TWO-PERSON GAME WITH HESITANT FUZZY PAYOFF: AN APPLICATION IN MADM

Hesitant Fuzzy Set (HFS) permits the membership function having a collection of probable values which are more effective for modelling the real-life problems. Multiple Attribute Decision Making (MADM) process apparently assesses multiple conflicting attribute in decision making. In traditional decision making problems, each player is moving independently whereas in reality it is seen that each player aims to maximize personal profit which causes a negative impact on other player. MADM problem treats with candidate to the best alternative corresponding to the several attributes. Here, we present an MADM problem under hesitant fuzzy information and then transforming it into two-person matrix game, referred to herein as MADM game. The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is one of the prominent approach for solving the MADM problems. In this work, we develop the TOPSIS based on Ordered Weighted Aggregation (OWA) operator and hybrid hesitant fuzzy normalized Euclidean distance. Then the two approaches, namely Hybrid Hesitant Fuzzy Ordered Weighted Aggregation-TOPSIS (HHFOWA-TOPSIS) and the Linear Programming Problem (LPP) are applied to solve the formulated MADM game. For solving MADM game, we apply LPP by considering the various values of α,ψ, and HHFOWA-TOPSIS for finding the optimal alternative according to their scores. An investment selection problem is included to explain the feasibility and superiority of our formulated approaches. A comparison analysis is drawn among the obtained results which are derived from the two approaches. LPP and HHFOWA-TOPSIS provide the best alternative for the proposed problem. Finally, conclusions about our findings and outlooks are described. Mathematics Subject Classification. 91A05, 91A35, 03E72, 06D72, 62C86. Received August 9, 2020. Accepted September 20, 2021.

