of which are directly accessible. Moreover, we assume the explicit form/expression/model information of the cost function is unknown, but its value can be measured by the local player. To solve this problem, a non-model based distributed NE seeking algorithm is proposed, which requires no gradient information but the measurements of player's local cost function. A leaderfollowing consensus technique is adopted with a row-stochastic adjacency matrix, which simplifies the implementation and increases the application range of the algorithm as compared to the doubly-stochastic matrix. Moreover, the algorithm is able to work with uncoordinated step-sizes, allowing the players to choose their own preferred step-sizes, which makes the algorithm more distributive. The convergence of the proposed algorithm is rigorously studied for both scenarios of diminishing and constant step-sizes, respectively. It is shown that players' actions converge to the exact NE almost surely for the case of diminishing step-size, and to an approximated NE with a gap depending on the step-size selection for the case of constant step- size. Numerical examples are provided to verify the algorithm's effectiveness.

This paper presents the design of adaptively distributed Nash Equilibrium (NE) seeking algorithms in noncooperative games for heterogeneous general linear multi-agent systems (MASs) under unknown unmodeled dynamics and bounded disturbances. Different from existing works that only consider single or multiple integrators, we aim to steer agents' outputs of MASs with nonidentical dynamics to the NE in a distributed way and not needing known information on the Lipschitz and monotone constants of pseudo-gradients as well as the algebraic connectivity of the graph. To overcome difficulties brought by heterogeneous dynamics and NE seeking requirements, we first present an adaptively distributed NE seeking algorithm that can tune on-line the edges of graphs to solve the studied problem. By leveraging monotone and matrix properties, the global asymptotic convergence to the NE is obtained. Moreover, this design is extended to develop another adaptively distributed NE seeking algorithm to tackle the impact of unknown dynamics and disturbances. Two exam -ples with numerical simulation results are provided to illustrate the effectiveness of the developed NE seeking algorithms.

This article is concerned with the hybridNash equilibrium (NE) seeking problem over a network in a partial-decision information scenario. Each agent has access to both its own cost function and local decision information of its neighbors. First, an adaptive gradient-based algorithm is constructed in a fully distributed manner with the guaranteed convergence to the NE, where the network communication is required. Second, in order to save communication cost, a novel event-triggered scheme, namely, edge-based adaptive dynamic event-triggered (E-ADET) scheme, is proposed with online-tuned triggering parameter and threshold, and such a scheme is proven to be fully distributed and free of Zeno behavior. Then, a hybrid NE seeking algorithm, which is also fully distributed, is constructed under the E-ADET scheme. By means of the Lipschitz continuity and the strong monotonicity of the pseudogradient mapping, we show the convergence of the proposed algorithms to the NE. Compared with the existing distributed algorithms, our algorithms remove the requirement on global information, thereby exhibiting the merits of both flexibility and scalability. Finally, two examples are provided to validate the proposed NE seeking methods.

Dear Editor, This letter is concerned with the distributed Nash equilibrium (NE) seeking in an N-player game over random graphs. We develop a distributed stochastic forward-backward (DSFB) algorithm based on local information exchange between agents. We prove that the DSFB algorithm can converge to an NE almost surely, and analyze the convergence rate of the proposed algorithm. Compared with the existing works on distributed NE seeking, the communication graph in this letter is supposed to be time-varying and stochastic, which makes the NE seeking algorithm more suitable for practical scenarios, but brings a great challenge in both the design and convergence analysis of the algorithm. Besides, by establishing a variational inequality on NE, we relax the co-coercivity or strong monotonicity assumption on the extended pseudo-gradient.

An attack-resilient distributed Nash equilibrium (NE) seeking problem is addressed for noncooperative games of networked systems under malicious cyber-attacks, i.e., false data injection (FDI) attacks. Different from many existing distributed NE seeking works, it is practical and challenging to get resilient adaptively distributed NE seeking under unknown and unbounded FDI attacks. An attack-resilient NE seeking algorithm that is distributed (i.e., independent of global information on the graph's algebraic connectivity, Lipschitz and monotone constants of pseudo-gradients, or number of players), is presented by means of incorporating the consensus-based gradient play with a distributed attack identifier so as to achieve simultaneous NE seeking and attack identification asymptotically. Another key characteristic is that FDI attacks are allowed to be unknown and unbounded. By exploiting nonsmooth analysis and stability theory, the global asymptotic convergence of the developed algorithm to the NE is ensured. Moreover, we extend this design to further consider the attack-resilient NE seeking of double-integrator players. Lastly, numerical simulation and practical experiment results are presented to validate the developed algorithms' effectiveness.

We consider a gossip approach for finding a Nash equilibrium in a distributed multi-player network game. We extend previous results on Nash equilibrium seeking to the case when the players' cost functions may be affected by the actions of any subset of players. An interference graph is employed to illustrate the partially-coupled cost functions and the asymmetric information requirements. For a given interference graph, we design a generalized communication graph so that players with possibly partially-coupled cost functions exchange only their required information and make decisions based on them. Using a set of standard assumptions on the cost functions, interference and communication graphs, we prove almost sure convergence to a Nash equilibrium for diminishing step sizes. We then quantify the effect of the second largest eigenvalue of the expected communication matrix on the convergence rate, and illustrate the trade-off between the parameters associated with the communication and the interference graphs. Finally, the efficacy of the proposed algorithm on a large-scale networked game is demonstrated via simulation.

