Fuzzy Matrix Games Research Articles

Solving I-fuzzy two person zero-sum matrix games: Tanaka and Asai approach

This paper aims to design a method for solving a two person zero-sum matrix game with I-fuzzy goals and I-fuzzy pay-offs, where the symmetric triangular I-fuzzy numbers prescribe the entries of the pay-offs matrix. The most common approaches in the literature to solve matrix games with fuzzy goals and fuzzy pay-offs employ ranking function or defuzzification technique, like in Bector and Chandra (Fuzzy mathematical programming and fuzzy matrix games, vol 169. Springer, Berlin, 2005) and Vijay et al. (Fuzzy Optim Decis Making 6:299–314, 2007). Our proposed approach in this work differs from the existing approaches in the sense that it is devoid of ranking or defuzzification function. It also provides precise degrees of belief and disbelief in achieving the goals set by each player. An essential concept of ‘almost positive I-fuzzy number’ introduced by Aggarwal et al. (Notes Intuit Fuzzy Sets 23:85–101, 2017) is employed to study matrix games in I-fuzzy setting. Solving such a game is shown to be equivalent to solving a pair of crisp non-linear programming problem. In this way, our approach is unique for solving such a game. Some numerical examples are included to illustrate the proposed approach.

Intuitionistic interval-valued hesitant fuzzy matrix games with a new aggregation operator for solving management problem

In our daily life, we encounter many problems with uncertainty and vagueness in nature. Mathematical formulations and solutions of these problems are not easy and appear to be a challenging task to the researchers. Crisp sets and fuzzy sets suffer to deal with these. Hesitant fuzzy set—a protracted version of fuzzy set—comes into the fore to bridge over the gap. The set of all possible values of membership of hesitant fuzzy set might be considered as a set of possible intervals. Non-membership functions are also added therein to get intuitionistic interval-valued hesitant fuzzy numbered sets. In the literature, several aggregation operators exist, and here we consider a new one which is easy to apply in our formulated problems. Here, a matrix game whose payoffs are intuitionistic interval-valued hesitant fuzzy numbers is solved using our proposed aggregation operator. A tangible management problem with numerical values is demonstrated here to verify the applicability of the new aggregation operator over the matrix game.

Multiple attribute decision making with triangular intuitionistic fuzzy numbers based on zero-sum game approach

For many decision problems with uncertainty, triangular intuitionistic fuzzy number is a useful tool in expressing ill-known quantities. This paper develops a novel decision method based on zero-sum game for multiple attribute decision making problems where the attribute values take the form of triangular intuitionistic fuzzy numbers and the attribute weights are unknown. First, a new value index is defined for triangular intuitionistic fuzzy numbers on the basis of the centroid. Thereby, a new ranking approach is presented for comparing triangular intuitionistic fuzzy numbers. We formulate a multiple attribute decision making problem as a two-person zero-sum game with payoffs of triangular intuitionistic fuzzy numbers. Then, following the new ranking approach, the fuzzy matrix game is converted as a pair of crisp linear programming models, and the optimal strategies are objectively derived by solving such models. Therefore, the ranking order of alternatives is determined by the expected scores of alternatives. An example of video monitoring system selection is demonstrated to illustrate the effectiveness of the proposed methodology.

On solving two-person zero-sum fuzzy matrix games via linear programming approach

In this paper, a two-person zero-sum matrix game with fuzzy numbers payoff is introduced. Using the fuzzy number comparison introduced by Rouben's method (1991), the fuzzy payoff is converted into the corresponding deterministic payoff. Then, for each player, a linear programming problem is formulated. Also, a solution procedure for solving each problem is proposed. Finally, a numerical example is given for illustration.

Recurrent Neural Network Model: A New Strategy to Solve Fuzzy Matrix Games.

