Nash Equilibrium Seeking For Games Research Articles

Distributed Robust Nash Equilibrium Seeking for Mixed-Order Games by a Neural-Network-Based Approach

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 article. 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 multiagent consensus. By constructing appropriate Lyapunov functions, we analytically prove the 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.

Distributed Nash Equilibrium Seeking for General Networked Games With Bounded Disturbances

This paper is concerned with anti-disturbance Nash equilibrium seeking for games with partial information. First, reduced-order disturbance observer-based algorithms are proposed to achieve Nash equilibrium seeking for games with first-order and second-order players, respectively. In the developed algorithms, the observed disturbance values are included in control signals to eliminate the influence of disturbances, based on which a gradient-like optimization method is implemented for each player. Second, a signum function based distributed algorithm is proposed to attenuate disturbances for games with second-order integrator-type players. To be more specific, a signum function is involved in the proposed seeking strategy to dominate disturbances, based on which the feedback of the velocity-like states and the gradients of the functions associated with players achieves stabilization of system dynamics and optimization of players' objective functions. Through Lyapunov stability analysis, it is proven that the players' actions can approach a small region around the Nash equilibrium by utilizing disturbance observer-based strategies with appropriate control gains. Moreover, exponential (asymptotic) convergence can be achieved when the signum function based control strategy (with an adaptive control gain) is employed. The performance of the proposed algorithms is tested by utilizing an integrated simulation platform of virtual robot experimentation platform (V-REP) and MATLAB.

Robust Distributed Nash Equilibrium Seeking for Games Under Attacks and Communication Delays

In this article, a noncooperative game in which the engaged players are of double integrator-type dynamics, is investigated. The dynamics of the players are considered to be subject to unknown time-varying disturbances and unmodeled terms. Besides, the communication topology among the players suffers from attacks and time-varying communication delays. To find a Nash equilibrium for such games, a distributed robust Nash equilibrium seeking algorithm is proposed. By utilizing average dwell-time and time-ratio constraints to tackle the attacks, the closed-loop system is modeled as a hybrid system with memory. To analyze the stability of the distributed switched algorithm, a Lyapunov functional is constructed, by which we show the uniform global asymptotic stability of the equilibrium. Finally, an example is given to illustrate our results.

Distributed Nash Equilibrium Seeking for Games in Systems With Bounded Control Inputs

Noticing that actuator limitations are ubiquitous in practical engineering systems, this article 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 first considered and two saturated control strategies are designed to seek the Nash equilibrium of the game. Then, second-order integrator-type systems are further considered. In this case, a centralized seeking strategy is first proposed without considering the boundedness of the control inputs, followed by a distributed counterpart. By further adapting a saturation function into the distributed Nash equilibrium seeking strategy, the boundedness of the control input is addressed. 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 analytically studied by Lyapunov stability analysis. Finally, by considering the connectivity control of mobile sensor networks, the proposed methods are numerically verified.

Stochastic Nash Equilibrium Seeking for Games with General Nonlinear Payoffs

We introduce a multi-input stochastic extremum seeking algorithm to solve the problem of seeking Nash equilibria for a noncooperative game whose N players seek to maximize their individual payoff functions. The payoff functions are general (not necessarily quadratic), and their forms are not known to the players. Our algorithm is a nonmodel-based approach for asymptotic attainment of the Nash equilibria. Different from classical game theory algorithms, where each player employs the knowledge of the functional form of his payoff and the knowledge of the other players' actions, a player employing our algorithm measures only his own payoff values, without knowing the functional form of his or other players' payoff functions. We prove local exponential (in probability) convergence of our algorithms. For nonquadratic payoffs, the convergence is not necessarily perfect but may be biased in proportion to the third derivatives of the payoff functions and the intensity of the stochastic perturbations used in the algorithm. We quantify the size of these residual biases. Compared to the deterministic extremum seeking with sinusoidal perturbation signals, where convergence occurs only if the players use distinct frequencies, in our algorithm each player simply employs an independent ergodic stochastic probing signal in his seeking strategy, which is realistic in noncooperative games. As a special case of an N-player noncooperative game, the problem of standard multivariable optimization (when the players' payoffs coincide) for quadratic maps is also solved using our stochastic extremum seeking algorithm.

Stochastic Nash Equilibrium Seeking for Games with General Nonlinear Payoffs

AbstractIn this paper, we propose a multi-input stochastic extremum seeking algorithm to solve Nash equilibrium seeking for a noncooperative game whose N players seek to maximize their general nonlinear payoff functions. Our algorithm is a non-model based approach for asymptotic, locally stable attainment of the Nash equilibria in probability. Different from classical game theory algorithms, where each player employs the knowledge of the functional form of his payoff and the knowledge of the other players's actions, a player employing our algorithm measures only his own payoff values, without knowing the functional form of his or other players’ payoff functions. For the general nonlinear payoffs, the convergence is not perfect but is biased in proportion to the third derivatives of the payoff functions and is dependent on the intensity of stochastic perturbation. We quantify the size of these residual biases. Compared to the deterministic extremum seeking with sinusoidal perturbation signals, the advantage of our algorithm lies in that each player independently employs his seeking strategy by choosing an ergodic stochastic probing signal and there is no requirement on different frequencies for different players.