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.


Introduction
Basic decision-making issues related to game hypotheses have large-scale implementations in science, engineering, management science and sociology. Fuzzy set (FS) (Zadeh 1965) gives assistance to a brilliant and wonderful role B Sankar Kumar Roy sankroy2006@gmail.com Ankan Bhaumik la.ankan@gmail.com 1 Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, West Bengal 721102, India to investigate the inner situations of issues identified with regular day-to-day existence's. Although it has a few constraints to deal with imprecise information, hazy and murky data when various kinds of fuzziness, murkiness and unpredictability yield up at the same time. Analysts have nurtured the fuzzy sets and stretched out the FS to intuitionistic fuzzy set (IFS) (Atanassov 1986), hesitant fuzzy set (HFS) (Torra 2010), etc. HFS effectively actualized the problems' environments where FS, IFS flunk to depict issues about fuzziness, uncertainty. HFS portrays a group of values from [0, 1] instead of single one to every member of corresponding set.
Artificial neural frameworks or artificial neural nets are physical cell frameworks generally able to acquire, store and use exploratory information toward knowledge. The learning is acquired in stable mappings inserted in network framework. Neurons or nodes are the basic unit or element of net. In brain-neuron system, i.e., in neural net systems, activity starts at networks' polarization, then the firing rate of neuron is considered through a set of input connections using synapses on cells and the corresponding dendrite; then neurons are given internal resting space and consequently neuron's axonal projections are done. In artificial neural nets, every processing element is marked by an activity level, an output cost or value, a group of input links, a bias cost, i.e., an artificial resting stage of corresponding neuron, and a bunch of output links. Each of these characteristics of the unit is expressed mathematically by means of real numbers. Thus, every connection possesses weight, may be positive or negative, i.e., synaptic influence which decides the impact of the approaching contribution on the enactment level of the unit. Weights determine excitatory or inhibitory initiation. Artificial neural network may be classified as the generalization of brain-style computational methods in mathematical sciences, mainly in applied sciences. McCulloch and Pitts (1943) and Hebb (1949) originated the idea of brain-style computation. Minsky and Papert (1969) proposed artificial intelligence as symbol processing and it became a dominant theme in artificial intelligence in 1970s.
Contemporarily, von Neumann and Morgenstern (1944) pioneered Game Theory. A game constitutes with players, their turns and returns. These turns are called strategies and returns and quantitatively are termed as payoffs. Players assume pure strategies randomly, and when with some probabilities, these strategies are defined to be mixed one. Matrix game and related duality hypothesis in linear programming problem (LPP) have a linkage in crisp form of intricacies of issues, yet reality wants various dubious natures. Because of the vulnerability, imprecision attributes of frameworks in question and the vagueness, ambiguity of adjudications of decision players, we understand inclusion of aversion, hesitance environments in game problems. Campos (1989) first illuminated fuzzy matrix game. Li (2013Li ( , 2014 solved matrix games with fuzzy payoffs. Bhaumik et al. (2017), Bhaumik and Roy (2021), Bhaumik et al. (2021), Jana and Roy (2019), Roy and Mula (2016) and Roy and Jana (2021) understood matrix games and effectively applied them in real-life circumstances. Several articles (cf. Azam and Yao 2015;Collins and Hu 2008;Tang and Li 2020;Xia 2019) have been published on fuzzy game theory.
Hirota and Pedrycz (1993) discussed on logic-based neural networks. Neural nets have been applied in fuzzy logicsystem, soft-computing (cf. Hayashi 1992, 1995;Jang 1993;Lin and Lee 1991), function approximation (Wu and Er 2000), fuzzy modeling (cf. Sugeno and Yasukawa 1993;Takagi and Sugeno 1985), etc., but hybrid-neural net has not been applied in matrix game using logic-gate switching circuits. Also, matrix game under hesitant triangular intuitionistic fuzzy environment has not been discussed using neural net in literature. These can be considered as the research gaps from the others in literature. The main aims of this study are as follows: (i) To construct a game model using artificial neural nets. (ii) To apply switching circuit gates in neural nets. (iii) To compute a quick geometric way for defuzzification of a set of hesitant fuzzy elements.
The rest of the paper is set as: Preliminaries related to triangular intuitionistic fuzzy set, triangular norm and conorm, hesitant fuzzy set are talked about in short in Sect. 2. Section 3 describes classical matrix game shortly. In Sect. 4, we discuss on neural network model with two subsections: one for biological neural network and another for artificial neural network, briefly. In Sect. 5, logic-gate switching circuits are defined with the elementary gate operators, and in Sect. 6, we develop fuzzy logic-gate switching circuit-oriented artificial neural network model. Matrix games with artificial neural network and logic-gate switch are described in Sect. 7. A numerical problem is simulated and exercised in Sect. 8. Section 9 is confined under the consideration of results and the corresponding discussion to the problem having some comparative analyses with others, and Sect. 10 finishes up this work with extent of future research scopes.

