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This paper studies distributed Nash equilibrium (NE) seeking under Denial-of-Service (DoS) attacks and quantization. The players can only exchange information with their own direct neighbors. The transmitted information is subject to quantization and packet losses induced by malicious DoS attacks. We propose a quantized distributed NE seeking strategy based on the approach of dynamic quantized consensus. To solve the quantizer saturation problem caused by DoS attacks, the quantization mechanism is equipped to have zooming-in and holding capabilities, in which the holding capability is consistent with the results in quantized consensus under DoS. A sufficient condition on the number of quantizer levels is provided, under which the quantizers are free from saturation under DoS attacks. The proposed distributed quantized NE seeking strategy is shown to have the so-called maximum resilience to DoS attacks. Namely, if the bound characterizing the maximum resilience is violated, an attacker can deny all the transmissions and hence distributed NE seeking is impossible.

This paper explores distributed Nash equilibrium seeking problems for games in which the players have limited knowledge on other players' actions. In particular, the involved players are considered to be high-order integrators with their control inputs constrained within a pre-specified region. A linear transformation for the players' dynamics is firstly utilized to facilitate the design of bounded control inputs incorporating multiple saturation functions. By introducing consensus protocols with adaptive and time-varying gains, the unknown actions for players are distributively estimated. Then, a fully distributed Nash equilibrium seeking strategy is exploited, showcasing its remarkable properties: i) ensuring the boundedness of control inputs; ii) avoiding any global information/parameters; and iii) allowing the graph to be directed. Based on Lyapunov stability analysis, it is theoretically proved that the proposed distributed control strategy can lead all the players' actions to the Nash equilibrium. Finally, an illustrative example is given to validate effectiveness of the proposed method.

Distributed Nash equilibrium seeking for games in uncertain networked systems without a prior knowledge about control directions is explored in this paper. More specifically, the dynamics of the players are supposed to be first-order or second-order systems in which the control directions are unknown and there are parametric uncertainties. To achieve Nash equilibrium seeking in a distributed way, Nussbaum function based strategies are proposed through separately designing an optimization module and a state regulation module. The optimization module generates a reference trajectory, that can search for the Nash equilibrium, for the state regulation module. The state regulator is designed to steer the players' actions to the reference trajectory. An adaptive law is included in the state regulation module to compensate for the uncertain parameter in the players' dynamics and the Nussbaum function is included to address the unavailability of the control directions. Fully distributed implementations of the proposed algorithms are discussed and investigated. Through our analytical explorations, we show that the proposed seeking strategies can drive the players' actions to the Nash equilibrium asymptotically without requiring the homogeneity of the players' unknown control directions based on Barbalat's lemma. A numerical example is given to support the theoretical analysis of the proposed algorithms.

In practical applications, decision-makers with heterogeneous dynamics may be engaged in the same decision-making process. This motivates us to study distributed Nash equilibrium seeking for games in which players are mixed-order (first- and second-order) integrators influenced by unknown dynamics and external disturbances in this paper. To solve this problem, we employ an adaptive neural network to manage unknown dynamics and disturbances, based on which a distributed Nash equilibrium seeking algorithm is developed by further adapting concepts from gradient-based optimization and multi-agent consensus. By constructing appropriate Lyapunov functions, we analytically prove convergence of the reported method. Theoretical investigations suggest that players' actions would be steered to an arbitrarily small neighborhood of the Nash equilibrium, which is also testified by simulations.

The design of Nash equilibrium seeking strategies for games in which the involved players are of second-order integrator-type dynamics is investigated in this paper. Noticing that velocity signals are usually noisy or not available for feedback control in practical engineering systems, this paper supposes that the velocity signals are not accessible for the players. To deal with the absence of velocity measurements, two estimators are designed, based on which Nash equilibrium seeking strategies are constructed. The first strategy is established by employing an observer, which has the same order as the players' dynamics, to estimate the unavailable system states (e.g., the players' velocities). The second strategy is designed based on a high-pass filter and is motivated by the incentive to reduce the order of the closed-loop system which in turn reduces the computation costs of the seeking algorithm. Extensions to Nash equilibrium seeking for networked games are provided. Taking the advantages of leader-following consensus protocols, it turns out that both the observer-based method and the filter-based method can be adapted to deal with games in distributed systems, which shows the extensibility of the developed strategies. Through Lyapunov stability analysis, it is analytically proven that the players' actions can be regulated to the Nash equilibrium point and their velocities can be regulated to zero by utilizing the proposed velocity-free Nash equilibrium seeking strategies. A numerical example is provided for the verifications of the proposed algorithms.

This paper considers the design of fully distributed Nash equilibrium seeking strategies for multi-agent games. To develop fully distributed seeking strategies, two adaptive control laws, including a node-based control law and an edge-based control law, are proposed. In the node-based adaptive strategy, each player adjusts their own weight on their procurable consensus error dynamically. Moreover, in the edge-based algorithm, the fully distributed strategy is designed by adjusting the weights on the edges of the communication graph adaptively. By utilizing LaSalle's invariance principle, it is shown that the Nash equilibrium is globally asymptotically stable by both strategies given that the players' objective functions are twice-continuously differentiable, the partial derivatives of the players' objective functions with respect to their own actions are globally Lipschitz, the pseudo-gradient vector of the game is strongly monotone and the communication network is undirected and connected. In addition, we further show that the edge-based method can be easily adapted to accommodate time-varying communication conditions, in which the communication network is switching among a set of undirected and connected graphs. In the last, a numerical example is given to illustrate the effectiveness of the proposed methods.

