Saddle-point Equilibrium Research Articles

Carbon emission reduction dynamic decision-making in the electricity supply chain with different carbon emission allowance principles

To address the impacts of typical principles for allocating initial carbon emission allowance, namely, the grandfathering and benchmarking principles on long-term emission reduction strategies, this paper presents four differential game models for an electricity supply chain consisting of a single electricity supplier and a single electricity retailer. The equilibrium solutions for different models are obtained and comparatively analyzed. The results show that the power supplier exhibits greater emission reduction efforts and, therefore, that the retailer has higher profits under the benchmarking principle than under the grandfathering principle. Moreover, under the two principles, the level of the power supplier’s emission reduction efforts is related to the decision-making modes of the electricity supply chain. Under specific conditions, the power supplier’s profit is also higher under the benchmarking principle than under the grandfathering principle. In addition, this study extends the models to investigate the constrained differential game models and obtains the game equilibrium solutions, which can be proved as saddle point equilibria.

Zero-sum games subject to uncertain stochastic noncausal systems considering polynomial control functions

An uncertain stochastic noncausal system (USNS) is an uncertain stochastic singular system that is expected to be regular but not impulse-free. This study investigates zero-sum games (ZSGs) under the constraint of nonlinear uncertain stochastic noncausal systems (USNSs) considering polynomial control functions. Firstly, a method is introduced to convert USNSs into subsystems, including forward uncertain stochastic difference equations as well as backward uncertain stochastic difference equations. Equilibrium equations are then derived to determine the saddle-point equilibrium solution of the zero-sum game (ZSG) for USNSs. Subsequently, a unified framework is developed to find the saddle-point equilibrium and equilibrium value of the ZSG. Additionally, a numerical example and an analysis of the treatment of industrial wastewater are provided to demonstrate the effectiveness of the proposed framework.

Optimistic value models of saddle point equilibrium control problems for uncertain jump systems

The saddle point equilibrium control problem is of great importance in differential game theory. In this paper, we investigated the optimistic value model for the saddle point equilibrium control problem based on multi-dimensional uncertain differential equations with jumps. The equilibrium equation for the proposed model is presented. Then a linear quadratic game optimistic value model and a bang-bang game optimistic value model are discussed, respectively. Finally, some applications are analysed by the results obtained in this paper.

Saddle-Point Equilibrium Strategy for Linear Quadratic Uncertain Stochastic Hybrid Differential Games Based on Subadditive Measures

This paper describes a kind of linear quadratic uncertain stochastic hybrid differential game system grounded in the framework of subadditive measures, in which the system dynamics are described by a hybrid differential equation with Wiener–Liu noise and the performance index function is quadratic. Firstly, we introduce the concept of hybrid differential games and establish the Max–Min Lemma for the two-player zero-sum game scenario. Next, we discuss the analysis of saddle-point equilibrium strategies for linear quadratic hybrid differential games, addressing both finite and infinite time horizons. Through the incorporation of a generalized Riccati differential equation (GRDE) and guided by the principles of the Itô–Liu formula, we prove that that solving the GRDE is crucial and serves as both a sufficient and necessary precondition for identifying equilibrium strategies within a finite horizon. In addition, we also acquire the explicit formulations of equilibrium strategies in closed forms, alongside determining the optimal values of the cost function. Through the adoption of a generalized Riccati equation (GRE) and applying a similar approach to that used for the finite horizon case, we establish that the ability to solve the GRE constitutes a sufficient criterion for the emergence of equilibrium strategies in scenarios extending over an infinite horizon. Moreover, we explore the dynamics of a resource extraction problem within a finite horizon and separately delve into an H∞ control problem applicable to an infinite horizon. Finally, we present the conclusions.

Dynamic Modeling of Environmental Subsidies

In this research, the intertemporal optimal management of subsidies offered by the environmental regulator and the dynamic conflict between two groups of economic agents involved in environmental quality are discussed. First the environmental model is examined in its optimal control management form with two state variables. The analysis of the improved model, inspecting the social planner’s decision (policy) variable, the variable which influences not only environmental quality but the national budget stock, reveals that is dependent on the growth of the national budget stock. A negative growth rate leads to the long run saddle point equilibrium, while an incremental growth rate, but less than the model’s discount rate, leads to an interesting complicated limit cycle equilibrium, for which under certain parameter values the orbit’s phase portrait can be drawn. For the dynamic game model between the social planner and natural resource exploiters, the equilibrium conditions are examined. It is rather a richer than the point equilibrium for which the cyclical strategies have great importance. Therefore, the conditions for that rich equilibrium are found. As a continuation, the paper concludes that the equilibrium condition is that the players’ discount rates are both greater than the national budget interest rate. Finally, the certain values of the equilibrium strategies and national budget stock are provided.

