Robust Nash Equilibrium Research Articles

Robust asset-liability management games for n players under multivariate stochastic covariance models

This paper investigates a non-zero-sum stochastic differential game among n competitive CARA asset-liability managers, who are concerned about the potential model ambiguity and aim to seek the robust investment strategies. The ambiguity-averse managers are subject to uncontrollable and idiosyncratic random liabilities driven by generalized drifted Brownian motions and have access to an incomplete financial market consisting of a risk-free asset, a market index and a stock under a multivariate stochastic covariance model. The market dynamics permit not only stochastic correlation between the risky assets but also path-dependent and time-varying risk premium and volatility, depending on two affine-diffusion factor processes. The objective of each manager is to maximize the expected exponential utility of his terminal surplus relative to the average among his competitors under the worst-case scenario of the alternative measures. We manage to solve this robust non-Markovian stochastic differential game by using a backward stochastic differential equation approach. Explicit expressions for the robust Nash equilibrium investment policies, the density generator processes under the well-defined worst-case probability measures and the corresponding value functions are derived. Conditions for the admissibility of the robust equilibrium strategies are provided. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium investment strategies and draw some economic interpretations from these results.

A Distributed Robust Two-Time-Scale Switched Algorithm for Constrained Aggregative Games

Robust Nash equilibrium (NE) seeking for a constrained aggregative game is explored. Our setting has the following three features: 1. Local feasibility sets are considered. 2. Each player has first-order dynamics that are influenced by unknown time-varying disturbances and unmodeled terms. 3. The communication topology among players may be influenced by attacks. The existence of unknown disturbances results in the trajectories of the players going out of the feasible sets even though the initial values start from inside. To make the ideas clear, we first use an example that exhibits the three features to introduce the problem formulation to be studied. Subsequently, a two-time-scale distributed, projected algorithm is proposed. The simulation results are also shown. To further illustrate the effectiveness of the proposed algorithm through theoretical analysis, such an algorithm is generalized into a two-time-scale hybrid system by using an average dwell-time automaton and a time-ratio monitor to impose restrictions on the frequency and relative activation time of the attacks. To analyze stability of such a hybrid system, the reduced system and the boundary layer system are presented and corresponding Lyapunov functions are constructed. Correspondingly, uniform (uniform global) asymptotic stability is obtained under some mild assumptions. Also, some results are given for systems without attacks. Finally, two aggregative games are used to show the effectiveness of the proposed algorithm and theoretical analysis.

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.

Optimal Deployment in Moving Target Defense against Coordinated Cyber–Physical Attacks via Game Theory

This work proposes a method for the intelligent deployment of distributed flexible AC transmission system (D-FACTS) devices. In recent years, in the field of moving target defense (MTD) strategies to detect coordinated cyber–physical attacks (CCPAs), establishing how to deploy D-FACTS devices has become an important research point. Although some research results have been proposed, the obtained deployment solutions are unintelligent due to not carefully considering smart attackers’ behaviors. A method for achieving the intelligent deployment of D-FACTS devices is proposed in this paper. First, the basic concept of corrupting CCPAs is summarized; second, based on considering practical constraints and the basic concept, a protected transmission line set is confirmed; and third, a zero-sum game model is formulated, and a robust Nash equilibrium solution is computed. Due to the game’s characteristics, this solution reflects the smart attackers’ sense of action. Relying on the solution, those lines that are most likely to be tripped form a new protected transmission line set. Finally, a comprehensive algorithm using a metric proposed in previous studies is proposed for finding an intelligent solution for the deployment of D-FACTS devices. We validated our results through extensive simulations using IEEE 14-bus, 30-bus, and 118-bus power systems provided by MATPOWER and the real-world load profiles from New York State. Our work, in tracking the targets that attackers are most likely to attack, opens up new ideas for the intelligent deployment of D-FACTS devices.

