Distributed Nash Equilibrium Searching via Fixed-Time Consensus-Based Algorithms

The development of circuit testing and verification methods is commonly driven by formal analysis centered on an abstract mathematical model of the error or defect the method is designed to detect. Hardware Trojans, however, confound attempts to develop simple representative models due to the varieties of their physical embodiments in a circuit and the creative nature of a rational human adversary. Since it is nonetheless desirable to have a mathematical framework for determining the effectiveness of hardware Trojan detection methods, we present a game theoretic framework for so doing. Modeling the Trojan maker and detection method designer as opposing players in a 2-person strategic game is a necessary step in our process. However, the ultimate utility of the approach depends on an accurate security economic model of both players that can correctly consider the players' incentives, empirically-derived detection method efficacy metrics, a comprehensive taxonomy of hardware Trojans, and the places in the design cycle of the circuit where the Trojan insertion and detection occur. In this paper, we present such a security economic model and the resulting game, which we call the Trust Game. We illustrate the value of this game primarily in the context of how it may guide the development of new hardware Trojan detection methods. We solve a representative game, illustrating the value of two common solution concepts, the iterated elimination of dominated strategies and Nash equilibrium. We further show that this framework has utility to both of the opposing players in the game. Finally, we recommend the development of standardized Trust Games that can be used to quickly measure the efficacy of both new hardware Trojans and hardware Trojan detection methods.

ABSTRACTThis paper investigates the leader-following fixed-time output feedback consensus problem for second-order multi-agent systems with input saturation. By combing fixed-time control technique and bi-limit homogeneous systems theory, a class of bounded fixed-time consensus protocols are developed for leader-following multi-agent systems. The protocol design is divided into two parts. First, when all the state information of the followers are measurable, a state feedback consensus protocol is designed to achieve fixed-time consensus. Then, when the velocity information is unmeasurable, an observer-based fixed-time consensus protocol is proposed. With the help of Lyapunov stability theorem and the property of a homogeneous function, it is theoretically shown that the states of all followers can track that of the leader in fixed-time in the presence of input saturation. Finally, numerical simulation is carried out to illustrate the effectiveness of theoretical results.

In this paper, we consider distributed Nash equilibrium (NE) seeking in potential games over a multi-agent network, where each agent can not observe the actions of all its rivals. Based on the best response dynamics, we design a distributed NE seeking algorithm by incorporating the non-smooth finite-time average tracking dynamics, where each agent only needs to know its own action and exchange information with its neighbours through a communication graph. We give a sufficient condition for the Lipschitz continuity of the best response mapping for potential games, and then prove the convergence of the proposed algorithm based on the Lyapunov theory. Numerical simulations are given to verify the result and illustrate the effectiveness of the algorithm.

Stability and robustness analysis of minmax solutions for differential graphical games

Game theory studies mathematical models of interactions amongst self-interested entities. A means a description of the outcome of a game, and it is important that it should be defined in such a way that a solution always exists (every game should have an outcome). Nash's famous theorem that mixed-strategy equilibria are guaranteed to exist, resulted in Nash equilibrium being the prominent solution concept in game theory. As a result, computational challenges of the form a game, find a solution, have the property that we are searching for something whose existence is guaranteed (they are total search problems). Moreover, these solutions belong to the complexity class NP, since it is usually straightforward to check whether a proposed solution is correct (an incorrect one will admit a profitable deviation by one or more of the players, and this is usually easy to find). However, in versions of the problem that appear to be computationally hard, we cannot apply NP-completeness, due to a result of Megiddo saying that total search problems cannot be NP-complete unless NP is equal to co-NP. In this tutorial, which is intended for people familiar with NP-completeness, I give an overview of the alternative notions of computational hardness that apply to game-theoretic solution concepts. I discuss the complexity class PPAD (introduced by Papadimitriou) which captures the computational complexity of various classes of games that don’t seem to be solvable in polynomial time. I also mention the complexity classes PLS and FIXP, and the kinds of games that they apply to. Suppose, alternatively, that we have a polynomial-time algorithm that applies to some given class of games. A follow-up question is whether there exist algorithms that find a solution via processes that reflect decentralised selfish behaviour. This is because a solution concept arguably remains unrealistic if it can be efficiently computed, but only using a highly centralised algorithm. In the second half of the tutorial I present some results on learning dynamics for equilibrium computation, and mention recent work on communication complexity and query complexity. I discuss some research directions and open problems, such as the following. What are the prospects for proving that PPAD is as hard as NP? How about algorithms that find improved approximate Nash equilibria? 2-player games are easy to solve in practice, using the Lemke-Howson algorithm, so is there a satisfying mathematical sense in which 2-player games are easy to solve? (For example, a sense in which Lemke-Howson works most of the time?)