Preliminaries
We introduce here the basic definitions and some properties on triangular intuitionistic fuzzy set, triangular norm and conorm, and hesitant fuzzy set. Let X denotes a universe of discourse. A fuzzy setȂ in X is distinguished by a membership function μȂ : X → [0, 1]. A fuzzy setȂ in X can be demonstrated as follows: Membership degrees μȂ(x) ofȂ are crisp numbers.
Arithmetic Operations on Triangular IFNs: Letφ = (ϕ, ϕ, ϕ); φ , ρφ andθ = (ϑ, ϑ, ϑ); θ , ρθ appear for two triangular IFNs, then the addition, subtraction, multiplication, division and scalar multiplication of the numbers are conveyed as below: Addition: where "∧ and "∨ individually denote min and max operators in fuzzy sense. Scalar Multiplication: For any real number k, where the assumption ofφ > 0 or < 0 is decided by the extension principle of fuzzy set (Li 2013 Definition 2.4 Complement of fuzzy set: Considering a fuzzy sentence p, we describe its complement as some sentence fulfilling the uniformity: is named as hesitant fuzzy element (HFE), an essential unit of HFS is a set fitted with various merits in [0, 1] representing the conceivable membership degrees to component x ∈ X .

Example 2.1 A H F
Furthermore, in order to aggregate hesitant fuzzy information, Xu and Xia (2011) defined some new operations on h, h 1 and h 2 with λ > 0 as below:

Classical matrix game
In this part, we describe some basics on classical game theory. A matrix game is communicated as A = (a i j ) where i = 1, 2, . . . , p, j = 1, 2, . . . , q with components as real numbers and the corresponding matrix is termed as payoff matrix. We think about two players. Players I and II play row i and column j, individually, and the results to players I and II are a i j and −a i j , respectively, in case of zerosum concept. Strategies that advantage player's individual adjustments are picked by players. Expecting the game with arrangement of unadulterated techniques S 1 and S 2 and that of blended or mixed strategies X and Y for players I and II individually, we characterize, Here X i (i = 1, 2, . . . , p) and Y j ( j = 1, 2, . . . , q) are probabilities in which the players I and II sort-out their pure strategies α i ∈ S 1 (i = 1, 2, . . . , p) and β j ∈ S 2 ( j = 1, 2, . . . , q) individually and game is enunciated as Basically, we wish to get the most favorable strategy(ies) for players' and the value of considered game. The estimation of the game is characterized to be the maximum ensured gain to maximizing player I or the minimum conceivable loss to minimizing player II, and here the best strategic procedures are utilized by both players. If a player records the most exceedingly awful potential results of all things considering his or her prospective strategies, the individual in question will pick that strategy, the most reasonable for the person in question. This idea is observed as maximin and minimax principle. When maximin for player I and minimax for player II be equal, then the existence of a saddle point in corresponding game is certified (von Neumann and Morgenstern 1944 noted the term saddle point).
Expect that player I uses any mixed strategy from X . Clearly, player I's normal increase floor is min(X t AY ) and if shortly be denoted by v, we need to maximize v, . Such X * and v * , respectively, called player I's maximin strategy and game value, are obtained from the accompanying LPP in Model 1 as: What's more, with same contention, player II's optimal or minimax strategy, say Y * ∈ Y , and game value, state w * are depicted from Model 2 as: 4 Neural network model

Biological network system
A typical neuron or nerve cell belongs to the vertebrate nervous system which contains the nucleus (genetic informer) and offers to two sorts of cell processes, axon and dendron. Axon acts as transmitting element or output element, whereas dendron as input element. Branches of the axon of one neuron communicate with signals to other neuron at a site called the synapse. Synapses are the elementary signal processing devices.
Though the brain with its nervous system makes up for the slow rate of operation with a few factors, the brain is an exceptionally perplexing, nonlinear, parallel data handling framework. From early stage of childhood to adult stage, synapses are modified gradually through the learning process. And these motivate the scientists to use neural networks and the related sciences in artificial intelligence like pattern recognition, perception and motor controlling in fuzzy sets and systems. Thus neural networks motivate to generate fuzzy rules from examples (cf. Sugeno and Yasukawa 1993;Wang and Mendel 1992).