Zero-sum stochastic games in continuous-time with risk-sensitive average cost criterion on a countable state space

We consider zero-sum stochastic games in continuous time with controlled Markov chains and with risk-sensitive average cost criterion. Here the transition and the cost rates may be unbounded. We prove the existence of the value of the game and a saddle-point equilibrium in the class of all stationary strategies under a Lyapunov stability condition. This is accomplished by establishing the existence of a principal eigenpair for the corresponding Hamilton-Jacobi-Isaacs (HJI) equation. This, in turn, is established by using a nonlinear version of Krein-Rutman theorem. We then obtain a characterization of the saddle-point equilibrium in terms of the corresponding HJI equation. Finally, we use a controlled population system to illustrate our results.

Game-based distributed optimal formation tracking control of underactuated AUVs based on reinforcement learning

This paper investigates game-based distributed optimal time-varying formation tracking control problem based on backstepping technique and reinforcement learning (RL) for autonomous underwater vehicles (AUVs) with time-varying disturbances and input saturation. The proposed method enables the multi-AUV to track a desired trajectory while holding the time-varying formation. To accomplish this, we adopt a game between leader and followers to enhance robustness of the system to time-varying disturbances. The approximate optimal solution of the established Hamilton–Jacobi–Isaacs (HJI) equation is obtained by RL which performs online via the single critic Neural Network (SCNN). Hence, the optimal solution corresponds to the saddle point equilibrium of the tracking game. In addition, the command filter is used to construct the unknown hydrodynamic parameter and time-varying disturbances. Furthermore, we introduce a quasi-norm to confront input saturation. The effectiveness of the proposed approach is verified by the simulation results provided.

A new two-scroll 4-D hyperchaotic system with a unique saddle point equilibrium, its bifurcation analysis, circuit design and a control application to complete synchronization

A new two-scroll 4-D hyperchaotic system with a unique saddle point equilibrium, its bifurcation analysis, circuit design and a control application to complete synchronization

Dual reference effect and dynamic control of quality improvement and low-carbon effort under nonlinear demand

Businesses often engage in quality improvement efforts to enhance their competitiveness in the market and increase their product sales. However, in a low-carbon economy, it is crucial for enterprises to incorporate green development into their business strategies. This paper uses optimal control theory to study the dynamic control problem of enterprise quality improvement and low carbon input under consumer reference quality and reference low carbon, analyzes and compares the input decisions under two conditions of profit maximization by the enterprise and social welfare maximization by the government, explores the system steady-state equilibrium and performs simulation analysis. It is shown that (i) there is a unique saddle-point equilibrium of the system in both cases and it depends on the discount rate and memory parameters (ii) the higher the consumer loyalty to product quality or low-carbon attributes, the corresponding instantaneous investment rate increases (iii) there is a cross-effect of reference quality and reference low-carbon on the firm's decision, and the impact of product quality on the firm's profit is higher than that of low-carbon attributes (iv) the social welfare maximization conditions are all higher than the investment in the firm's profit maximization conditions. Based on this, enterprises can make investment decisions more rationally, while also helping the government conduct macroeconomic regulation and control.

Sign‐indefinite linear quadratic differential game for stochastic systems via static output feedback strategy

AbstractSign‐indefinite linear quadratic differential game for stochastic systems by means of static output feedback (SOF) strategy is investigated. First, the saddle point equilibrium condition of external disturbance and control strategy is established based on stochastic cross‐coupled algebraic matrix equations (SCCAMEs). Second, the numerical algorithms for solving the SCCAMEs are discussed. In particular, the difference between the existing algorithm and the proposed one is addressed. In order to show the validity for the several algorithms, a modified practical numerical example is demonstrated. In particular, it is shown that the optimality for the cost functional cannot be attained via the existing algorithm. Furthermore, it is claimed that the new proposed algorithm by combining the numerical algorithm that consist of stochastic Ricatti and Lyapunov equations with the Newton method is useful and reliable because the setting of the initial condition is quite easy task and fast convergence is attained.