investment policy, military battles, sport competitions, and various types of competition. In the real-life problem, because of absence of available data, the payoffs of the matrix games are not in precise form. The DMs are behaved here as the players. A two-person zero-sum game is the normal game displayed by including two players, where the matrix games are considered with crisp payoffs. At that point, one player wins and whereby other player losses when two players pick the pure strategies 1 = { : = 1, 2, . . . , } and 2 = { : = 1, 2, . . . , } respectively.
In conventional MADM process, it is assumed that the determination of alternatives corresponding to attributes are exactly known by DM. But in real-life situations, DM does not know exactly the determination of alternatives corresponding to the attributes in uncertain environment. The uncertainty in determinations may be of various types such as probabilistic, fuzzy, fuzzy-probabilistic etc. Here we deal with uncertainty of hesitant fuzzy (Torra and Narukawa [49]) type which is the generalized version of fuzzy set (Zadeh [55]). Here DM chooses a hesitant fuzzy MADM problem. The most useful tool to tackle such type of problems is the hesitant fuzzy set.
Numerous approaches have been discussed for analyzing fuzzy MADM problems by many researchers. We can divide them into two parts. First one is the methods that apply in different ways of ranking fuzzy numbers. So the score of each alternative is computed. Then the best alternative is selected by the score function. The second one depends on various ordering of fuzzy numbers. In traditional MADM methods, TOPSIS has been paid great attention by researchers and scientists. Originally, the concept of TOPSIS was introduced by Hwang and Yoon [15]. The basic theory of TOPSIS is that the selected alternative should have the shortest distance from the positive ideal solution (PIS) and largest distance from the negative ideal solution (NIS). Due to its flexibility and comprehensibility, it is widely applied in various fields. Yoon and Kim [54] analyzed the behavioral TOPSIS in decision making. Hatami-Marbini and Kangi [14] proposed an extension of fuzzy TOPSIS for a group decision making with an application to Tehran stock exchange. Wang and Chen [50] studied an MADM based on intervalvalued intuitionistic fuzzy sets and solved it by simplex algorithm and extended TOPSIS. Liang and Xu [27] worked out a new extension of TOPSIS method for Multiple Criteria Decision Making (MCDM) with hesitant Pythagorean fuzzy sets. Aggarwal [2] elaborated hesitant fuzzy information sets and its application in group decision making. Lalotra and Singh [24] described knowledge measure of hesitant fuzzy set and its application in multi-attribute decision-making. Xiong et al. [52] presented a novel MAGDM approach with proportional hesitant fuzzy sets. Xia and Xu [51] proposed hesitant fuzzy information aggregation in decision making. Liao and Xu [28] depicted some new hybrid weighted aggregation operators under hesitant fuzzy MCDM environment. Jana and Roy [18] worked on soft matrix game by using hesitant fuzzy MCDM approach. Jin et al. [19] offered ordered weighted averaging aggregation on convex poset. Also, Sun et al. [48] considered an innovative TOPSIS approach based on hesitant fuzzy correlation coefficient and its applications. Chen and Hwang [10] unfolded the most appropriate method for solving MADM problem under fuzzy environment. Farhadinia and Xu [13] presented a novel distance-based multiple attribute decision-making with hesitant fuzzy sets. Mo et al. [34] unfolded feature-based hesitant fuzzy aggregation method for satisfaction with life scale. Bhaumik and Roy [5] proposed different aggregation operators in intuitionistic interval-valued hesitant fuzzy matrix games for solving management problems. Mula et al. [35] analyzed bi-matrix games with birough payoff elements and solved with the help of birough programming approach. Chen and Larbani [11] described a new method for analyzing a fuzzy MADM problem by reducing it into a two-person zero-sum game by using -cut. Also, Roy [38] elaborated multi-criteria decision making and fuzzy set theory in game environment. Jana and Roy [16] studied solution of matrix games with generalized trapezoidal fuzzy payoffs. Campos [9] offered fuzzy linear programming models to solve fuzzy matrix games. Also, Bhaumik et al. [6] presented robust ranking technique for analyzing a triangular intuitionistic fuzzy matrix game. Roy and Mondal [41] worked out an approach for solving fuzzy interval-valued matrix game. Bector et al. [4] offered duality in linear programming with fuzzy parameters and matrix games with fuzzy payoffs. Roy and Bhaumik [43] unfolded intelligent water management problem with the help of triangular type-2 intuitionistic fuzzy two-person zero-sum games approach. Larbani [25] analyzed bimatrix game with fuzzy payoffs by introducing nature as third player. Roy and Mula ([39,40]) proposed rough set approach in rough bimatrix game environment. Aggarwal and Khan [3] elaborated multi-objective fuzzy matrix games via multi-objective linear programming approach. Bigdeli and Hassanpour [8] discussed an approach to solve multi-objective linear production planning games with fuzzy parameters. Roy and Mula [42] analyzed matrix game with rough payoffs using genetic algorithm. Moreover, Larbani [26] viewed on non-coopertaive fuzzy games in normal form. Jana and Roy [17] discussed about dual hesitant fuzzy matrix games based on new similarity measure. Bhaumik et al. [7] studied Prisoners' dilemma game theory by using TOPSIS with the help of hesitant interval-valued intuitionistic fuzzy-linguistic term set approach. Ezimadu [12] viewed cooperative advertising decisions in a manufacturer-distributor-retailer supply chain by using game theory. Singh and Lalotra [47] presented generalized correlation coefficients of the hesitant fuzzy sets with their application to clustering analysis. Achemine et al. [1] analyzed -equilibria in bi-matrix games with uncertain payoffs. More recently, Roy and Jana [44] discussed on multi-objective linear production linear production planning game in triangular hesitant fuzzy set. Recently, Khalilpourazari and Doulabi [20] designed a hybrid reinforcement learning based algorithm with application in prediction of the COVID-19 pandemic in Quebec. Khalilpourazari et al. [23] proposed gradient-based grey wolf optimizer with Gaussian walk and its application in modelling and prediction of the COVID-19 pandemic. Khalilpourazari and Doulabi [21] analyzed robust modelling and prediction of the COVID-19 pandemic in Canada. Khalilpourazari and Pasandideh [22] designed emergency flood evacuation plans using robust optimization and artificial intelligence. From the recent literature survey, we notice that a lot of papers has been discussed on MADM game under fuzzy environment. But to the best of our knowledge, till now no one did consider the MADM game with hesitant fuzzy environment. A complete notation list include sets, parameters and decision variables is shown in Table 1.
However, there are various distances, similarity and correlation coefficient measure for HFS and its extension has been developed to make the decision with the help of TOPSIS. So it is essential to construct new approach to tackle this situation. The goal of this study is to develop an MADM game under hesitant fuzzy environment and to find the best alternative for the real-life application. Inspired by the above research works, we propose an HHFOWA-TOPSIS to solve the hesitant fuzzy MADM game. In this game the two players are denoted as actual DM and Nature. So here DM is the player I and Nature is the player II. Player I attempts to maximize the minimum payoffs whereas player II efforts to minimize the maximum lose. An extensive comparison of different characteristics between the present study and related works in these direction is shown in Table 2. The major contributions of this paper are summarized in three aspects, which are as follows: Fuzzy set Campos [9] Fuzzy set Chen and Larbani [11] Fuzzy MADM Liang and Xu [27] Hesitant fuzzy set Sakawa and Yano [45] Fuzzy set