Robust game-theoretic optimization for energy management in community-based energy system

• A system scheme is proposed to optimize the ESS capacity and energy consumption for multi-community in an individual-oriented way considering the uncertainty of distributed generation. • A robust non-cooperative game approach is formulated for the proposed system scheme, oriented from the pursuit of the economical operation facing with the uncertainty. • To solve the optimization problem, the distributed algorithm is proposed by combining PSO and C&CG algorithm, which can converge to the robust Nash equilibrium. It is significant to schedule energy consumption in community-based energy system consisting of distributed generation, energy storage, multi-community. In this paper, a robust game-theoretic approach is proposed to optimize the storage capacity and energy consumption considering the uncertainty of distributed generation. Energy management with energy storage optimization is studied by considering the cost of distributed generations, cost of energy storage, and bidirectional energy trading. In order to search the robust solution for Nash equilibrium and optimal storage capacity, a distributed algorithm combining column and constraint generation and particle swarm optimization is designed. Simulation results show that the proposed approach has a well performance in reducing operation cost of energy system and tackling the uncertainty of distributed generation.

Distributed multi-agent collision avoidance using robust differential game

This paper proposes a novel robust differential game scheme to solve the collision avoidance problem for networked multi-agent systems (MASs), subject to linear dynamics, external disturbances and limited observation capabilities. Compared with the existing differential game approaches only considering obstacle avoidance objectives, we explicitly incorporate the trajectory optimization target by penalizing the deviation from reference trajectories, based on the artificial potential field (APF) concept. It is proved that the strategies of each agent defined by individual optimization problems will converge to a local robust Nash equilibrium (R-NE), which further, with a fixed strong connection topology, will converge to the global R-NE. Additionally, to cope with the limited observation for MASs, local robust feedback control strategies are constructed based on the best approximate cost function and distributed robust Hamilton–Jacobi–Isaacs (DR-HJI) equations, which does not require global information of agents as in the traditional Riccati equation form. The feedback gains of the control strategies are found via the ant colony optimization (ACO) algorithm with a non-dominant sorting structure with convergence guarantees. Finally, simulation results are provided to verify the efficacy and robustness of the novel scheme. The agents arrived at the targeted position collision-free with a reduced arrival time, and reached the targeted positions under disturbance.

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.

Optimal Bi-Level Bidding and Dispatching Strategy Between Active Distribution Network and Virtual Alliances Using Distributed Robust Multi-Agent Deep Reinforcement Learning

The deregulated active distribution network (ADN) would incorporate numerous autonomous stakeholders, including some emerging distributed virtual alliances (DVAs) like virtual microgrids and virtual power plants. Those DVAs would autonomously participate in energy market trading through bidding among themselves and dispatching conducted by the ADN. In this paper, the optimal bidding and dispatching model for DVAs and ADN in the day-ahead market is first developed as a stochastic dynamic programming model with the risk of misconduct considered, and then re-formulated as a Markov Decision Process to be solved by a new Distributed Robust Multi-Agent Deep Deterministic Policy Gradient algorithm based on the concept of robust Nash equilibrium (RNE). This algorithm is a fully distributed online optimization that would converge to RNE. It is an effective risk-averse method to obtain the optimal bidding strategies of DVAs and the optimal dispatching decisions of distribution system operator (DSO). Its high computational performance is demonstrated in the case studies, and the strategic decisions of DVAs and DSO are thoroughly analyzed.

A robust game-theoretic optimization model for battery energy storage in multi-microgrids by considering of renewable based DGs uncertainty

