Pareto Nash Equilibrium Research Articles

A Cross-Layer Game-Theoretic Approach to Resilient Control of Networked Switched Systems Against DoS Attacks.

This article investigates the resilient control strategies of networked switched systems (NSSs) against denial-of-service (DoS) attacks and external disturbance. In the network layer, both the defender and the attacker allocate energy over multiple channels. Considering the impact of switching characteristic in the physical layer on the network layer, a dynamic regulating factor is proposed to adjust the total energy of the defender. To optimize the signal-to-interference-noise ratio and energy consumption simultaneously at each player's side, a multiobjective game problem is formulated. Furthermore, a nondominated sorting genetic algorithm framework is employed, incorporating the knee point selection mechanism to attain the Pareto-Nash equilibrium, based on which the optimal defense strategy can be derived to achieve resilience against DoS attacks. In the physical-layer, taking the dynamic packet loss caused by DoS attacks and external disturbance into account, an H∞ minimax controller containing control inputs and the switching signal is designed to guarantee the optimal performance for NSSs through the dynamic game-theoretic approach. Finally, the networked continuous stirred tank reactor system is provided to verify the effectiveness of the proposed method.

Fuzzy Weighted Pareto–Nash Equilibria of Multi-Objective Bi-Matrix Games with Fuzzy Payoffs and Their Applications

Based on our previous research, this paper further discusses the multi-objective bi-matrix game with fuzzy payoffs (MBGFP), which is a special case of the fuzzy constrained multi-objective game with fuzzy payoffs. We first prove that any bi-matrix game with interval payoffs (BGIP) has at least one Pareto–Nash equilibrium. Then, with the help of BGIP, we obtain the necessary and sufficient conditions for the existence of fuzzy Pareto–Nash equilibrium of MBGFP. Secondly, based on the bilinear programming method for calculating Nash equilibrium in crisp bi-matrix games, we established a bilinear programming method with parameters for calculating fuzzy Pareto–Nash equilibrium. By considering the importance of each objective to the players, MBGFP is transformed into a bi-matrix game with fuzzy payoffs (BGFP). Furthermore, we obtained the necessary and sufficient conditions for the existence of fuzzy weighted Pareto–Nash equilibrium and its calculation method. Finally, a practical example is used to illustrate the effectiveness of our proposed calculation method.

Existence and stability of fuzzy Pareto-Nash equilibria for fuzzy constrained multi-objective games with fuzzy payoffs

<abstract><p>A new class of fuzzy constrained multi-objective games with fuzzy payoffs (FCMGFPs) is considered in this paper. First, Berge's maximum theorem for the fuzzy-vector-valued function is obtained. Based on this theorem and the Fan-Glicksberg fixed point theorem, the existence theorem of the fuzzy Pareto-Nash equilibrium for the FCMGFP is established. Second, the abstract rationality function for the FCMGFP is given by using a nonlinear scalarization function of interval vectors. Finally, a series of results, such as structural stability ($ (\gamma, \epsilon) $-stability) and robustness to $ \epsilon $-equilibrium ($ (\gamma, \epsilon) $-robustness), are obtained.</p></abstract>

Generic Stability of the Weakly Pareto-Nash Equilibrium with Strategy Transformational Barriers

The object of this paper is to establish a new model with strategy transformational barriers for a class of generalized multileader multifollower multiple objective games (GMLMFMOG) and further deduce some new results of the weakly Pareto-Nash equilibrium (WPNE) with strategy transformational barriers for the GMLMFMOG. First, we investigate the existence of the WPNE with strategy transformational barriers for the GMLMFMOG by using the Kakutani-Fan-Glicksberg fixed point theory. Next, we study the generic stability of the GMLMFMOG with strategy transformational barriers in Hausdorff space. Finally, we obtain that the majority of the WPNE with strategy transformational barriers for the GMLMFMOG are essential on the meaning of Baire’s category. In addition, we demonstrate that there is at least an essential component for the GMLMFMOG with strategy transformational barriers.