This paper aims to investigate the fuzzy constrained matrix game (MG) problems using the concepts of recurrent neural networks (RNNs). To the best of our knowledge, this paper is the first in attempting to find a solution for fuzzy game problems using RNN models. For this purpose, a fuzzy game problem is reformulated into a weighting problem. Then, the Karush-Kuhn-Tucker (KKT) optimality conditions are provided for the weighting problem. The KKT conditions are used to propose the RNN model. Moreover, the Lyapunov stability and the global convergence of the RNN model are also confirmed. Finally, three illustrative examples are presented to demonstrate the effectiveness of this approach. The obtained results are compared with the results obtained by the previous approaches for solving fuzzy constrained MG.

Methods for solving matrix games with cross-evaluated payoffs

In the traditional fuzzy matrix game, given a pair of strategies, the payoffs of one player are usually associated with themselves, but not linked to the payoffs of the other player. Such payoffs can be called self-evaluated payoffs. However, according to the regret theory, the decision makers may care more about what they might get than what they get. Therefore, one player in a matrix game may pay more attention to the payoffs of the other player than his/her payoffs. In this paper, motivated by the pairwise comparison matrix, we allow the players to compare their payoffs and the other ones to provide their relative payoffs, which can be called the cross-evaluated payoffs. Moreover, the players’ preference about the cross-evaluated payoffs is usually distributed asymmetrically according to the law of diminishing utility. Then, the cross-evaluated payoffs of players can be expressed by using the asymmetrically distributed information, i.e., the interval-valued intuitionistic multiplicative number. Comparison laws are developed to compare the cross-evaluated payoffs of different players, and aggregation operators are introduced to obtain the expected cross-evaluated payoffs of players. Based on minimax and maximin principles, several mathematical programming models are established to obtain the solution of a matrix game with cross-evaluated payoffs. It is proved that the solution of a matrix game with cross-evaluated payoffs can be obtained by solving a pair of primal–dual linear-programming models and can avoid some unreasonable results. Two examples are finally given to illustrate that the proposed method is based on the cross-evaluated payoffs of players, and can directly provide the priority degree that one player is preferred to the other player in winning the game.

Dual hesitant fuzzy matrix games: based on new similarity measure

The dual hesitant fuzzy set is an effective mathematical approach to deal with the data which are imprecise, uncertain or incomplete information. Dual hesitant fuzzy set is an extension of hesitant fuzzy set which encloses fuzzy set, intuitionistic fuzzy set and hesitant fuzzy set as a special one. In this paper, the axiomatic definition of similarity measure between the dual hesitant fuzzy set is presented. A new similarity measure by considering membership and non-membership functions of dual hesitant fuzzy set is introduced. It is shown that the corresponding distance measure can be obtained from the proposed similarity measure. To check the utility, the proposed similarity measure is applied in a bidirectional approximate reasoning system into matrix game. Matrix game with precise data is hardly applicable in real-life decision-making problem. In view of more realistic sense, we choose the elements as dual hesitant fuzzy into the payoff of the matrix game, which is treated as dual hesitant fuzzy matrix game. Mathematical formulation of dual hesitant fuzzy matrix game with on restriction (DHFMGR) is described. Four algorithms are emerged on the proposed similarity measure, which are provoked to find the optimal value of the DHFMGR. A numerical example is incorporated to illustrate the applicability and feasibility of the proposed measure in dual hesitant fuzzy matrix game. The paper ends with the conclusions including an outlook for future study in this direction.

Solution of Matrix Games with Generalised Trapezoidal Fuzzy Payoffs

ABSTRACTIn this paper, it is shown that once linear ranking function is chosen, then the fuzzy matrix game (FMG) is converted into crisp one and then it is solved easily by the proposed method. The main aim of this paper is to reduce the computational complexity of the FMG using the ranking function. The applicability and feasibility of the FMG is illustrated with a numerical example. This paper finalises with the conclusion and to include an outlook for future study in this direction.