With the development of electricity market, in the future, various stakeholders such as battery energy storage system (BESS), multi-microgrid (MMG), photovoltaic (PV), and wind turbine (WT) clusters, can trade with the distribution network or each other to meet their power balance needs and to maximizing their profits. In order to predict the charging and discharging power of the battery, a Markov Decision Process (MDP) model of random variables is established. Then, a robust game-theoretic optimization (GO) model is established for MMG's economical operation by considering the renewable based DGs uncertainty. And suggestions for battery operation modes under different compensation coefficients are given. In addition, BESS capacity and demand response (DR) for each microgrid are considered in this proposed model, and an uncertainty adjustment parameter is introduced to adjust the level of conservatism of the robust solution against the modeled uncertainty. In order to search the robust Nash equilibrium solution, this paper proposes a two-level game model based on game theory to study the operation strategy of BESS in the distribution network by considering of renewable based DGs uncertainty. This model is decomposed into a master problem (1st level) and a sub-problem (2nd level), which are solved based on the column and constraint generation algorithm , particle swarm optimization (PSO) and MDP model for random variables of BESS. Simulation result shows that the proposed approach has a well performance in tackling the uncertainty of renewable based distributed energy generation.

Probably Approximately Correct Nash Equilibrium Learning

We consider a multiagent noncooperative game with agents' objective functions being affected by uncertainty. Following a data driven paradigm, we represent uncertainty by means of scenarios and seek a robust Nash equilibrium solution. We treat the Nash equilibrium computation problem within the realm of probably approximately correct learning. Building upon recent developments in scenario-based optimization, we accompany the computed Nash equilibrium with a priori and a posteriori probabilistic robustness certificates, providing confidence that the computed equilibrium remains unaffected (in probabilistic terms) when a new uncertainty realization is encountered. For a wide class of games, we also show that the computation of the so called compression set-which is at the core of scenario-based optimization-can be directly obtained as a byproduct of the proposed methodology. We demonstrate the efficacy of our approach on an electric vehicle charging control problem.

Omni-channel retailing: A data-driven distributionally robust approach for integrated fulfillment services under competition with traditional and online retailers

Omni-channel (OC) retailing strategy provides consumers with seamless shopping experience by integrating different sales channels. In this study, we formulate a price and delivery lead time (DLT) dependent demand function for an omni-channel retailer (OCR) that provides integrated fulfillment services including buy-online-pick-up-in-store (BOPS) and order-in-store-deliver-to-home (OSDH) services. To investigate the extent of OC retailing success and its effects on the pre-existing retailing business models, we examine a dynamic competition between traditional, online, and OC retailing business models through a Nash-Stackelberg game. Our study also addresses the demand uncertainty as a multiplicative variable to the market potential of a product. To hedge against demand uncertainty, due to lack of perfect knowledge about the demand probability distribution, we construct a data-driven ambiguity set for candidate distributions based on the Wasserstein metric, which is utilized to quantify the distances between chosen distributions and the empirical distribution derived from the available data. In this case, part of the problem adopts a distributionally robust Nash equilibrium to find the optimal retailers’ decisions. The obtained results demonstrate the potential OC retailing functionality in terms of demand attraction and profitability. In addition, our results showed that utilizing the distributionally robust approach would lead to shorter DLT and higher sales prices compared to the deterministic demand environment. Interestingly, the pre-existing retailers’ sales prices under demand uncertainty can be higher than the situation under deterministic demand, while they are less than the sales prices before OCR entrance to the market. Therefore, there is no price inflation in the market in the presence of OCR.

Data-driven game-based control of microsatellites for attitude takeover of target spacecraft with disturbance

This paper investigates the problem of using multiple microsatellites to control the attitude of a target spacecraft losing control ability. Considering external disturbance and unknown system dynamics, a data-driven robust control method based on game theory is proposed. Firstly, the attitude takeover control of the target using multiple microsatellites is modeled as a robust differential game among disturbance and multiple microsatellites, in which microsatellites can obtain the worst-case control policies. Subsequently, policy iteration algorithm is put forward to acquire the robust Nash equilibrium control policies of microsatellites with known dynamics, which is a basis of data-driven algorithm. Then, by employing off-policy integral reinforcement learning, a data-driven online controller without information about system dynamics is developed to get the feedback gain matrices of microsatellites by learning robust Nash equilibrium solution from online input-state data. To validate the effectiveness of the proposed control method, numerical simulations are provided.