This investigation
Hesitant fuzzy MADM -A game theory oriented MADM problem is developed under hesitant fuzzy environment and then it is converted into crisp MADM game by using -cut and maximin principle. -The idea of LPP is established to obtain optimal strategy of the crisp MADM game and a new score function is introduced to get the ranking order of alternatives. -The concept of HHFOWA-TOPSIS based on OWA and hybrid hesitant fuzzy normalized Euclidean distance is introduced to solve the hesitant fuzzy MADM game.
The rest of the paper is organized as follows: In Section 2, we keep some basic definitions of two-person zero-sum games, hesitant fuzzy sets, score function, aggregation operator, hybrid hesitant normalized Euclidean distance and membership function. In Section 3, we convert the basic MADM problem with hesitant fuzzy decision matrix into crisp MADM game by -cut and maximin principle. We develop the solution procedure for MADM game by incorporating two algorithms in Section 4. In Section 5, an investment selection problem is considered to check the applicability of LPP and HHFOWA-TOPSIS approach, and the results are discussed. In Section 6, we find a likeness of our outcomes with the similar papers. At last, conclusions are described in Section 7 with the suggestions of future study.

Preliminaries
Here we include the concept of matrix game, and some basic definitions of HFSs, score function, aggregation operator, hybrid hesitant normalized Euclidean distance and membership function.

Concept of two-person zero-sum game
Game theory is mainly connected with the strategies interaction of DMs and they are known that their strategies depend on one another. A two-person zero-sum game is the normal game which includes two players; and considering that both the players have pure and mixed strategies. In matrix game, non-cooperation is more interesting than cooperation due to realistic situation. In practical situation, the payoffs of cooperation and non-cooperation are in the sense of fuzzy understanding. A two-person zero-sum matrix game can be stated in the following payoff matrix as:   where is the payoff matrix of player I. All the elements in the matrix game are considered to be positive in this paper. Since this is the two-person zero-sum game so the payoffs of the player II is − .
Definition 2.1. The mixed strategy of player I is described by -tuple = ( 1 , 2 , . . . , ) of probabilities that add to 1 i.e., = 1, where is a vector whose each component is equal to 1. Similarly we define the mixed strategy of player II.
Definition 2.2. The mixed strategies of players I and II be and respectively, then the expected payoff of player I is defined as . So the expected payoff of player II is − .
3. Let * and * be the mixed strategies of players I and II respectively, are said to be optimal strategies if * ≤ * * and * * ≤ * for all mixed strategies and .
Corollary 2.4. (Parthasarathy and Raghavan [37]) Let * be the solution of optimization problem which is an optimal strategy of player I, whereas any solution * of the optimization problem is an optimal strategy of player II.
Corollary 2.5. (Parthasarathy and Raghavan [37]) Considering a payoff matrix , then on calculating the optimal strategies which are defined in equations (2.2) and (2.3), equivalent to solve (MOD-1) and (MOD-2) as: which then an optimal strategy of player I, whereas any solution * of the optimization problem is an optimal strategy of player II.
Corollary 2.6. ( [37]): Assuming a payoff matrix , calculating optimal strategies defined in equations (2.2) and (2.3) are equivalent to solve (MOD-1) and (MOD-2) as: Corollary 2.7. Problems (2.2) and (2.3) of players I and II can be rewritten in the following models respectively: It is easy to observe that from (MOD-3) and (MOD-4), we get = min } respectively. From Corollary 2.6, we discuss the subsequent steps for analyzing the game problem. First, we solve (MOD-1) and (MOD-2) and assume that ′ * and ′ * be the optimal solutions respectively. Secondly, to receive the optimal strategy * of player I, first we determine the value by the formula ∑︀ =1 ′ * = 1/ . Then applying the transformation ′ = / , we have the optimal strategy * = ′ * . Similarly, we have the optimal strategy for player II. So, by Corollary 2.6, the problem of calculating the sets of optimal strategies is converted to solve two LPPs i.e., (MOD-3) and (MOD-4). The two problems will be applied extensively to the rest of the paper.

