Two-person Zero-sum Game Research Articles

Solving MADM Problems based on Matrix Games with a New Defuzzification Method for Triangular Intuitionistic Fuzzy Numbers

Abstract: The paper discusses about a new defuzzification method called the Comprehensive Li-Samuel ranking for Triangular Intuitionistic Fuzzy Numbers (TrIFNs) which is improved from the Li-Samuel ranking method associated with membership function and Li-Samuel ranking method associated with non-membership function. The Comprehensive Li-Samuel ranking for TrIFNs is utilised to derive the weights for the Multiple Attribute Decision Making (MADM) problems under TrIFN. The MADM problem is solved based on two-person zero-sum game which is converted into a pair of LPP by using Comprehensive Li-Samuel ranking. The process of transforming decision matrix into a matrix game and deriving decision maker’s and attribute weights from the matrix game are discussed. Numerical illustration is given to justify the viability and effectiveness of the proposed method and comparisons are made with existing ranking methods. The comparison study reveals that the proposed method provides the best maximum optimal solution for the decision problem than the existing methods

A Two-Person Zero-Sum Game Approach for a Retirement Decision with Borrowing Constraints

A Two-Person Zero-Sum Game Approach for a Retirement Decision with Borrowing Constraints

Observability inequality of backward stochastic heat equations with Lévy process for measurable sets and its applications

This paper endeavours to directly establish the observability inequality of backward stochastic heat equations involving a Lévy process for measurable sets. As an immediate application, we attain the approximate controllability of forward stochastic heat equations featuring a Lévy process. Subsequently, we embark upon the formulation of a nuanced optimization problem, encompassing both the determination of an optimal actuator location and its associated minimum norm control. This formulation is elegantly cast as a two-person zero-sum game problem. We then proceed to articulate a set of conditions–both sufficient and necessary–required for optimizing the solution through the prism of a Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem.

Two-person zero-sum game with terminal state constraints

In this paper, the discrete-time two-person zero-sum game with terminal state constraints is considered. By applying the variational method, the maximum principle for zero-sum game with terminal constraint is derived along with the forward and backward difference equations (FBDEs). The main contributions of this paper are two-fold. On the one hand, by decoupled solving the FBDEs, the analytical solution to the zero-sum game with terminal state constraint is given. On the other hand, the necessary and sufficient conditions for the reachability of the system are obtained. In this way, the solvability of the zero-sum game with terminal state constraint is re-expressed by the reachability of the system.

Matrix norm methods for zero-sum fuzzy matrix games with payoffs of triangular fuzzy numbers

In this paper, we mainly consider the solution of two-person zero-sum fuzzy matrix games with payoffs of triangular fuzzy numbers. Contrary to the literature, we focus on developing the methods to solve the game directly based on only the norms of the payoff matrix holistically, without solving a linear programming problem or handling sub-games created by taking the components of fuzzy numbers separately. For this purpose, we first present fuzzy versions of 1-norm and ∞-norm with the help of a ranking function and develop the fuzzy matrix norm method to obtain an approximate solution of the zero-sum fuzzy matrix game. In addition to this approach, we provide the fuzzy extended matrix norm method as an enhanced version of the method, which involves the use of newly defined fuzzy matrix norms. Thus, we managed to avoid the complexity of the optimization process of the linear programming problem via the proposed matrix norm-based methods. Finally, we illustrate the implementation of the methods by considering several benchmark examples and a 3 × 3 fuzzy matrix game.

Formulation of Game Model as a Linear Programming Problem using Various Models

Game theory is the examination of strategic interactions between two or more individuals known as players who act based on their individual self-interest within a framework known as the game. Every player possesses a set of possible actions referred to as strategies, from which they make selections. In a two-person zero sum game, each of the two players has at least two strategies. In such a game problem where both players have no inferior strategies, we can determine the optimal mixed strategies of the game problem by converting it to a linear programming problem and solving it using the simplex method or variations of it. In this paper, consideration of some existing models along with our proposed model on the conversion of game problem to LPP was made. We compared the results across the various models considered. The results obtained revealed that our proposed model on the conversion of game problem to LPP produced a higher value of the game compared to the others considered; and thus, produced better performance.