On Nash–Stackelberg–Nash games under decision-dependent uncertainties: Model and equilibrium

In this paper, we discuss a class of two-stage hierarchical games with multiple leaders and followers, which is called Nash–Stackelberg–Nash (N–S–N) games. Particularly, we consider N–S–N games under decision-dependent uncertainties (DDUs). DDUs refer to the uncertainties that are affected by the strategies of decision-makers and have been rarely addressed in game equilibrium analysis. In this paper, we first formulate the N–S–N games with DDUs of complete ignorance, where the interactions between the players and DDUs are characterized by uncertainty sets that depend parametrically on the players’ strategies. Then, a rigorous definition for the equilibrium of the game is established by consolidating generalized Nash equilibrium and Pareto-Nash equilibrium. Afterward, we prove the existence of the equilibrium of N–S–N games under DDUs by applying Kakutani’s fixed-point theorem. Finally, an illustrative example is provided to show the impact of DDUs on the equilibrium of N–S–N games.

Self-organized Partition of Distribution Networks with Distributed Generations and Soft Open Points

Islanding operation relying on distributed generations (DGs) is anticipated to be a promising operational measure for resilience enhancement of distribution networks. In this paper, a self-organized network partition framework for controllable DGs (CDGs) is proposed for the distribution network with DGs and soft open points (SOPs) from a game-theoretic perspective. The labeled numbers of the nodes on paths between every two adjacent CDGs are used as the optimization variables in the partition formation, which can avoid a large number of invalid partition schemes. Two objectives including the weighted recovered load and the power loss are considered, which reflect the goals of maximizing the critical load recovery and minimizing the power loss, respectively. The power loss is determined through the AC power flow calculation using Newton-Raphson method considering the operational characteristics of SOPs. The partition problem is modeled as a discrete bi-objective game, which is further transformed into a continuous game model, and the existence of the Pareto-Nash equilibrium is also proved. A decentralized method is developed to solve the proposed bi-objective game relying on the local communications and a global information discovery scheme, which is more practical after the extreme events. Numerical studies on the IEEE 69-bus distribution system and a 135-bus multi-feeder system are performed to validate the effectiveness of the proposed method. The proposed model is applied in different network topologies, and compared with the centralized optimization models. The robustness of the method to communication failures is analyzed, and the scalability of the proposed model is also validated through testing it in a larger distribution network.

Characterizations of Pareto-Nash Equilibria for Multiobjective Potential Population Games

This paper studies the characterizations of (weakly) Pareto-Nash equilibria for multiobjective population games with a vector-valued potential function called multiobjective potential population games, where agents synchronously maximize multiobjective functions with finite strategies via a partial order on the criteria-function set. In such games, multiobjective payoff functions are equal to the transpose of the Jacobi matrix of its potential function. For multiobjective potential population games, based on Kuhn-Tucker conditions of multiobjective optimization, a strongly (weakly) Kuhn-Tucker state is introduced for its vector-valued potential function and it is proven that each strongly (weakly) Kuhn-Tucker state is one (weakly) Pareto-Nash equilibrium. The converse is obtained for multiobjective potential population games with two strategies by utilizing Tucker’s Theorem of the alternative and Motzkin’s one of linear systems. Precisely, each (weakly) Pareto-Nash equilibrium is equivalent to a strongly (weakly) Kuhn-Tucker state for multiobjective potential population games with two strategies. These characterizations by a vector-valued approach are more comprehensive than an additive weighted method. Multiobjective potential population games are the extension of population potential games from a single objective to multiobjective cases. These novel results provide a theoretical basis for further computing (weakly) Pareto-Nash equilibria of multiobjective potential population games and their practical applications.