Artificial neural network
In mathematics, biological structures of neural systems influence mathematical modeling to construct network functions as forward and backward calculations. And this leads to artificial neural network (ANN). ANN was found its roots almost 75 years ago in the works of McCulloch and Pitts (1943) and later by others (cf. Hayashi 1992, 1995;Kwan and Cai 1994). Definition 4.2 Hybrid neural net (Fuller 2000): In a simple net, as picturesquely in Fig. 1, all input criteria like signals as well as weights are reals. Signals interact with weights and pass through one to another layer using sigmoidal function. This straightforward neural net with increase, expansion and sigmoidal function is called regular (or standard) neural net. If triangular-norm, triangular-conorm or their combination is employed and used in next layer we call it hybrid neural net. Definition 4.3 Fuzzy hybrid neural net: When weights are crisp and signals are fuzzy, then hybrid neural net is termed as fuzzy hybrid neural net. A fuzzy hybrid neural net may not use multiplication, addition and sigmoidal function.
In ANN, signal flows or transfers on the basis of the net's activity, sometimes, termed as an activation or transfer function. The output of the flow persists if the value of activation function remains greater than some parameters, say, the threshold level.

Definition 4.4 Max fuzzy neuron (Kwan and Cai 1994):
The signal X i interfaced with the weight W i produces p i = W i X i , i = 1, 2. The input value p i is aggregated utilizing the most extreme conorm z = max{ p 1 , p 2 } = max{W 1 X 1 , W 2 X 2 }, and the j-th yield of the neuron is registered by where f is an initiation capacity and θ is known as the threshold level.
Definition 4.5 Min fuzzy neuron (Kwan and Cai 1994): The signal X i communicated with the weight W i produces p i = W i X i , i = 1, 2. The input value p i is amassed utilizing the minimum norm z = min{ p 1 , p 2 } = min{W 1 X 1 , W 2 X 2 }, and the j-th output of neuron is processed by y j = g j ( f (z − θ)) = g j ( f (min{W 1 X 1 , W 2 X 2 } − θ)), where f and θ are the same as in Definition 44.
In this paper, we consider Max fuzzy neuron and Min fuzzy neuron with t-norm and t-conorm in processing of the problem's solution (Fig. 2).

Logic-gate switching circuit
In algebra of switching circuits, electrical or electronic switching circuits are depicted mathematically or planned to get an outline for circuit having some criteria. In switching circuits, we can consider conductor-non-conductor, charged-uncharged, decidedly and contrarily polarized components. These days, semi-conductor components in switching circuits have more importance. In specific situation, switches are meant as so-called gates, or combination of gates. This can be treated as emblematic portrayal. In this way a gate (or combination of gates) is a polynomial p which has the elements x i for each i. We depict the gate as an acknowledgement of a switching function. In the event that, as for worth, p = 1 (or 0), we have current (or no current) through switching circuit p. Examples of switching gates with properties as output are given in Fig. 3 (X i s are treated here as input variables).
Since human thinking nowadays is not confined within 1-0 logical concept, a consequent fuzzy approach tends the switching circuit output toward linguistic variables like fuzzy sets. So the output in switching circuit also collaborates the crisp and fuzzy concept.

Fuzzy logic-gate switching circuit-oriented artificial neural network (FGSC-ANN) model
Here, we discuss the steps algorithmically to collaborate the fuzzy data through the artificial neural net. From the collected data, the required optimal value is obtained applying following Algorithm 1. Algorithm 1: Construction of FGSC-ANN using fuzzy numbers.
Step 1.1: Collection of input data.
Step 1.2: Weight assign with input-data according to problem, if required. Step 1.3: Weighted-data summation.
Step 1.4: Summed data are divided with two switches, namely, original and corresponding NOT gate. Step 1.5: All combinations are get together.
Step 1.6: All combined data set forms a geometrical figure, may be any polygon.
Step 1.7: Each vertex of the polygon is ranked according to their distances from centroid of the polygon.
Step 1.8: Signal flows through the minimum distance.
Step 1.9: Optimum vertex is obtained.