This brief studies a distributed Nash equilibrium (NE) seeking strategy for a multi-coalition noncooperative game with local decision sets. In the game model, each coalition consists of some agents and its cost funciton is determined by the sum of the local cost functions subject to the agents in the coalition, where each coalition serves as a virtual player and the actual decision-makers are the agents in the coalition. The cost of each coalition is to minimize its own cost function under the case that the agents in each coalition only know their local information and can not directly access the opponent’s decision information in other coalitions. To this end, a distributed seeking strategy that can be viewed a two-time scale system is developed to search the NE of the formulated multi-coalition game. The fast time-scale system is used to estimate each coalition’s pseudo-gradient by a connected interference graph that illustrates the interactions among the agents in each coalition. The slow time-scale system based on a projected psedudo-gradient dynamics is implemented to seek the NE, where the coalitions estimate other coalitions’ decisions by interacting only with their neighbors via a weight-balanced digraph. For this distributed NE seeking strategy, asymptotic convergence result is given by Lyapunov stability analysis. Finally, the theoretical results is demonstrated via a numerical example.

This article proposes a resilient distributed Nash equilibrium (NE) seeking algorithm for noncooperative games with multiple double-integrator agents who suffer from false data injection (FDI) attacks. A malicious attacker injects false data into agents’ actuators and sensors so that agents’ strategies deviate from the NE of the game with the compromised control inputs and interactive information. First, the robustness of the seeking algorithm against the FDI attacks is analyzed. Then, to mitigate the effect of the attacks on agents’ strategies, the false data injected in the actuators and sensors are regarded as extended states which can be observed by extended state observers (ESOs). Thus, a resilient NE seeking algorithm is proposed based on ESOs. The resilient algorithm can drive the system to converge to the NE without requiring any information about the nature of the attacks. An explicit criterion is given to ensure the effectiveness of the designed algorithms. An example is given to illustrate the results.

This article studies privacy-preserving distributed Nash equilibrium (NE) seeking for aggregative games over directed graphs, where agents' cost functions contain sensitive information. A novel differentially private algorithm using decaying Laplace noise is developed to address two key issues: 1) how to design a distributed algorithm over directed graphs that achieves linear convergence while satisfying differential privacy requirements and 2) how to characterize the tradeoff between convergence accuracy and the privacy budget. First, sufficient conditions for linear convergence are established through the appropriate design of constant step sizes and convex combination parameters. Second, the differential privacy properties of the algorithm are analyzed without assuming bounded gradients, and a quantitative relationship between convergence accuracy and privacy budget is characterized. Furthermore, under additional restrictions on adjacent functions, the cumulative privacy budget admits an explicit expression and remains finite over an unbounded horizon, while the proposed algorithm is proven to converge to the exact NE. Finally, the effectiveness of the proposed algorithm is validated through a Nash-Cournot game and comparative simulations, which demonstrate its superior convergence performance compared to existing methods.

In this paper, we consider the Nash equilibrium (NE) seeking problem for aggregative games and design a distributed heavy-ball algorithm to solve it. This algorithm has faster convergence rate than the well-known distributed first-order algorithms for aggregative games. In order to seek the NE, each player needs to exchange information with its neighbours as well as a central aggregation. For aggregative games, the aggregative term can be either linear or nonlinear in this paper. Furthermore, we consider the generalised Nash equilibrium seeking problem for aggregative games by taking into account the linear coupled constraints among players, and modify our initial algorithm to include game constraints.

We aim to address the Nash equilibrium (NE) seeking problem for multiple players over Markovian switching communication networks in this article, where a new type of distributed synchronous discrete-time algorithm is proposed and utilized. Specifically, each player in the present game model is assumed to employ a gradient-like projection algorithm to choose its action based upon the estimated ones for all the others. Under the mild condition that the union network of all communication network candidates is connected, we show that the players' actions could converge to an arbitrarily small neighborhood of the NE in the mean-square sense by adjusting the algorithm parameters. It is further found that the unique NE is mean-square stable when it is not restricted by any constraint set. In addition, we show that the proposed distributed discrete-time NE seeking algorithm can be utilized to deal with the energy trading problem in microgrids where each microgrid is modeled as a rational player using a purchase price as its action to buy energy from other microgrids with surplus supplies. The energy market allocates the excess energy according to the principle of proportional distribution. Some numerical simulations are finally presented to verify the validity of the present discrete-time NE seeking algorithm in solving the energy trading problem.

In this paper, the distributed Nash equilibrium seeking problem for the games with convex compact set constraint is studied. A distributed estimator based on the leader-follower consensus law is presented to estimate the states of the players. Each player communicates its estimations to its neighbors. The projections of the estimations for the states of the players onto the constraint sets are used in the Nash equilibrium seeking strategy. By synthesizing the projection term of the player's state and the estimator-projection-based gradient play term, the Nash equilibrium seeking algorithm is proposed. The pseudo-gradient of the cost function is only assumed to be strongly monotone in the compact constraint region not required to be globally strongly monotone. Also, the proposed Nash equilibrium seeking algorithm for the constrained games is feasible under any initial states of the players if the given conditions are satisfied. The convergence of the seeking algorithm is proved. Numerical simulations are provided to validate the effectiveness of the proposed seeking algorithm.

In this paper, we study the distributed Nash equilibrium (NE) seeking problem for a class of aggregative games with players described by uncertain perturbed nonlinear dynamics. To seek the NE, each player needs to construct a distributed algorithm based on information of its cost function and the exchanging information obtained from its neighbors. By combining the internal model principle and the average consensus technique, we propose a distributed gradient-based algorithm for the players. This paper not only assures the NE seeking of aggregative games but also achieves the disturbance rejection of external disturbances.