This paper considers the privacy-preserving Nash equilibrium seeking strategy design for a class of networked aggregative games, in which the players' objective functions are considered to be sensitive information to be protected. In particular, we consider that the networked game is free of central node and the aggregate information is not directly available to the players. As there is no central authority to provide the aggregate information required by each player to update their actions, a dynamic average consensus protocol is employed to estimate it. To protect the players' privacy, we perturb the transmitted information among the players by independent random noises drawn from Laplace distributions. By synthesizing the perturbed average consensus protocol with a gradient algorithm, distributed privacy-preserving Nash equilibrium seeking strategies are established for the aggregative games under both fixed and time-varying communication topologies. With explicit quantifications of the mean square errors, the convergence results of the proposed methods are presented. Moreover, it is analytically proven that the proposed algorithm is $\epsilon$-differentially private, where $\epsilon$ depends on the stepsize of the gradient algorithm and the scaling parameter of the random variables. The presented results indicate that there is a tradeoff between the convergence accuracy and the privacy level. Lastly, a numerical example is provided for the verification of the proposed methods.

This paper aims to accommodate games in which the players' dynamics are subject to un-modeled and disturbance terms. The un-modeled and disturbance terms are regarded as extended states for which observers are designed to estimate them. Compensating the players' dynamics with the observed values, the control laws are designed to achieve the robust seeking of the Nash equilibrium for networked games. Firstly, we consider the case in which the players' dynamics are subject to bounded disturbances only. In this case, the seeking strategy is developed by employing a smooth observer based on the Proportional-Integral (PI) control. By utilizing the designed strategy, we show that the players' actions would converge to a small neighborhood of the Nash equilibrium. Moreover, the ultimate bound can be adjusted to be arbitrarily small by tuning the control gains. Then, we further consider the case in which both an un-modeled term and a disturbance term coexist in the players' dynamics. In this case, we adapt the idea from the Robust Integral of the Sign of the Error (RISE) method in the strategy design to achieve the asymptotic seeking of the Nash equilibrium. Both strategies are analytically investigated via the Lyapunov stability analysis. The applications of the proposed methods for a network of velocity-actuated vehicles are discussed. Finally, the effectiveness of the proposed methods is verified via conducting numerical simulations.

Noticing that physical limitations are ubiquitous in practical engineering systems, this paper considers Nash equilibrium seeking for games in systems where the control inputs are bounded. More specifically, first-order integrator-type systems with bounded control inputs are firstly considered and two saturated control strategies are designed to seek for the Nash equilibrium of the game. Then, we further consider the Nash equilibrium seeking problem for games in second-order integrator-type systems. As this problem has rarely been investigated, we firstly propose a centralized seeking strategy without considering the boundedness of the control inputs, followed by a distributed counterpart. By further adapting a saturation function into the Nash equilibrium seeking strategy, a new seeking strategy is then designed for the considered second-order systems with bounded controls. In the proposed distributed strategies, consensus protocols are included for information sharing and the saturation functions are utilized to construct bounded control inputs. The convergence results are established through conducting Lyapunov stability analysis. Lastly, by considering the connectivity control of mobile sensor networks, the proposed methods are numerically verified.

This paper aims to reduce the communication and computation costs of the Nash equilibrium seeking strategy for the $N$-coalition noncooperative games proposed in [1]. The objective is achieved in two manners: 1. An interference graph is introduced to describe the interactions among the agents in each coalition. 2. The Nash equilibrium seeking strategy is designed with the interference graphs considered. The convergence property of the proposed Nash equilibrium seeking strategy is analytically investigated. It is shown that the agents' actions generated by the proposed method converge to a neighborhood of the Nash equilibrium of the graphical $N$-coalition noncooperative games under certain conditions. Several special cases where there is only one coalition and/or there are coalitions with only one agent are considered. The results for the special cases demonstrate that the proposed seeking strategy achieves the solution seeking for noncooperative games, social cost minimization problems and single-agent optimization problems in a unified framework. Numerical examples are presented to support the theoretical results.

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, an aggregate game approach is proposed for the modeling and analysis of energy consumption control in smart grid. Since the electricity user's cost function depends on the aggregate load, which is unknown to the end users, an aggregate load estimator is employed to estimate it. Based on the communication among the users about their estimations on the aggregate load, Nash equilibrium seeking strategies are proposed for the electricity users. By using singular perturbation analysis and Lyapunov stability analysis, a local convergence result to the Nash equilibrium is presented for the energy consumption game that may have multiple Nash equilibria. For the energy consumption game with a unique Nash equilibrium, it is shown that the players' strategies converge to the Nash equilibrium non-locally. More specially, if the unique Nash equilibrium is an inner Nash equilibrium, then the convergence rate can be quantified. Energy consumption game with stubborn players is also investigated. Convergence to the best response strategies for the rational players is ensured. Numerical examples are provided to verify the effectiveness of the proposed methods.