We study the economic interests of a wireless access point owner and his paying client, and model their interaction as a dynamic game. The key feature of this game is that the players have asymmetric information - the client knows more than the access provider. We find that if a client has a utility function (a temporal utility function that grows linearly), it is a Nash equilibrium for the provider to charge the client a constant price per unit time. On the other hand, if the client has a utility function (a utility function that is a step function), the client would be unwilling to pay until the final time slot of the file transfer. We also study an expanded game where an access point sells to a reseller,which in turn sells to a mobile client and show that if the client has a web browser utility function, that constant price is a Nash equilibrium of the three player game. Finally, we study a two player game in which the access point does not know whether he faces a web browser or file transferor type client, and show conditions for which it is not a Nash equilibrium for the access point to maintain a constant price.

In this note, we investigate the fixed time consensus in probability for stochastic multi-agent systems under undirected graph. First, a nonlinear consensus protocol with Gaussian white noise is given, and the concept of fixed-time consensus in probability is given. Second, we prove that multi-agent systems can achieve fixed-time consensus in probability under undirected topology by using graph theory, stochastic Lyapunov theory and probability theory. Finally, a simulation example is given to verify the effectiveness of the theory results.

Distributed Nash equilibrium seeking for aggregative games with second-order nonlinear players

Nash equilibria in discrete routing games with convex latency functions

This article proposes a game-based distributed decision optimization method for heterogeneous multiagent systems with unknown nonlinear dynamics. Due to the information exchange between agents in the network, the unknown nonlinear dynamics lead to the degradation of local and all-agent control performance, which causes the strategies of all agents to deviate from the Nash equilibrium under a given goal. To address this problem, an adaptive distributed algorithm is designed to seek Nash equilibrium by combining two optimization levels. Specifically, the decision layer uses a distributed consensus algorithm to achieve benefit evaluation and a gradient algorithm to generate reference signals. Then, the control layer uses the virtual reference signal from the decision layer and the neural network estimation information to design an adaptive control algorithm. The proposed method performs real-time adaptive optimization of the strategies and control performance of the decision and control layers, ensuring the successful implementation of the distributed Nash equilibrium search. The convergence of the proposed algorithm is proved in the Lyapunov sense. Finally, simulation examples demonstrate the performance and effectiveness of the proposed method.

Distributed Nash Equilibrium (NE) seeking for networked games has been widely investigated in recent years. In particular, convergence speed and communication cost are two of the critical concerns for the design of the seeking algorithms. To achieve fast convergence while saving communication resources, this paper investigates the appointed-time distributed NE seeking problem for networked games under a discrete-time communication scenario. By utilizing the idea of motion planning, a new continuous-time distributed NE seeking algorithm, which ensures convergence to NE at the prescribed time, is proposed based on sampled-data information exchange. Compared with the existing related works, the established algorithm has advantages on both the communication cost and convergence speed (i.e., appointed-time convergence). Finally, the effectiveness of the proposed method is verified via numerical simulations.

