Fuzzy Matrix Games Research Articles

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.

Pandemic management by using sentiment analysis and trapezoidal type-2 fuzzy linguistic intuitionistic fuzzy matrix games

In order to explore effective management strategies in the context of epidemics, this study introduces a novel concept: Trapezoidal type-2 fuzzy linguistic intuitionistic fuzzy set (TrT2FLIFS) and proposes a trapezoidal type-2 fuzzy linguistic intuitionistic fuzzy matrix game (TrT2FLIFMG). Subsequently, employing sentiment analysis based on the BosonNLP sentiment lexicon, the study extracts comment data from Weibo related to epidemics made by users and calculates their textual scores. These two methods are integrated and applied to policy selection in epidemic management, along with the introduction of a new ranking function to compare the importance of alternative policies. Finally, a comparative analysis with existing methods is conducted to validate the effectiveness of the proposed approach.

Type-2 interval-valued intuitionstic fuzzy matrix game and application to energy vehicle industry development

Game theory is a powerful tool in modeling actual decision-making problems. In many cases, it is difficult for players to give a crisp value of payoff. To conquer this issue, various extensions of fuzzy sets have been introduced to represent the payoffs. Although type-2 intuitionistic fuzzy (T2IF) set (T2IFS) can solve the game problem effectively, sometimes players prefer to give the primary (secondary) membership and non-membership degrees with some approximate ranges rather than crisp values. T2IFS cannot express such uncertain payoffs. Thus, this paper initiates a new concept called type-2 interval-valued intuitionistic fuzzy (T2IVIF) set (T2IVIFS) which has stronger ability than T2IFS and interval-valued intuitionistic fuzzy set (IVIFS) in representing uncertain information. Hamacher operations of T2IVIFSs are given. Hamming and Euclidean distances between T2IVIFSs are defined respectively. Some Hamacher aggregation operators for T2IVIFSs are developed and some desirable properties are analyzed. Afterwards, an effective method is proposed for solving T2IVIF matrix game, which can make game results more accurate and reliable. An example of new energy vehicle industry development is provided to validate the proposed T2IVIF matrix game. Comparative analyses reveal that the matrix game in T2IVIF environment has better flexibility and reliability than that in T2IF and IVIF environments.

On a value of a matrix game with fuzzy sets of player strategies

The present paper investigates a fuzzy matrix game with fuzzy sets of player strategies. We construct a game value with the help of Zadeh's extension principle in combination with an approach to fuzzy matrix games worked out by Liu and Kao. We prove that the FSs of players strategies in a fuzzy matrix game generate a game value in the form of a type-2 fuzzy set (T2FS) on the real line. Furthermore, the corresponding type-2 membership function is given. It is shown that the T2FS of the game value can be decomposed according to the secondary membership grades into a finite collection of fuzzy numbers. Each of them is the value of the corresponding fuzzy matrix game for crisp sets of player strategies. These sets are the corresponding cuts of the original fuzzy sets of player strategies. Illustrative examples are given.

Applications to biogas-plant implementation problem based on type-2 picture fuzzy matrix game under new minkowski type measures

To manage ambiguity and uncertainty information in real life problems, certain intellectuals have initiated numerous modifications of fuzzy sets. But, in certain situations, the decision-makers have been neglected to take a feasible decision. For instance, when an intellectual provides information in the form of a two-dimension or type-1 fuzzy set, then the prevailing theories have been neglected. For this, in this study, we elaborated the new idea of type-2 picture fuzzy sets (T2-PFS) and their algebraic laws have been given. Moreover, by using the T2-PFS, Hamacher aggregation operators are proposed and discussed their important properties. The Minkowski sorts of measures under the T2-PFS are also initiated. In the current study, we aim to utilize the matrix game in a type-2 picture fuzzy environment. By using the proposed operators and measures based on the Hausdorff metric. The elaborated idea has been defined with the biogas-plant implementation dilemma to verify the pertinence and authenticity. Lastly, to discuss the sensitivity analysis and geometrical shown of the initiated works, certain examples are illustrated to determine the supremacy and consistency of the proposed approaches.