ANN-based fuzzy matrix game
In ANNs, weighted interconnections are established by mathematical formulation, termed as rules. Rules are basically governed by two approaches, crisp approach and fuzzy approach. A mathematical model when uses fuzzy set in a way is termed as a fuzzy approach of model rather than the crisp set-oriented approach. When ambiguous, uncertain, imprecise conditions are applied in if-then relationship between the variables of the rules in ANNs, fuzzy neural networks (FNNs) are originated. FNNs are classified by using two models, namely Mamdani model (Mamdani and Assilian 1975) and Takagi-Sugeno model (cf. Sugeno and Yasukawa 1993;Takagi and Sugeno 1985), depending on the structures of if-then rules. If the antecedent (if part) and the consequent (then part) are fuzzy propositions like: where A i and B i are from linguistic fuzzy sets and k is the number of rules in the model, then the Mamdani model is useful to apply. When knowledge is acquired in quantitative or data-based information, then knowledge can be accessed through the Takagi-Sugeno-Kang (TSK) model.
Here we consider zero-order TSK fuzzy model with four rules and say it as Modified TSK Model.
Modified TSK Model: Assumption: If x and y are inputs, then output is z. R 1 : If x is X 1 and y is Y 1 then z is a 11 , i.e., the output is ((X 1 , Y 1 ), a 11 ) R 2 : If x is X 2 and y is Y 1 then z is a 21 , i.e., the output is ((X 2 , Y 1 ), a 21 ) R 3 : If x is X 1 and y is Y 2 then z is a 12 , i.e., the output is ((X 1 , Y 2 ), a 12 ) R 4 : If x is X 2 and y is Y 2 then z is a 22 , i.e., the output is If all of a 11 , a 12 , a 21 and a 22 are assumed as fuzzy numbers, the fuzzy game in matrix form can be written as the following with X 1 , X 2 as player I's strategies and Y 1 , Y 2 are that for player II, Here, for example, the payoff a 11 emerged as the outcome when player I plays his/her strategy X 1 and player II plays Y 1 .
Consider player I's strategies have weights w 1 and w 2 and player II's strategies have weights w 3 and w 4 , respectively. Therefore, according to the concept of game theory, discussed in Sect. 3, we must have Assuming that each rectangular game has a solution and assuming v I and v I I to be the game values for players I Algorithm 2: Construction of matrix game solution.
Step 2.1: Construction of rules of the matrix game according to the strategies of the players.
Step 2.2: Combination of rules to form the payoffs of the matrix game.
Step 2.3: Application of the concept of the saddle point or mixed strategy to the matrix game.
Step 2.4: Achieving optimum strategies and obtaining the game value.
Algorithm 3: Construction of ANN-logic gate-switching circuit-oriented matrix game solution.

Numerical simulation
Consider the existence of two business houses H 1 and H 2 . By selling their products, both these houses are aiming to increase their profits in terms of market shares. One wishes to maximize his gain and the other aims to cut his lost. The two houses consider various strategies. House H 1 considers X 1 : Advertisement, X 2 : Reducing the printed price. And house H 2 chooses Y 1 : Attracting packaging features, Y 2 : Giving promotion-pack to customers free of cost.
Again we consider that these two houses have their own managing bodies which call meetings regularly (say, every three months or every six months) to put some weights on their decisions. The decisions after each meeting may vary from the previous meeting's decisions or not. So, hesitant environment arises. Both the houses have efforts to increase their market shares considering the fact that when one house profits, the other loses. So the outcome, after applying strategies, is the profit percentage of the houses in terms of market shares. If we consider this problem as a game issue with two players I and II representing houses H 1 and H 2 , respectively, then the payoff matrix is as follows: Here, player I has strategies X 1 and X 2 ; player II has Y 1 and Y 2 . And the payoff elements are hesitant triangular intuitionistic fuzzy elementsČ i j , i, j = 1, 2 with their corresponding weights, separated by second semicolon, given below:  (8.3, 9.7, 10); 0.7, 0.2 ; 0.3, (7, 9, 10); 0.6, 0.2 ; 0.2} indicates that if player I assumes X 1 and player II considers Y 2 , then the profit will be 90% with minimum 80% to maximum 100% having 6% positive chance and 3% pessimistic chance if the managing body gives 5% weight to their decisions. In the following meeting the decisions remain same and no problem arises, but if weight is given 3%, then the profit percentage is 97, lying between 83 and 100 having 7% positive chance. The remaining member of the set can be depicted likewise. Now, using the regular neural net structure, we combine the data in hesitant fuzzy set and using the mean averaging operator, we get fromČ 11 , x 1 = (5. 49, 7.49, 9.12); 0.6, 0.3 .
These set of values of sixteen fuzzy numbers can be assigned as sixteen vertices of a solid figure as in Fig. 6. Now, inspired from the articles (cf. Coupland and John 2008;Wu et al. 2012), the centroid of the figure is computed using the formulae: The centroid of the figure is the triangular fuzzy number (153.25, 624.99, 1747.43); 0.2, 0.7 . Now, computing the Euclidean distances of all vertices from the centroid, the shortest distance arises for the vertexx 1 x 2x3 x 4 and the farthest for the vertex x 1 x 2 x 3 x 4 . Now considering the vertex x 1 x 2x3 x 4 , we form the payoff matrix, given in TABLE1.
Payoff matrices for player I and player II are diverse because of different weights, earmarked for the strategies of the players, and consequently we get different values of the game. But in each cases, optimal strategies for player I from player I's payoff matrix and optimal strategies for player II from corresponding payoff matrix are, respectively: 0). If we consider different weights to different strategies, we get the different resolutions. The whole procedure is picturesquely represented in Fig. 7 (Here,ǎ i j is hesitant triangular IF numbered-set and k a i j are its members), where the game model is performed through Algorithm 2.