Characterization of Nash Equilibrium Strategy for Heptagonal Fuzzy Games

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

Interval-valued intuitionistic fuzzy matrix games based on Archimedean t-conorm and t-norm

Fuzzy game theory has been applied in many decision-making problems. The matrix game with interval-valued intuitionistic fuzzy numbers (IVIFNs) is investigated based on Archimedean t-conorm and t-norm. The existing matrix games with IVIFNs are all based on Algebraic t-conorm and t-norm, which are special cases of Archimedean t-conorm and t-norm. In this paper, the intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm are employed to aggregate the payoffs of players. To derive the solution of the matrix game with IVIFNs, several mathematical programming models are developed based on Archimedean t-conorm and t-norm. The proposed models can be transformed into a pair of primal–dual linear programming models, based on which, the solution of the matrix game with IVIFNs is obtained. It is proved that the theorems being valid in the exiting matrix game with IVIFNs are still true when the general aggregation operator is used in the proposed matrix game with IVIFNs. The proposed method is an extension of the existing ones and can provide more choices for players. An example is given to illustrate the validity and the applicability of the proposed method.

Intelligent Water Management: a Triangular Type-2 Intuitionistic Fuzzy Matrix Games Approach

Matrix games with fuzzy payoffs have spread itself nowadays in diverse fields. Fuzzy game theory with triangular type-1 fuzzy numbers are visited more by researchers. In this paper, we consider matrix games with payoffs as triangular type-2 intuitionistic fuzzy numbers, i.e., Triangular Type-2 Intuitionistic Fuzzy Matrix Game (TT2IFMG) as a new and rare concept. A new ranking function is used to get relevant solutions of TT2IFMG. We are living in times of unprecedented scientific-technical advancement, yet facing several critical global problems that threaten human welfare and our ecosystem. Water management, a burning problem of the Earth now-a-days, is treated here under the scanner of TT2IFMG environment where we discuss some policy-management toward the free and fair accession of water against its limited resources.

Research on brand valuation models based on triangular fuzzy matrix game

Market competition has entered into the era of brand competition with economic growth. Brand evaluation represents an important approach for brand management and value enhancement. In this paper, an analysis was made, based on the triangular fuzzy matrix game theory, on the profits and losses incurred by participating enterprises and evaluating institutions due to their adoption of different game strategies. The applications of triangular fuzzy matrix game theory were expanded to the brand valuation field, and a theoretical method for evaluating the current evaluation models was proposed.

Solving multi-objective fuzzy matrix games via multi-objective linear programming approach

A class of multi-objective fuzzy matrix games is studied and it is shown that solving such a game is equivalent to solving a pair of multi-objective linear programming problems. This work generalizes an earlier study of Fernandez et al. [7] from crisp scenario to fuzzy scenario on the lines of Bector et al. [4]. Further certain difficulties with similar studies reported in the literature are also discussed.

Two Person Zero-Sum Fuzzy Matrix Games and Extensions

In this work, fuzzy Linear programming problems (FLPP) and their implementations is being done in two person zero-sum fuzzy matrix game theory, based on Bector, Chandra and Vijay [2] model. Verification of solution of all possible type of two person zero-sum game is done by solving examples by numerical method. Generalized fuzzy set first introduced by Atanassov (1986) which is called I-fuzzy set has been studied and using this knowledge, application in (FLPP) and two person zero-sum matrix game with I-fuzzy goals is given. At the end I gave useful MATLAB codes for solving all three type of two person zero-sum fuzzy matrix games, with fuzzy goals, with fuzzy pay-offs and with both fuzzy goals and fuzzy pay-offs.

Matrix games with payoffs of belief structures

Imprecise matrix games, such as interval-valued matrix games and fuzzy matrix games, have attracted much interest for a long time. Most of the previous studies on imprecise matrix games mainly focus on the fuzzy uncertainty of payoffs. However, the uncertainties of nonspecificity and discord involved in payoffs are not well addressed so far. The purpose of this paper is to study the matrix game with such types of uncertainties. In order to achieve that purpose, we present a matrix game model with payoffs of belief structures so as to integrate discord and nonspecificity. The proposed model can be used to express more imprecise interactions between players in the reality. Besides, as another main contribution of the study, an effective method for solving matrix games with belief structures payoffs is developed to help us find the equilibrium points of the kind of games. At last, an example is given to illustrate the proposed model and method.