Multi-objective search game: Long-term vs short-term

This study focuses on investigating a multi-objective search allocation game problem, specifically addressing the distribution of a search budget (player 1) across a searched area to detect an evading target (player 2). The game is defined as a two-person zero-sum game, as an optimal search strategy must consider the target’s optimal evasion strategy. Traditionally, the sole objective function in search theory literature has been to maximize the detection probability at the end of the time horizon. Our work extends the problem by proposing a multi-objective formulation that aims to strike a balance between long-term and short-term search objectives. In this context, the goal is to maximize the detection probability of the target at the end of each time period (e.g., hourly) throughout the time horizon. The main challenge in this problem lies in handling a multitude of objective functions, where the number of objective functions is an input data. To the best of our knowledge, no prior research has explored the multi-objective search allocation game. Therefore, we meticulously study the significance of this problem by highlighting the disparities and conflicts among these objectives. Subsequently, we propose a solution approach based on the epsilon-constraint algorithm, as well as a heuristic method that decomposes the problem into smaller multi-objective subproblems. We conduct a series of experiments to evaluate two key aspects of the methods: computation time and quality.

Optimal control and zero-sum game subject to multifactor uncertain random systems with jump

A differential equation incorporating random matrices, multiple Liu processes, and multiple V jump processes is employed to portray a multifactor uncertain random system with jump. The existence and uniqueness theorem for such an equation is proved. Based on the theorem, problems of optimal control and two-person zero-sum game subject to multifactor uncertain random systems with jump are considered. An equation of optimality is provided for solving a problem of optimal control. Equilibrium equations are proposed to identify the saddle-point of a two-person zero-sum game problem. As an extension, we generalize the obtained results to the problems subject to differential equations including random matrices, multiple Liu processes, and multiple V-n jumps processes. Finally, a portfolio selection game problem is analysed utilizing the acquired theoretical results.

Robust location parameter estimation in the presence of adversary

In classical parameter estimation settings, sensor observation models are often assumed to be known. However, when the sensors themselves become unreliable, the traditional observation models may no longer hold. It is then expected that estimation performance would degrade due to the abnormal behavior of sensor observations. We formulate the estimation problem as a two-person zero-sum game and propose a mini-max estimator with the optimization goal to minimize the worst possible estimation error. We show that there exists a saddle-point solution for a single sensor observation. We then apply our result and characterize the estimation performance for networks with multiple sensors.

Strengthening an integrated risk management approach to counter terrorism: Incorporating security, game theory, and economic diversification

This paper looks at the in-depth investigation into improving risk management tactics against terrorism using a multifaceted strategy that incorporates security measures, game theory concepts, and economic diversity. The goal of the study is to provide a comprehensive and strong framework that tackles both immediate security issues and long-term viable counterterrorism strategies in Nigeria. To find vulnerabilities and anticipate future threats, the integration of security measures include improving intelligence collecting, international cooperation, and border restrictions. By incorporating game theory ideas, it is possible to analyse the tactical exchanges between security forces and terrorists, providing insights into how the opposition makes decisions and allowing for the deployment of preventive measures. The use of the two-person zero sum game, the prisoner’s dilemma which depicts the Nash Equilibrium named after the Nobel Laureate economist John Nash were used in explaining how the various terrorists attack can be curbed. Different strategies were also applied so as to enable one understand how the terrorists think, possibly attack their victims and also how their network works. The study discovered that the terrorists attack mainly the communities that produce agricultural products in commercial quantities which affects the agricultural economic diversification. The study also highlights the importance of economic diversification as a key element in mitigating terrorism's underlying causes. By encouraging economic expansion and development in fragile and volatile areas, this study also promotes economic growth, discouraging the extremist ideologies and strengthening local communities in Nigeria as a whole.