Uncertain stochastic hybrid differential game system with V-n jumps: Saddle point equilibrium strategies and application to advertising duopoly game

This paper focuses on the saddle point equilibrium strategies for uncertain stochastic hybrid differential game systems under chance theory. We consider systems with dynamics described by both an uncertain differential equation with V-n jumps (UDEV) and a stochastic differential equation (SDE), with linear and quadratic performance index functions. Based on chance theory and the equation of optimality of uncertain stochastic hybrid optimal control model, the saddle equilibrium equation for general uncertain stochastic hybrid differential game system is established. Furthermore, three classes of differential game models are discussed by using equilibrium equation. Finally, an advertising duopoly game problem as an application is provided, the corresponding optimal profit difference value and advertising strategies are analyzed.

Security defense decision method based on potential differential game for complex networks

Most defense strategies in complex networks are developed from the defense perspective, overlooking the key attack-defense characteristics in cybersecurity. A defense decision algorithm is ineffective when dealing with dynamic attacking behaviors, and when based on attack-defense analysis, stochastic uniform network models are generally used to model the target network, while most networks are large and complex. Thus, the algorithms and their results do not well suit small-world, scale-free, and high-aggregation networks. In this study, considering the structural characteristics of complex networks and the attack-defense characteristics of cybersecurity, potential differential game theory is integrated with complex networks, and a global optimal defense decision algorithm is proposed according to the overall network defense objective. Based on the evolutionary analysis of network security states, a network attack-defense potential differential game model is constructed. Adversarial analysis is carried out on the overall attack-defense strategy, and a defense decision algorithm is designed based on a saddle point equilibrium strategy. Simulation tests are carried out on small-world and scale-free networks to evaluate the effectiveness of the proposed method by comparing its performance with that of random defense strategies and classic decision algorithms.

A differential game approach for real-time security defense decision in scale-free networks

The current network security defense decision method is based on the principle of attack–defense confrontation and generally uses a random network model to project the real network. Most of the existing complex network defense decision methods consider a single defender while ignoring the confrontation and dependence between network attack and defense behaviors. In addition, the pertinence and dynamics of the defense effect are also insufficient. This article comprehensively considers the structure of a scale-free network and characteristics of network attack and defense behavior and combines a complex network with differential game theory to study the real-time defense decision problem. First, a propagation model based on a scale-free network is designed to realize the real-time fluctuation analysis of a network security state. Then, the attack and defense strategy are described, and the calculation method of revenue based on the confrontation analysis of the attack and defense behavior and a change in the security state of the network nodes based on the saddle point equilibrium strategy are introduced. Finally, based on the average connection degree of the real word Internet, a simulated network is constructed to conduct experiments. By comparison with random defense strategies and several typical complex network defense decision methods, the effectiveness and advancement of the proposed method are verified.

Dynamic event-triggered adaptive control for zero-sum games of nonlinear stochastic systems

This paper considers solving the zero-sum games (ZSGs) of nonlinear stochastic systems under the architecture of dynamic event-triggered mechanism (DETM). The adaptive dynamic programming (ADP) approach with great approximate computing power is proposed to seek the saddle-point equilibrium solution of ZSGs. First, the Hamilton–Jacobi–Isaacs equation of stochastic version is constructed which is the key to solving the two-player ZSGs. Then, the ADP method is employed to search the approximated solution under the dynamic triggering mechanism. On the foundation of static event-triggered mechanism (SETM), the DETM is derived through introducing an internal dynamic invariable to further lower the update frequency for the strategies. Additionally, the stability of the closed system is demonstrated in detail via the Lyapunov theory. Finally, two representative examples are provided to present the availability and reliability of this approach from different perspectives.

Zero-sum games involving teams against teams: Existence of equilibria, and comparison and regularity in information

Many emerging problems involve teams of agents taking part in a game. Such problems require a stochastic analysis with regard to the correlation structures among the agents belonging to a given team. In the context of standard Borel state, measurement and action spaces (thus, including real vector spaces and finite spaces), this paper makes the following contributions for two teams of finitely many agents taking part in a zero-sum game: (i) An existence result will be presented for saddle-point equilibria in zero-sum games involving teams against teams when common randomness is assumed to be available in each team with an analysis on conditions for compactness of strategic team measures to be presented. (ii) Blackwell’s ordering of information structures is generalized to n-player teams with standard Borel spaces, where correlated garbling of information structures is introduced as a key attribute; (iii) building on this result Blackwell’s ordering of information structures is established for team-against-team zero-sum game problems. (iv) Finally, continuity of the equilibrium value of team-against-team zero-sum game problems in the space of information structures under total variation is established.