Behavioral Strategies between Government and Real Estate Developers on Prefabricated Buildings Based on Triangular Fuzzy Matrix Game

The subsidies provided by the local government to real estate developers are an important and common practice to promote the development of prefabricated buildings worldwide. However, there is a lack of current research on how government subsidies affect the decision-making of real estate developers with respect to adopting prefabricated construction methods. This study developed a fuzzy game model integrating classical game theory with the triangular fuzzy number approach between the local government and a real estate developer, with different behavioral strategies regarding whether or not to develop prefabricated buildings. It analyzed the Nash equilibriums under the circumstances of pure and mixed strategies and probed the influencing factors of the game equilibrium results via numerical simulation. The research conclusions are as follows: (1) the government should encourage real estate developers to actively participate in prefabricated building by using financial subsidies, fund rewards, process supervision and economic penalties comprehensively; (2) the pure strategy (no manufacturing process supervision, active development) between the government and real estate developers is feasible when the benefit of “active development” for real estate developers is larger than that of “passive development”; (3) positive incentives, such as offering financial subsidies, reducing the cost of prefabricated building development and improving the popularity of prefabricated construction in the building market, should be taken to motivate real estate developers; and (4) negative incentives, such as increasing economic punishment and enhancing manufacturing process supervision, can also facilitate real estate developers to actively participate in the development of prefabricated buildings. Incentives are more effective when the economic penalties are greater than the incremental costs of developing prefabricated buildings. Some policy implications are given to guide prefabricated building market development. Finally, the main problems that need further study in the future are highlighted.

Fuzzy matrix game: A fast approach using artificial hybrid neural-net logic-gate switching circuit

Introducing neuro-fuzzy concept in decision-making problems, makes a new way in artificial intelligence and expert systems. Sometimes, neural networks are used to optimize certain performances. In general, knowledge acquisition becomes difficult when problem’s variables, constraints, environment, decision maker’s attitude and complex behavior are encountered with. A sense of fuzziness prevails in these situations, sometimes numerically and sometimes linguistically. Neural networks (or neural nets) help to overcome this problem. Neural networks are explicitly and implicitly hyped to draw out fuzzy rules from numerical information and linguistic information. Logic-gate and switching circuit mobilize the fuzzy data in crisp environment and can be used in artificial neural network, also. Game theory has a tremendous scope in decision making; consequently, decision makers’ hesitant characters play an important role in it. In this paper, a game situation is clarified under artificial neural network through logic-gate switching circuit in hesitant fuzzy environment with a suitable example, and this concept can be applied in future for real-life situations.

The decision making method of financial institutions in industrial cluster upgrading based on interval-valued intuitionistic trapezoidal fuzzy number game matrix

The decision problem of financial institutions supporting the upgrading of industrial clusters is ambiguous, and it is difficult to satisfy the basic feature of uncertainty by the payment function constructed by precise values. To cope with the fuzziness and uncertainty of game matrix, this paper firstly proposes a basic theory of interval-valued intuitionistic trapezoidal fuzzy number (IITFN) based on score function and accuracy function to construct an IITFN fuzzy matrix game (FMG) model. Next, expected utility theory is used to offer the acquisition of mixed strategy optimal solution of IITFN-FMG. Finally, it is applied to analyze the conditions and strategies of financial institutions to support the upgrading of industrial clusters. The main innovations of this paper are: The introduction of interval-valued intuitionistic trapezoidal fuzzy number to characterize the fuzzy features in the decision-making process of financial institutions and cluster enterprises, the proposed construction of a game matrix based on IITFN, and the corresponding solution method to obtain the conditions and strategies for financial institutions to support the upgrading of industrial clusters, and the proposed method and countermeasures to break the “prisoner’s dilemma” in the cooperation between financial institutions and cluster enterprises.

Modified Ranking Function to Compute Fuzzy Matrix Games

Game theory problems (GTP) frequently occur in Economy, Business Studies, Sociology, Political Science, Military Activities, and so on are some of the subjects covered. To tackle the uncertainty in Games, the analysis of games in which the payoffs are represented by fuzzy numbers (FN) will benefit from fuzzy set theory (FST). The purpose of this paper is to develop an efficient technique for solving constraint matrix games (MG) with payoff trapezoidal fuzzy numbers (TFN). The description of the new ranking method is introduced for a constrained matrix with TFN and values. Stock market forecasting has been one of the most important research areas for decades. Stock market values are volatile, non-linear, complicated and chaotic. Based on a ranking function (RF), we used a new algorithm to solve the fuzzy game problem (FGP) employing TFN and also to try to get a desirable gain. Centered on the latest proposed ranking algorithm, the Fuzzy decision method is designed to analyze possible stock opportunists. The paper considers a zero-sum game for two persons in which TFN are fuzzy payoffs. A ranking method (RM) is proposed to convert TFN into crisp numbers (CN) and it is used to solve FGP. The fuzzy game (FG) issue with concept strategies pure minimax maximin is presented. This problem is converted into the crisp problem by a new RF and then solved using the arithmetic (oddment) method. With the help of numerical examples, the suggested technique is explained. This paper finalizes the conclusion and includes an outlook for future study in this direction