An Alternative Approach for Solving Fuzzy Matrix Games

In this paper, two-person matrix games is considered whose elements of pay-off matrix are triangular fuzzy numbers (TFNs). To solve such game a new methodology based on -cut of TFN is developed for each of the players. In this methodology, two auxiliary bi-objective linear programming (BOLP) models are derived. Then using average weighted approach these two BOLP models are decomposed into two auxiliary crisp linear programming (LP) problems. Finally, the value of the matrix game for each player is obtained by solving two corresponding auxiliary LP problems using the existing simplex method. Validity and applicability of this methodology are illustrated with practical example compared to existing methods.

Vague linguistic matrix game approach for multi-criteria decision making with uncertain weights

Vague linguistic numbers are especially adaptive to describe the evaluations of alternatives in multi-criteria decision making. Such description reveals the DM's preference and hesitant degree simultaneously. For handling the uncertainty in setting the weights of criteria, a fuzzy matrix game related with the initial decision matrix is established. This fuzzy game model is then transformed to a linear programming. By solving this programming, the weights of criteria are obtained. The weighted arithmetic average operator of vague linguistic numbers yields the aggregated ranking values of alternatives. An example illustrates the procedure of the proposed method.

A Fictitious Play Algorithm for Matrix Games with Fuzzy Payoffs

Fuzzy matrix games, specifically two‐person zero‐sum games with fuzzy payoffs, are considered. In view of the parametric fuzzy max order relation, a fictitious play algorithm for finding the value of the game is presented. A numerical example to demonstrate the presented algorithm is also given.

Solving constrained matrix games with payoffs of triangular fuzzy numbers

Constrained matrix games with payoffs of triangular fuzzy numbers (TFNs) are a type of matrix games with payoffs expressed by TFNs and sets of players’ strategies which are constrained. So far as we know, no study have yet been attempted for constrained matrix games with payoffs of TFNs since there is no effective way to simultaneously incorporate the payoffs’ fuzziness and strategies’ constraints into classical and/or fuzzy matrix game methods. The aim of this paper is to develop an effective methodology for solving constrained matrix games with payoffs of TFNs. In this methodology, we introduce the concepts of Alpha-constrained matrix games for constrained matrix games with payoffs of TFNs and the values. By the duality theorem of linear programming, it is proven that players’ gain-floor and loss-ceiling always have a common interval-type value and hereby any Alpha-constrained matrix game has an interval-type value. Moreover, using the representation theorem for the fuzzy set, it is proven that any constrained matrix game with payoffs of TFNs always has a TFN-type fuzzy value. The auxiliary linear programming models are derived to compute the lower and upper bounds of the interval-type value and optimal strategies of players for any Alpha-constrained matrix game. In particular, the mean and the lower and upper limits of the TFN-type fuzzy value of any constrained matrix game with payoffs of TFNs can be directly obtained through solving the derived three linear programming models with data taken from only 1-cut and 0-cut of payoffs. Hereby the TFN-type fuzzy value of any constrained matrix game with payoffs of TFNs are easily computed. The proposed method in this paper is compared with other methods and its validity and applicability are illustrated with a numerical example.

Nash equilibrium strategy for fuzzy non-cooperative games

In this article, two-person zero-sum games are investigated in the fuzzy environment. Several models constructed by Maeda in the symmetrical fuzzy environment are extended to the models in the asymmetric fuzzy environment. The existence of equilibrium strategies for these extended models is proposed in the asymmetric fuzzy environment. However, in some cases, Nash equilibrium strategies may not exist. Therefore, two special cases are presented for which Nash equilibrium strategies do exist. In order to investigate the existence of (weak) Pareto Nash equilibrium strategies for fuzzy matrix games, we introduce the concept of crisp bi-matrix games with parameters. By solving the parametric bi-matrix games, we obtain the (weak) Pareto Nash equilibrium strategies for the fuzzy matrix games.