Robust Secrecy Competition With Aggregate Interference Constraint in Small-Cell Networks

In this article, we address the security issue in a tiered small-cell network aiming at security optimization for small-cell users (SUEs) to defend against eavesdropping. Meanwhile, the transmissions from small-cell base stations (SBSs) are subject to the aggregate interference constraints of macro-cell users (MUEs). In particular, we consider two-fold information uncertainties in small cells, i.e., the uncertainties regarding the eavesdroppers and interference channels to the MUEs. As such, the SBSs compete for robust secrecy rate with robust protection for the MUEs. We adopt the generalized robust Nash equilibrium problem (GRNEP) formulation, for which we confirm the existence of equilibrium and analyze the condition for the uniqueness with variational inequality-assisted analysis. Furthermore, to solve for the equilibrium, we introduce the pricing mechanism and decompose the original GRNEP as a nonlinear complementarity problem with a priced NEP, where the former provides solution of price coefficients and the latter for resource allocation strategies based on given prices. Finally, extensive simulation results are provided to demonstrate the impacts of the interference constraint and uncertainties upon the security performance of an individual SUE and the overall network, which also corroborate the effectiveness of our proposal in security provisioning for the SUEs and interference protection for the MUEs.

Strong robustness to incomplete information and the uniqueness of a correlated equilibrium

We define and characterize the notion of strong robustness to incomplete information, whereby a Nash equilibrium in a game $$\mathbf{u}$$ is strongly robust if, given that each player knows that his payoffs are those in $$\mathbf{u}$$ with high probability, all Bayesian–Nash equilibria in the corresponding incomplete-information game are close—in terms of action distribution—to that equilibrium of $$\mathbf{u}$$ . We prove, under some continuity requirements on payoffs, that a Nash equilibrium is strongly robust if and only if it is the unique correlated equilibrium. We then review and extend the conditions that guarantee the existence of a unique correlated equilibrium in games with a continuum of actions. The existence of a strongly robust Nash equilibrium is thereby established for several domains of games, including those that arise in economic environments as diverse as Tullock contests, all-pay auctions, Cournot and Bertrand competitions, network games, patent races, voting problems and location games.

Distributed robust Nash equilibrium seeking for aggregative games under persistent attacks: A hybrid systems approach

An aggregative game involving double-integrator agents is investigated. In this game, the dynamics of all players are subjected to unknown time-varying disturbances and unmodeled terms, and are also influenced by networked attacks on the communication topology. To seek a Nash equilibrium for such games, a novel distributed robust Nash equilibrium seeking algorithm is proposed and the complete closed-loop system is modeled as a hybrid system using an average dwell-time automaton and a time-ratio monitor to constrain attacks. Then, to analyze stability for the proposed algorithm, a Lyapunov function is constructed and uniform asymptotic stability is obtained under a local Lipschitz assumption on the gradients of the payoff functions. In addition, global uniform asymptotic stability is obtained under a global Lipschitz assumption on the gradients. Finally, an example is used to illustrate the results.

Robust Nash equilibria in vector-valued games with uncertainty

We study a vector-valued game with uncertainty in the pay-off functions. We reduce the notion of Nash equilibrium to a robust set optimization problem and we define accordingly the notions of robust Nash equilibria and weak robust Nash equilibria. Existence results for the latter are proved and a comparison between the former and the analogous notion in Yu and Liu (J Optim Theory Appl 159:272–280, 2013) is shown with an example. The proposed definition of weak robust Nash equilibrium is weaker than that already introduced in Yu and Liu (2013). On the contrary, the robust Nash equilibrium we introduce is not comparable with the notion of robust equilibrium in Yu and Liu (2013), that is defined componentwise. Nevertheless, by means of an example, we show that our notion has some advantages, avoiding some pitfalls that occurs with the other.