Linguistic Pythagorean hesitant fuzzy matrix game and its application in multi-criteria decision making

Linguistic Pythagorean hesitant fuzzy matrix game and its application in multi-criteria decision making

Hexadecagonal Fuzzy Numbers: Novel Ranking and Defuzzification Techniques for Fuzzy Matrix Game Problems

With the help of various different representations of a hexadecagonal fuzzy number, this paper investigates the uncertainty associated with ambiguity and imprecision in the results of 16-component game scenarios. Numerous membership functions for alpha-cuts are established in terms of symmetrical and asymmetrical situations. We look at the centroid methodology, as well as the mean of the alpha-cut technique, the mean of the bounded area removal method and the bounded area included by the fuzzy number. A new centroid-based technique is used to rank two hexadecagonal fuzzy numbers. Defuzzification approaches have also been used to numerical examples based on fuzzy game theory in order to illustrate the effectiveness of the techniques.

Analysis of Probability Distribution due to Aspiration Level in Fuzzy Matrix Game

In this paper we studied zero-sum matrix game. These games have been represented in form of fuzzy linear programming problems which consists the aspiration level and tolerance level. The aspiration levels are determined on the basis of payoff matrix instead of using probability distribution over the strategies. For a particular tolerance level, the aspiration level is studied and the result are illustrated graphically.

A new method to solve intuitionistic fuzzy matrix game based on the evaluation of different experts

In the word of uncertain competitive situations everything is in the state of flux. Under such situations knowing the exact outcomes of mixed strategies adopted by a player is nearly impossible. It is highly rational to assume that no two experts will project the similar fuzzy payoffs for mix of strategies used. Aggregation of expert’s judgement becomes utmost important before solving such competitive situations. Considering this the present paper proposes a method to solve intuitionistic fuzzy game problems by using aggregation operators on payoff judgments of more than one expert. The proposed method significantly adds to the existing literature by overcoming the limitation of Li’s existing method that considers only one expert’s opinion for solving intuitionistic fuzzy game problems. Illustrative example has been given for showing the superiority of the proposed method.

A q-rung orthopair fuzzy non-cooperative game method for competitive strategy group decision-making problems based on a hybrid dynamic experts\u2019 weight determining model

How to select the optimal strategy to compete with rivals is one of the hottest issues in the multi-attribute decision-making (MADM) field. However, most of MADM methods not only neglect the characteristics of competitors’ behaviors but also just obtain a simple strategy ranking result cannot reflect the feasibility of each strategy. To overcome these drawbacks, a two-person non-cooperative matrix game method based on a hybrid dynamic expert weight determination model is proposed for coping with intricate competitive strategy group decision-making problems within q-rung orthopair fuzzy environment. At the beginning, a novel dynamic expert weight calculation model, considering objective individual and subjective evaluation information simultaneously, is devised by integrating the superiorities of a credibility analysis scale and a Hausdorff distance measure for q-rung orthopair fuzzy sets (q-ROFSs). The expert weights obtained by the above model can vary with subjective evaluation information provided by experts, which are closer to the actual practices. Subsequently, a two-person non-cooperative fuzzy matrix game is formulated to determine the optimal mixed strategies for competitors, which can present the specific feasibility and divergence degree of each competitive strategy and be less impacted by the number of strategies. Finally, an illustrative example, several comparative analyses and sensitivity analyses are conducted to validate the reasonability and effectiveness of the proposed approach. The experimental results demonstrate that the proposed approach as a CSGDM method with high efficiency, low computation complexity and little calculation burden.

Strategic Management in Marketing: A Game Theoretic Approach

In marketing, a real-world dilemma emerging between two rivals, McDonald's and Burger King, is investigated. Both firms use three strategies: discounted pricing, status quo, and aggressive commercial. In such cases, ambiguity is a determining factor. To deal with confusion in payoffs, octagonal fuzzy numbers are used. To rank fuzzy numbers, the average of odd positions, average of even positions, and quartile deviations are used. To solve the reduced modelled two competitors zero sum fuzzy matrix games, the proposed ranking methods are used. Finally, the findings are compared to current approaches that are quite similar to the proposed approach.