Two-person zero-sum risk-sensitive stochastic games with incomplete reward information on one side

This study considers zero-sum risk-sensitive discrete-time stochastic games with incomplete reward information on one side. We show the existence of the value function, derive a new Shapley equation, and prove that the value function solves the Shapley equation, which is used to construct an optimal policy for the informed player. For the uninformed player, we construct an optimal policy by introducing a dual game. Finally, we give an example to illustrate the effects of the reward information and risk-sensitive parameters on the value function and optimal policies.

An Optimization Method for Collaborative Radar Antijamming Based on Multi-Agent Reinforcement Learning

Attacking a naval vessel with multiple missiles is an important way to improve the hit rate of missiles. Missile-borne radars need to complete detection and antijamming tasks to guide missiles, but communication between these radars is often difficult. In this paper, an optimization method based on multi-agent reinforcement learning is proposed for the collaborative detection and antijamming tasks of multiple radars against one naval vessel. We consider the collaborative radars as one player to make their confrontation with the naval vessel a two-person zero-sum game. With temporal constraints of the radar’s and jammer’s recognition and preparation interval, the game focuses on taking a favorable position at the end of the confrontation. It is assumed the total jamming capability of a shipborne jammer is constant and limited, and the shipborne jammer allocates the jamming capability in the radar’s direction according to the radar threat assessment result and its probability of successful detection. The radars work collaboratively through prior centralized training and obtain a good performance by decentralized execution. The proposed method can make radars collaborate to detect the naval vessel, rather than only considering the detection result of each radar itself. Experimental results show that the proposed method in this paper is effective, improving the winning probability to 10% and 25% in the two-radar and four-radar scenarios, respectively.

An Overview of Generative Adversarial Networks

Generative adversarial network (GAN), put forward by two-person zero-sum game theory, is one of the most important research hotspots in the field of artificial intelligence. With a generator network and a discriminator network, GAN is trained by adversarial learning. In this paper, we aim to discusses the development status of GAN. We first introduce the basic idea and training process of GAN in detail, and summarize the structure and structure of GAN derivative models, including conditional GAN, deep convolution DCGAN, WGAN based on Wasserstein distance and WGAN-GP based on gradient strategy. We also introduce the specific applications of GAN in the fields of information security, face recognition, 3D and video technology, and summarize the shortcomings of GAN. Finally, we look forward to the development trend of GAN.

Solving zero-sum two-person game with triangular fuzzy number payoffs using new fully fuzzy linear programming models

Solving zero-sum two-person game with triangular fuzzy number payoffs using new fully fuzzy linear programming models

Emergency response scheme selection with T-spherical hesitant probabilistic fuzzy TODIM-TPZSG approach

This paper aims to investigate an innovative framework to handle emergency response scheme selection (ERSS) issues by integrating TODIM and TPZSG (two-person zero-sum game) methods under novel T-spherical hesitant probabilistic fuzzy set (T-SHPFS) environments. First, T-SHPFS is defined as an extension of the existing tools, which can depict the complex assessment information including several possible values of the various membership functions’ degrees and the associated statistical uncertainty information. Concomitantly, T-SHPFS’s normalization method, comparison laws, operation rules, cross-entropy measure and Hausdorff distance are explored. Then, an objective attribute weight determining model is constructed, considering the credibility of T-SHPF evaluations and the divergence degrees between attribute assessments simultaneously. Next, an integrated TODIM-TPZSG decision-making approach is developed to select the most desirable emergency response scheme. Finally, an illustrative example concerning the selection of the best medical waste disposal method during the COVID-19 epidemic is conducted to verify the effectiveness of the proposed TODIM-TPZSG method. Sensitivity analysis and comparisons between the TODIM-TPZSG and other representative methods are also provided to demonstrate the superiorities of the proposed method. The results reveal that the developed T-SHPFSs give DMs more assessment freedom; the proposed TODIM-TPZSG approach considers the decision makers’ psychological behaviors; the ranking results of the proposed method can reflect the specific divergence degrees among the alternatives; and the needed computation burden and computational complexity are low and less affected by the number of alternatives and criteria than most current ERSS methods.