Accurate and efficient detection of the radio-frequency spectrum is a challenging issue in wireless sensor networks (WSNs), which are used for multi-channel cooperative spectrum sensing (MCSS). Due to the limited battery power of sensors, lifetime maximization of a WSN is an important issue further sensing quality requirements. The issue is more complex if the low-cost sensors cannot sense more than one channel simultaneously, because they do not have high-speed Analogue-to-Digital-Convertors which need high-power batteries. This paper proposes a novel game-theoretic sensor selection algorithm for MCSS that extends the network lifetime assuming the quality of sensing and the limited ability of sensors. To this end, an optimization problem is formulated using the “max–min” method, in which the minimum remaining energy of sensors is maximized to keep energy balancing in the WSN. This paper proposes a coalition game to solve the problem, in which sensors act as game players and decide to make disjoint coalitions for MCSS. Each coalition senses one of the channels. Other nodes, that decide to sense none of the channels, turn off their sensing module to reserve energy. First, a novel utility function for the coalitions is proposed based on the remaining energy and consumption energy of sensors besides their detection quality. Then, an algorithm is designed to reach a Nash-Equilibrium (NE) coalition structure. The existence of at least one NE, converging toward one of the NEs, and the computational complexity of the proposed algorithm are discussed. Finally, simulations are presented to demonstrate the ability of the proposed algorithm, assuming the systems using IEEE802.15.4/Zigbee and IEEE802.11af.

This paper investigates the Nash equilibrium (NE) seeking problem of the game in which the dynamics of all players are affected by external disturbances and the communication network among players may be influenced by persistent attacks. To solve this problem, a distributed robust switched algorithm is proposed. This distributed switched strategy consists of two modules: One is the internal-model based dynamics that can be used to compensate the effects caused by external disturbances; the other is gradient-based optimization module which can utilize the local information to estimate the players' decisions and dynamically seek the optimal solution. By utilizing average dwell-time and time-ratio constraints to characterize the network attacks, the proposed switched strategy is modeled into a hybrid dynamical system. Moreover, employing the hybrid systems approach, we prove that the decisions of all players can exponentially converge to the NE of the considered game. Finally, an example is provided to illustrate the theoretical results.

We study the computation of equilibria of two-strategy anonymous games, via algorithms that may proceed via a sequence of adaptive queries to the game's payoff function, assumed to be unknown initially. The general topic we consider is query complexity, that is, how many queries are necessary or sufficient to compute an exact or approximate Nash equilibrium. We show that exact equilibria cannot be found via query-efficient algorithms. We also give an example of a 2-strategy, 3-player anonymous game that does not have any exact Nash equilibrium in rational numbers. Our main result is a new randomized query-efficient algorithm that finds a $$On^{-1/4}$$-approximate Nash equilibrium querying $$\tilde{O}n^{3/2}$$ payoffs and runs in time $$\tilde{O}n^{3/2}$$. This improves on the running time of pre-existing algorithms for approximate equilibria of anonymous games, and is the first one to obtain an inverse polynomial approximation in poly-time. We also show how this can be used to get an efficient PTAS. Furthermore, we prove that $$\varOmega n \log {n}$$ payoffs must be queried in order to find any $$\epsilon $$-well-supported Nash equilibrium, even by randomized algorithms.

In this article, the prescribed-time distributed Nash equilibrium (NE) seeking problem for multicoalition games with heterogeneous general linear players is investigated. In such games, there exist multiple coalitions that consist of heterogeneous players. The players have to seek the NE of the entire coalition while being limited to access only their individual cost functions. The proposed algorithm involves a two-layer structure composed of the NE seeking part and the tracking control part. To address the challenge of global information dependency, the leader-following-based estimates are designed. The NE seeking part is designed based on the distributed estimates and concurrently performed over the weight-balanced digraphs. Subsequently, by introducing the time-varying parametric Lyapunov equation, the strategies of the players are driven to the NE solution within a manually prescribed settling time, and the prescribed-time convergence is proved. Finally, the proposed algorithm is executed on the heterogeneous coalitions made up of quadrotor unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs) in a group formation scenario. Both the simulation and the experiment results are presented to show the effectiveness of the algorithm.