Strongly essential set of vector Ky Fan's points problem and its applications

In this paper, several existence results of strongly essential set of the solution set for Ky Fan's section problems and vector Ky Fan's point problems are obtained. Firstly, two kinds of strongly essential sets of Ky Fan's section problems are defined, and some further results on existence of the strongly essential component of solutions set of Ky Fan's section problems are proved, which generalize the conclusion in ^{[<span class="xref"><a href="#b22" ref-type="bibr">22</a></span>]}, and further generalize the conclusions in ^{[<span class="xref"><a href="#b21" ref-type="bibr">21</a></span>,<span class="xref"><a href="#b28" ref-type="bibr">28</a></span>]}. Secondly, based on the above results, two classes of stronger perturbations of vector-valued inequality functions are proposed respectively, and several existence results of the strongly essential component of set of vector Ky Fan's points are obtained. By comparing several metrics, we give some strong and weak relationships among the various metrics involved in the text. The main results of this paper actually generalize the relevant conclusions in the current literature. Finally, as an application, we obtain an existence result of the strongly essential component of weakly Pareto-Nash equilibrium for multiobjective games.

Nash Equilibrium Strategy for Bi-matrix Games with L-R Fuzzy Payoffs

In this paper, bi-matrix games are investigated based on L-R fuzzy variables. Also, based on the fuzzy max order several models in non-symmetrical L-R fuzzy environment is constructed and the existence condition of Nash equilibrium strategies of the fuzzy bi-matrix games is proposed. At last, based on the Nash equilibrium of crisp parametric bi-matrix games, we obtain the Pareto and weak Pareto Nash equilibrium strategies of the fuzzy bi-matrix games.

Characterization of Nash Equilibrium Strategy for Heptagonal Fuzzy Games

In this paper, the Nash equilibrium strategy of two-person zero-sum games with heptagonal fuzzy payoffs is considered and the existence of Nash equilibrium strategy is studied. Also, based on the fuzzy max order several models in heptagonal fuzzy environment is constructed and the existence of their equilibrium strategies is proposed. In the sequel, the existence of Pareto Nash equilibrium strategies and weak Pareto Nash equilibrium strategies is investigated for fuzzy matrix games. Finally, the relation between Pareto Nash equilibrium strategy and parametric bi-matrix games is established.

Refinement of pure Pareto Nash equilibria in finite multicriteria games using preference relations

In this paper, we consider a noncooperative finite multicriteria two-person game G. We study the problem of refinement of Pareto Nash equilibria of G and we propose an approach based on modeling the preferences of the players by two binary relations. This approach follows three main steps: first, we associate to G another game $$\bar{G}$$ defined by the two sets of strategies and two binary relations over the set of the strategy profiles, where each binary relation expresses the preferences of the corresponding player. Second, we define a Noncooperative Equilibrium $$\textit{NCE}$$ for the game $$\bar{G}$$ and we prove that every $$\textit{NCE}$$ of $$\bar{G}$$ is a Pareto Nash equilibrium of G. Third, we propose a procedure for finding the set of $$\textit{NCE}$$ of $$\bar{G}$$ independently of how the binary relations are constructed. Moreover, we give three ways to model the preferences of the players by using scalarization and the outranking methods ELECTRE I and PROMETHEE II. All the steps of the proposed approach are completely illustrated through an accompanying example.

A Substructural Modal Logic of Utility

Abstract We introduce a substructural modal logic of utility that can be used to reason aboutoptimality with respect to properties of states. Our notion of state is quite general, and is able to represent resource allocation problems in distributed systems. The underlying logic is a variant of the modal logic of bunched implications, and based on resource semantics, which is closely related to concurrent separation logic. We consider a labelled transition semantics and establish conditions under which Hennessy—Milner soundness and completeness hold. By considering notions of cost, strategy and utility, we are able to formulate characterizations of Pareto optimality, best responses, and Nash equilibrium within resource semantics. We also show that our logic is able to serve as a logic for a fully featured process algebra and explain the interaction between utility and the structure of processes.

On the Stability and Levitin–Polyak Well-Posedness of Parametric Multiobjective Generalized Games

We consider parametric multiobjective generalized games. For such a game defined on topological vector spaces, sufficient conditions for the lower semicontinuity of a set of approximate weak Pareto–Nash equilibrium points as well as for the Levitin–Polyak well-posedness are proved under compactness assumptions. For the case where a game is defined on metric spaces, full characterizations of the Levitin–Polyak well-posedness are established in terms of measures of noncompactness.