Power control scheme for spectral coexisting multistatic radar and massive MIMO communication systems under uncertainties: A robust Stackelberg game model

In this paper, we study the problem of robust Stackelberg game-based power control (RSG-PC) for spectral coexisting multistatic radar and massive multiple-input multiple-output (MIMO) communication systems. Recognizing that the path propagation gains cannot be assumed to be constant, the column-wise model is employed to describe the uncertainty of the parameters. The primary objective of the RSG-PC scheme is to minimize the worst-case radiated power of each radar with uncertain path gains for a specified signal-to-interference-plus-noise ratio (SINR) requirement for target detection, while the massive MIMO communication base station (MIMO-CBS) is protected from the interference of radar transmissions. The hierarchical competition between the massive MIMO-CBS and multiple radars are modeled as a robust single-leader multi-follower Stackelberg game. In the considered game model, the massive MIMO-CBS plays as the leader to decide the prices first through the maximization of its own utility. The radars act as the followers, who compete with each other in a non-cooperative Nash game according to the released interference prices from the massive MIMO-CBS subsequently. Based on the theoretical findings on the existence and uniqueness of the robust Nash equilibrium (RNE) in the formulated game, we present an iterative power control approach that guarantees the convergence to the RNE. Finally, the numerical simulations are provided to verify that our RSG-PC scheme can provide efficient power resource utilization, guarantee a predefined target detection performance, and also protect the quality of service (QoS) of massive MIMO-CBS with uncertainties in path propagation gains.

Robust non-zero-sum investment and reinsurance game with default risk

This paper investigates a non-zero-sum stochastic differential game between two competitive CARA insurers, who are concerned about the potential model ambiguity and aim to seek the robust optimal reinsurance and investment strategies. The ambiguity-averse insurers are allowed to purchase reinsurance treaty to mitigate individual claim risks; and can invest in a financial market consisting of one risk-free asset, one risky asset and one defaultable corporate bond. The objective of each insurer is to maximize the expected exponential utility of his terminal surplus relative to that of his competitor under the worst-case scenario of the alternative measures. Applying the techniques of stochastic dynamic programming, we derive the robust Nash equilibrium reinsurance and investment policies explicitly and present the corresponding verification theorem. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium reinsurance and investment strategies and draw some economic interpretations from these results.

Game theoretical security detection strategy for networked systems

In this paper, a game theoretical analysis method is presented to provide the optimal security detection strategies for heterogeneous networked systems. A two-stage game model is firstly established, in which the attacker and defender are considered as two players. In the first stage, the two players make decisions on whether to execute the attack/monitoring actions or to keep silence for each network unit. In the second stage, two important strategic varibles, i.e. the attack intensity and detection threshold, are cautiously determined. The necessary and sufficient conditions to ensure the existence of the Nash equilibriums for the game with complete information are rigorously analyzed. The results reflect that with limited resources and capacities, the defender (attacker) tends to perform defense (attack) actions and further allocate more defense (less attack) resources to the units with larger assets. Besides, Bayesian and robust Nash equilibrium analysis is provided for the game with incomplete information. Finally, a sampling based Nash equilibrium verification and calculation approach is proposed for the game model with continuous kernels. Thus the convexity restrictions can be relaxed and the computational complexity is effectively reduced, with comparison to the existing recursive calculation methods. Numerical examples are given to validate our theoretical results.

Bayesian and Robust Nash Equilibria in Hydrodominated Systems Under Uncertainty

In this paper, we model strategic interaction of multiple producers in hydrodominated power systems under uncertainty as an equilibrium problem with equilibrium constraints (EPEC), reformulated as a stochastic mixed-integer linear program with disjunctive constraints. We model strategic hydropower producers who can affect the market price by submitting strategic bids in quantity, price, and ramp rate. The bids are submitted to the system operator who minimizes the dispatch cost. We take into account the hydrospecific constraints and uncertainty in the system. Solving the problem results in finding Nash equilibria. We discuss two types of Nash equilibria under uncertainty: Bayesian and robust Nash equilibria. Large EPEC instances can be solved using a decomposition method—Modified Benders Decomposition Approach. This method eliminates the problem of tuning the disjunctive parameter and reduces the memory requirements, resulting in improved computation time.