Differentially-private distributed Nash equilibrium seeking via a probabilistic mapping mechanism

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.

Distributed Nash equilibrium seeking with stochastic event-triggered mechanism

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.

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.

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 investigates the problem of distributed Nash equilibrium (NE) seeking in noncooperative games within a directed communication network. For promoting the efficiency of communication among players, a gradient-based dynamic event-triggered mechanism is proposed, where Zeno behavior is excluded. Moreover, based on the Lyapunov stability theory, we derive sufficient conditions for exponential convergence and demonstrate that the seeking strategy proposed facilitates the convergence of players' actions toward the NE. To illustrate the effectiveness of the proposed strategies, simulation results are presented in a system consisting of five agents.

The distributed Nash equilibrium (NE) seeking problem for multicoalition games has attracted increasing attention in recent years, but the research mainly focuses on the case without agreement demand within coalitions. This article considers a class of networked games among multiple coalitions where each coalition contains multiple agents that cooperate to minimize the sum of their costs, subject to the demand of reaching an agreement on their state values. Furthermore, the underlying network topology among the agents does not need to be balanced. To achieve the goal of NE seeking within such a context, two estimates are constructed for each agent, namely, an estimate of partial derivatives of the cost function and an estimate of global state values, based on which, an iterative state updating law is elaborately designed. Linear convergence of the proposed algorithm is demonstrated. It is shown that the consistency-constrained multicoalition games investigated in this article put the well-studied networked games among individual players and distributed optimization in a unified framework, and the proposed algorithm can easily degenerate into solutions to these problems.

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.

Distributed Nash equilibrium (NE) seeking problems for networked games have been widely investigated in recent years. Despite the increasing attention, communication expenditure is becoming a major bottleneck for scaling up distributed approaches within limited communication bandwidth between agents. To reduce communication cost, an efficient event-triggered and compressed distributed NE seeking (ETC-DNES) algorithm is proposed in this article to obtain an NE for games over directed graphs, where the communication efficiency is improved by event-triggered exchanges of compressed information among neighbors. ETC-DNES saves communication costs in both transmitted bits and rounds of communication. Furthermore, our method only requires the row-stochastic property of the adjacency matrix, unlike previous approaches that hinged on doubly stochastic communication matrices. We provide convergence guarantees for ETC-DNES on games with restricted strongly monotone mappings and testify its efficiency with no sacrifice on the accuracy. The algorithm and analysis are extended to a compressed algorithm with stochastic event-triggered mechanism, i.e., stochastic event-triggered and compressed distributed NE seeking (SETC-DNES) algorithm. In SETC-DNES, we introduce a random variable in the triggering condition to further enhance the algorithm efficiency. We demonstrate that SETC-DNES guarantees linear convergence to the NE while achieving even greater reductions in communication costs compared to ETC-DNES. Finally, numerical simulations illustrate the effectiveness of the proposed algorithms.

In this article, we study the problem of Nash equilibrium (NE) seeking of N -player games with high-order integrator agents over jointly strongly connected networks. The same problem was studied recently over static, undirected, and connected networks. Since a jointly strongly connected network can be directed and disconnected at every time instant, our result strictly includes the existing results as special cases. Moreover, in addition to some analytic conditions on the payoff functions, the existing results also rely on the satisfaction of an inequality involving some Lipschitz constant and the smallest nonzero eigenvalue of the Laplacian of the network graph. By adopting a different approach, our result does not rely on the satisfaction of this inequality. Furthermore, our result can also be extended to the case where the additive disturbances are imposed on the input of each agent. Our design is illustrated by two numerical examples.

Distributed Nash equilibrium seeking under quantization communication

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.

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.

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.

This paper considers a distributed Nash equilibrium (NE) seeking problem with limited communication capacity. A fully distributed NE seeking algorithm is proposed with quantized information, including projected pseudo-gradient dynamics, distributed decision estimation and adaptive quantization. Based on a proposed encoder-decoder scheme, the algorithm is able to converge to the theoretical NE without any errors caused by quantization. Finally, a numerical simulation is provided to validate the effectiveness of our algorithm.KeywordsDistributed Nash equilibrium seekingQuantization communicationProjected pseudo-gradient dynamicsConsensus