Type-2 intuitionistic fuzzy matrix games based on a new distance measure: Application to biogas-plant implementation problem

In reality, to deal with uncertain information, researchers have introduced several extensions of fuzzy sets. But, in most cases, two-dimensional fuzzy sets (considering elements with their membership degree) are considered. However, it fails to capture all types of incomplete information. Occasionally, the inadequacy of the data depends upon the large span of source or time or both. In that case, two-dimensional fuzzy sets or type-1 fuzzy sets (T1FSs) are not sufficient to portray the situation. Type-2 fuzzy set (T2FS), a three-dimensional extent of the fuzzy set, appears here to bridge this gap. Type-2 intuitionistic fuzzy set (T2IFS) is a novel extension of T2FS. It can express the fuzzy preference of the decision-makers toward their decisions under different parameters. It also considers both the acceptance and non-acceptance of decision-makers. In the present study, we aim to develop matrix games in T2IF environment. To do this, we first define Hamacher aggregation operators in T2IF environment with their properties. We also propose the Minkowski distance of T2IFSs based on the Hausdorff metric. Then a similarity measure of T2IFS is formed. Next, we solve the matrix games by utilizing the proposed distance measure. The proposed methodology has been delineated with the biogas-plant implementation problem to check the applicability and validity.

Solving matrix games based on Ambika method with hesitant fuzzy information and its application in the counter-terrorism issue

The hesitant fuzzy set has been studied as a powerful tool to describe the decision makers’ judgements under uncertain environment and applied to many domains. For solving the matrix games whose payoffs are expressed by the hesitant fuzzy information, the paper proposes the Ambika method of hesitant fuzzy matrix games (HFMGs). In this paper, firstly, the formal representation of HFMGs is established to meet the conditions of two-person finite zero-sum games. Secondly, after a new method of adding elements to the shorter hesitant fuzzy elements (HFEs), i.e. the hesitant fuzzy elements with possibility, is developed to keep the same length of HFEs, a weighting method based on the position of element in the HFEs is proposed. Then the hesitant fuzzy bi-objective nonlinear programming models for both players are established for HFMGs. Thirdly, according to the proposed value and ambiguity indexes, the Ambika method of HFMGs is developed to find the optimal solutions of mixed strategies by solving the converted linear programming models. Finally, as the illustration of the proposed method, a numerical example about how to choose the optimal solutions for a state security department is given in the counter-terrorism issue.

Matrix Game with Payoffs Represented by Triangular Dual Hesitant Fuzzy Numbers

Matrix Game with Payoffs RepresentedDue to the complexity of information or the inaccuracy of decision-makers’ cognition, it is difficult for experts to quantify the information accurately in the decision-making process. However, the integration of the fuzzy set and game theory provides a way to help decision makers solve the problem. This research aims to develop a methodology for solving matrix game with payoffs represented by triangular dual hesitant fuzzy numbers (TDHFNs). First, the definition of TDHFNs with their cut sets are presented. The inequality relations between two TDHFNs are also introduced. Second, the matrix game with payoffs represented by TDHFNs is investigated. Moreover, two TDHFNs programming models are transformed into two linear programming models to obtain the numerical solution of the proposed fuzzy matrix game. Furthermore, a case study is given to to illustrate the efficiency and applicability of the proposed methodology. Our results also demonstrate the advantage of the proposed concept of TDHFNs.

Fuzzy Multi-objective Matrix Game: An Entropic Uniformity Approach

This paper presents a Multi-Objective Matrix Game model with fuzzy goals, assuming that each player has a fuzzy goal of the payoffs and this model can be used to reduce a generalized regular Max-Min problem. For finding smooth and uniform distribution of the solution to this model we are applying an entropic uniformity approach. Applicability of the entropic uniformity approach can be illustrated with a practical example.

Solving Triangular Intuitionistic Fuzzy Matrix Game by Applying the Accuracy Function Method

In this paper, the matrix game based on triangular intuitionistic fuzzy payoff is put forward. Then, we get a conclusion that the equilibrium solution of this game model is equivalent to the solution of a pair of the primal–dual single objective intuitionistic fuzzy linear optimization problems ( I F L O P 1 ) and ( I F L O D 1 ) . Furthermore, by applying the accuracy function, which is linear, we transform the primal–dual single objective intuitionistic fuzzy linear optimization problems ( I F L O P 1 ) and ( I F L O D 1 ) into the primal–dual discrete linear optimization problems ( G L O P 1 ) and ( G L O D 1 ) . The above primal–dual pair ( G L O P 1 ) – ( G L O D 1 ) is symmetric in the sense the dual of ( G L O D 1 ) is ( G L O P 1 ) . Thus the primal–dual discrete linear optimization problems ( G L O P 1 ) and ( G L O D 1 ) are called the symmetric primal–dual discrete linear optimization problems. Finally, the technique is illustrated by an example.