Results and discussion
In this work, we contemplate fuzzy matrix game with respect to ANN and fuzzy logic gate switching circuit. Defuzzification technique using the centroid concept is applied and achieved a fine result to the matrix game problems.
Here we notice that the weights assigned to the strategies of the players or decision makers, when changed, give an interesting resolution. As the weights are changed, the crisp value of the game is changed, simultaneously. When we consider player I's strategies with weights w 1 = 0.5, w 2 = 0.5 and player II's strategies with weights w 3 = 0.5, w 4 = 0.5,

Comparative analysis
Our proposed study, in comparison with others, has some significant advantages. Our study is based on hesitant fuzzy sets and elements by considering artificial neural network and logic gate switching circuits. This kind of idea has not been thought of before. For this reason, the result-relative comparison is beyond the scope. The superiority of the proposed study can be obtained from the following discussions.
(i) Esfahlani et al. (2019) considered the intuitive neurorehabilitation video game employing the fusion of artificial neural network (ANN), inverse kinematics and fuzzy logic algorithms. Particularly, this paper manifests an approach to rectify incorrect positioning through real-time visual feedback on the screen of video game. But in our proposed study, we have considered the extended version of fuzzy sets as hesitant fuzzy sets.
We have developed game model using artificial neural nets applying switching circuit gates in neural nets by defuzzifying of hesitant fuzzy elements. (ii) Abu-Khalaf et al. (2008) in their paper presented an application of neural networks to find closed-form representation of feedback strategies for a zero-sum game that appears in the H ∞ control. Uncertain environments are not considered in this paper, whereas, in our proposed study, hesitant fuzzy environment is assumed in real-life problematic situations. (iii) In the paper by Mansoori et al. (2019), the fuzzy constrained matrix game problems were assumed in triangular fuzzy numbers using the concepts of recurrent neural network (RNN). In our proposed work, we have considered artificial hybrid neural-net logic-gate switching circuit in fuzzy matrix game under hesitant triangular intuitionistic fuzzy environment.

Conclusion
A few articles have been publicized using game theory in different fields (cf. Bhaumik et al. 2020;Jana and Roy 2018;Roy and Bhaumik 2018;Roy 2010;Sadeghi and Zandieh 2011) successfully. Hesitant fuzzy concept is also an important tool to represent the decision makers' hesitance characteristics and successfully has been applied in different aspects (cf. Chen and Xu 2014;Chen et al. 2013;Rodriguez et al. 2012;Torra and Narukawa 2009;Yu et al. 2013). The major objectives of this work, to explore the potentiality of the neuro-fuzzy systems in modeling game phenomenon and to access its behavioral structures through artificial neural network and logic-gate switching circuits, are fulfilled. From the model description, our suggested methodology is unique in the following manners: (i) This is (probably) the main endeavor to explain fuzzy matrix game using max fuzzy neuron and min fuzzy neuron in hybrid fuzzy neural network. (ii) This is the fast approach to combine the hesitant fuzzy elements using neural network. (iii) The applied defuzzification method is unique in the sense that it can be applied easily. (iv) In future research works, this model can be applied in marketing, finance, medical sciences, engineering, etc.
The analysis of the results indicates that the rendition of FGSC-ANN model in game theory would be significantly improved if the input data are transformed into the normal or real domain prior to model formulation. The results of the proposed study highly encourage the researchers with a suggestion that ANN is viable for modeling daily life problems in the light of game theory.
Funding The authors have not disclosed any funding.
Data Availability Enquiries about data availability should be directed to the authors.