The Worst-Case Weighted Multi-Objective Game with an Application to Supply Chain Competitions

In this paper, we propose a worst-case weighted approach to the multi-objective n-person non-zero sum game model where each player has more than one competing objective. Our “worst-case weighted multi-objective game” model supposes that each player has a set of weights to its objectives and wishes to minimize its maximum weighted sum objectives where the maximization is with respect to the set of weights. This new model gives rise to a new Pareto Nash equilibrium concept, which we call “robust-weighted Nash equilibrium”. We prove that the robust-weighted Nash equilibria are guaranteed to exist even when the weight sets are unbounded. For the worst-case weighted multi-objective game with the weight sets of players all given as polytope, we show that a robust-weighted Nash equilibrium can be obtained by solving a mathematical program with equilibrium constraints (MPEC). For an application, we illustrate the usefulness of the worst-case weighted multi-objective game to a supply chain risk management problem under demand uncertainty. By the comparison with the existed weighted approach, we show that our method is more robust and can be more efficiently used for the real-world applications.

On Nash Equilibrium Strategy of Two-person Zero-sum Games with Trapezoidal Fuzzy Payoffs

In this paper, we investigate Nash equilibrium strategy of two-person zero-sum games with fuzzy payoffs. Based on fuzzy max order, Maeda and Cunlin constructed several models in symmetric triangular and asymmetric triangular fuzzy environment, respectively. We extended their models in trapezoidal fuzzy environment and proposed the existence of equilibrium strategies for these models. We also established the relation between Pareto Nash equilibrium strategy and parametric bi-matrix game. In addition, numerical examples are presented to find Pareto Nash equilibrium strategy and weak Pareto Nash equilibrium strategy from bi-matrix game.

Existence and stability of weakly Pareto-Nash equilibrium for generalized multiobjective multi-leader–follower games

In this paper, we mainly study a kind of generalized multiobjective multi-leader---follower games in Hausdorff topological spaces. We establish an existence theorem of weakly Pareto-Nash equilibriu...

Model of Intrusion Detection Based on the Game Theory

With the development of network application, network security is facing greater pressure. Based on the characteristics of intrusion detection in the wireless network of the Ad hoc working group, the article introduces the game theory, proposes a game model of network security, concluds the Nash equilibrium in the stage game, repeats game, the pareto Nash equilibrium, more attack both income and payment, so that they get the best choice.

A Primary Market for Optimizing Power Peak-Load Demand Limits

In this paper the authors propose a market-based mechanism for organizational units of Commercial and Industrial power consumers or companies in a consortium to reduce their peak power demand. The market mechanism requires participants’ bids to indicate the value they associate with power needed to run various services, and the power quantity requirement for these services over a time horizon. The market resolution produces peak demand allocation, i.e., determination of the peak demand and the associated cost that the units need to pay. The global peak-demand is then derived by optimizing individual participants’ peak demand. The market mechanism is based on decision optimization, and guarantees the formally defined properties of Pareto optimality, Nash equilibrium and benefit distribution fairness.

THE EXISTENCE OF WEAKLY PARETO-BERGE EQUILIBRIUM POINTS IN MULTIOBJECTIVE LARGE GAMES

This paper considers noncooperative multi-objective games with multi-players (multi-objective large game).According to Berge equilibrium in normal games,we introduce the notion of weakly Pareto-Berge equilibrium in multi-objective large games.By generalizing section theorem,we show the existence of weakly Pareto-Berge equilibrium in multi-objective large games.As a special case,we obtain the existence of weakly Pareto-Nash equilibrium points in multi-objective large games.

Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications

In this paper, maximal element theorem on Hadamard manifolds is established. First, we prove the existence of solutions for maximal element theorem on Hadamard manifolds. Further, we prove that most of problems in maximal element theorem on Hadamard manifolds (in the sense of Baire category) are essential and that, for any problem in maximal element theorem on Hadamard manifolds, there exists at least one essential component of its solution set. As applications, we study existence and stability of solutions for variational relation problems on Hadamard manifolds, and existence and stability of weakly Pareto–Nash equilibrium points for n -person multi-objective games on Hadamard manifolds.