Analysis and Control of Macroeconomic Trends Based on the Leontief Model

In the present study, we consolidate the forty-five industrial sectors delineated in the U.S. input-output tables, as disseminated by the OECD, into three overarching sectors: advanced manufacturing, modern services, and a residual category termed ’others.’ We adopt gross fixed capital formation as a proxy for the proportion of net profit allocated to investment, positing that the investment requisite for augmenting output is commensurate with the requisite capital intensity. This framework enables us to forecast the trajectory of total output and GDP, taking into account the interplay of multiple determinants. In addressing the inherent linear control dynamics of the input-output model, we apply classical control theory to regulate the advanced manufacturing sector. By deriving control equations that accommodate multifactorial influences, we substantiate the efficacy of this control mechanism through rigorous numerical analysis. Moreover, we reconceptualize the dynamic input-output system as a game-theoretic model characterized by a saddle-point equilibrium. By leveraging the saddle-point equilibrium strategy, we pioneer an innovative approach to resolving the complexities of dynamic input-output analysis. This methodological innovation not only enhances the precision of our predictions but also contributes a novel perspective to the literature on economic modeling and control theory.

Continuous-time zero-sum games for Markov decision processes with risk-sensitive finite-horizon cost criterio on a general state space

In this manuscript, we study continuous-time risk-sensitive finite-horizon time-homogeneous zero-sum dynamic games for controlled Markov decision processes (MDP) on a Borel space. Here, the transition and payoff functions are extended real-valued functions. We prove the existence of the game’s value and the uniqueness of the solution of Shapley equation under some reasonable assumptions. Moreover, all possible saddle-point equilibria are completely characterized in the class of all admissible feedback multi-strategies. We also provide an example to support our assumptions.

Discrete-time zero-sum games for Markov chains with risk-sensitive average cost criterion

We study zero-sum stochastic games for controlled discrete time Markov chains with risk-sensitive average cost criterion with countable/compact state space and Borel action spaces. The payoff function is nonnegative and possibly unbounded for countable state space case and for compact state space case it is a real-valued and bounded function. For countable state space case, under a certain Lyapunov type stability assumption on the dynamics we establish the existence of the value and a saddle point equilibrium. For compact state space case we establish these results without any Lyapunov type stability assumptions. Using the stochastic representation of the principal eigenfunction of the associated optimality equation, we completely characterize all possible saddle point strategies in the class of stationary Markov strategies. Also, we present and analyze an illustrative example.

Towards carbon neutrality: A dynamic game approach replacing fossil energy with new marine energy

Based on the dynamic game perspective, this paper establishes a duopoly energy manufacturer substitution game model to explore whether the replacement of traditional fossil energy by marine new energy can reduce carbon emissions and effectively promote carbon neutrality. The results of the study found that the energy output growth rate of new marine energy will increase, and that of traditional fossil energy manufacturers decreases gradually in the game of new marine energy to conventional fossil energy. Research shows that the new marine energy is re-placing the traditional fossil energy, and the total output growth rate of the energy market depends on the technology accumulation and market game effect in this alternative process. Also, it is positive and keeps increasing in the alternative rate of new marine energy to traditional fossil energy. Furthermore, the substitution factor and discount factor positively affect the energy substitution rate. However, the depreciation rate has the opposite effect, and the market scale has no effect on the rate of new marine energy. In the process of marine new energy substitution, there is always a steady-state path that tends to the saddle point equilibrium. The improvement of technology R&D input efficiency is conducive to new energy substitution, reducing the use of fossil energy, and contributing to the realization of carbon neutrality.

Abatement technology innovation, worker productivity and firm profitability: A dynamic analysis

Based on the empirical evidence that improving environmental quality can raise worker productivity, this paper develops a dynamic analysis model to investigate the relationship between abatement technology innovation, worker productivity, and firm profitability in a Cournot competitive market. The main characteristic of this study is that we assume that (1) abatement technology innovation can boost firm profitability by increasing worker productivity; (2) each firm's abatement technology is an endogenous variable, whose state dynamic equation is linear and has an exogenous decay rate of abatement technology; and (3) emissions taxes and the wage rates are exogenous parameters. Our main results show that, first, the system admits a unique saddle-point equilibrium under firm decisions and government regulations, respectively. Second, under firm decisions, the output level (price) monotonically decreases (increases) in the abatement technology, whereas an increase in the rate of emissions taxes leads to an increase (decrease) in firm profitability (worker productivity). Furthermore, under government regulations, a negative relationship exists between output (price) and abatement technology; worker productivity is positively related to improvement in abatement technology (product market competition). Finally, government regulations are better than firm decisions for promoting abatement technology, worker productivity, output levels, and firm profitability.