Secure Downlink Transmission Strategies against Active Eavesdropping in燦OMA Systems: A Zero-Sum Game Approach

Non-orthogonal multiple access technology (NOMA), as a potentially promising technology in the 5G/B5G era, suffers from ubiquitous security threats due to the broadcast nature of the wireless medium. In this paper, we focus on artificial-signal-assisted and relay-assisted secure downlink transmission schemes against external eavesdropping in the context of physical layer security, respectively. To characterize the non-cooperative confrontation around the secrecy rate between the legitimate communication party and the eavesdropper, their interactions are modeled as a two-person zero-sum game. The existence of the Nash equilibrium of the proposed game models is proved, and the pure strategy Nash equilibrium and mixed-strategy Nash equilibrium profiles in the two schemes are solved and analyzed, respectively. The numerical simulations are conducted to validate the analytical results, and show that the two schemes improve the secrecy rate and further enhance the physical layer security performance of NOMA systems.

On max-min solutions of fuzzy games with nonlinear memberships functions

In this paper, we deal with two-person zero-sum games with fuzzy goals. We investigated the cases where the membership functions of the players are nonlinear. We examined how the solutions should be if the membership functions of players were exponential functions. In case players' membership functions are exponential, we developed a new method for the maximin solution according to a degree of attainment of the fuzzy goals. An application was made to show the effectiveness of the method.

Zero-sum semi-Markov games with state-action-dependent discount factors

Semi-Markov model is one of the most general models for stochastic dynamic systems. This paper deals with a two-person zero-sum game for semi-Markov processes. We focus on the expected discounted criterion with state-action-dependent discount factors. The state and action spaces are both Polish spaces, and the reward rate function is ω-bounded. We first construct a fairly general model of semi-Markov games under a given semi-Markov kernel and a pair of strategies. Next, based on the standard regularity condition and the continuity-compactness condition for semi-Markov games, we derive a “drift condition” on the semi-Markov kernel and suppose that the discount factors have a positive lower bound, under which the existence of the value function and an optimal pair of stationary strategies of our semi-Markov game are proved by using a general Shapley equation. Moreover, in the scenario of finite state and action spaces, a value iteration-type algorithm for approximating the value function and an optimal pair of stationary strategies is developed. The convergence and the error bound of the algorithm are also proved. Finally, we conduct numerical examples to demonstrate the main results.

A Note on Linear Complementarity via Two-Person Zero-Sum Games

The matrix [Formula: see text] of a linear complementarity problem can be viewed as a payoff matrix of a two-person zero-sum game. Lemke’s algorithm can be successfully applied to reach a complementary solution or infeasibility when the game satisfies the following conditions: (i) Value of [Formula: see text] is equal to zero. (ii) For all principal minors of [Formula: see text] (transpose of [Formula: see text]) value is non-negative. (iii) For any optimal mixed strategy [Formula: see text] of the maximizer either [Formula: see text] or [Formula: see text] for each coordinate [Formula: see text].

Multi-objective optimal scheduling of microgrid with electric vehicles

With the increasing global attention to environmental protection, microgrids with efficient usage of renewable energy have been widely developed. Currently, the intermittent nature of renewable energy and the uncertainty of its demand affect the stable operation of a microgrid. Additionally, electric vehicles (EVs), as an impact load, could severely affect the safe dispatch of the microgrid. To solve these problems, a multi-objective optimization model was established based on the economy and the environmental protection of a microgrid including EVs. The linear weighting method based on two-person zero-sum game was used to coordinate the full consumption of renewable energy with the full bearing of load, and balance the two objectives better. Moreover, the adaptive simulated annealing particle swarm optimization algorithm (ASAPSO) was used to solve the multi-objective optimization model, and obtain the optimal solution in the unit. The simulation results showed that the multi-objective weight method could diminish the influence of uncertainty factors, promoting the full absorption of renewable energy and full load-bearing. Additionally, the orderly charging and discharging mode of EVs could reduce the operation cost and environmental protection cost of the microgrid. Therefore, the improved optimization algorithm was capable of improving the economy and environmental protection of the microgrid.