Hesitant fuzzy number and hesitant fuzzy matrix
Hesitant fuzzy set is an extremely valuable tools to handle the circumstances where there are some problems to calculate the membership function of an element to a set. Here we introduce the definition of a triangular hesitant fuzzy number in the space of matrices. Suppose three HFEs are considered byh,h 1 andh 2 . Torra and Narukawa [49] defined some operations on them, which can be stated as: Now some operations on hesitant fuzzy elementsh,h 1 andh 2 are expressed based on the relationship between HFEs and intuitionistic fuzzy value as: Distance and similarity measures are most important in the applications of MADM. By using distance measure, we reduce the MADM game with hesitant fuzzy payoffs to crisp payoffs. Definition 2.10. Leth 1 andh 2 be two hesitant fuzzy elements, then the hybrid hesitant normalized Euclidean distance measure withinh 1 andh 2 is stated as where (h) is the number of the elements inh, and ℎ In most of the cases, (h 1 ) ̸ = (h 2 ) thus, assume that = max{ (h 1 ), (h 2 )}. Hence it is essential to extend the smaller one until both of them have similar length when we compare between them. To expand the smaller one, the suitable path is to add the same value many times in it. So the choice of this value basically depends on the DM risk preferences. An optimist bears graceful outcome and may add the maximum value, whereas a pessimist looks for unfavourable results and may attach the minimum value. As for example, leth 1  MADM game: The development scenario for game theory is to integrate MADM for tackling the decision making problems. Game theory plays a significant role in various collective negotiations among the various participants. As for example, various negotiations take place when a company (DM) wants to invest lump-sum amount of money but the collective risks are taken together. Then by using game theory, the DM and Nature (collective risks) can arrive at the optimal solution of this issue. So the best path is to combine an MADM with game to establish a decision making model.
Aggregation operator: OWA operator was introduced by Yager [53]. It is a well-known aggregation method, which has been extended and studied by many researchers. Merigo [33] introduced a model between the weighted average and induced OWA operator. Inspired by the idea of OWA operator based on the existing hesitant fuzzy information aggregation and distance measure, we propose a new hesitant fuzzy distance measure, called Hybrid Hesitant Fuzzy Ordered Weighted Average (HHFOWA) operator. It can be defined as follows: (

Problem environment
First, we reduce the initial hesitant fuzzy MADM problem into a two-person game with hesitant fuzzy payoffs and thereafter we solve the formulated MADM game by LPP and HHFOWA-TOPSIS approach. This MADM ame is converted into two-person crisp MADM game by using -cut and maximin principle, and then it is solved by LPP. Also the initial hesitant fuzzy MADM game is solved by HHFOWA approach for comparison purpose. The -cut is emerged in the procedure of Sakawa and Yano [45] for analyzing multi-objective non-linear problem with fuzzy information.

Statement of the problem
Now we present a hesitant fuzzy MADM problem which is described in Table 3 as: Where the player I has alternatives ( , = 1, 2, . . . , ) corresponding to attributes ( , = 1, 2, . . . , ); stands for ranking of -th alternative corresponding to -th attribute. The main intention of player I is to choose the best alternative in accordance with the obtainable information in the hesitant fuzzy MADM problem. Here, player I assumes that˜= [˜] × = ( , , ) is a hesitant fuzzy MADM problem.

Model formulation
Here, we convert the basic hesitant fuzzy MADM problem into two-person zero-sum game by using -cut and maximin principle. Now we consider the following subsection for describing our approach. In this way we achieve an interval MADM decision matrix with the help of equation (3.1), which is shown in Table 4.

Two-person zero-sum game
Now, we have an interval MADM decision matrix (i.e., Table 4). Player I does not have any message over the feature of this decision matrix due to uncertainty. Here we reduce the interval MADM decision matrix to a twoperson zero-sum game. So player I has to accept one of the criteria i.e., the maximin principle of decision making    Table 5.
Player I is the maximizing player and player II is the minimizing player. From definition, the expected payoff of the player I is = ∑︀ and for player II is − . So the basic hesitant fuzzy MADM problem is reduced into matrix game with crisp payoffs. Now we derive the optimal strategies of player I and player II and the value of the game.

Solution procedure
In this section, we give a detailed discussion of LPP and HHFOWA-TOPSIS approach. The procedure for solving the hesitant fuzzy MADM game consists of two algorithms. Algorithm 4.6 is for obtaining the optimal strategy of crisp MADM game and ranking order of each alternative. Then Algorithm 4.7 is based on HHFOWA-TOPSIS approach to get the optimal ranking order of alternatives.

LPP
At first we convert a two-person crisp matrix game (i.e., Table 5) into a problem for solving two free optimization problems. Then by Theorem 1, to find a solution for crisp game, then we derive the optimal solution of the consequent pair of optimization models and finally we obtain the mixed strategies. The consequent pair of optimization models is as follows: Proof. We want to prove that (MOD-5) and (MOD-7) are same. So our problem is ince ∈˜, it can be expressed as { + (1− ) }, 0 ≤ ≤ 1, then (MOD-9) can be stated as (Model-10).

where =
, and = ( − ). We want to show that set of constraints of (MOD-10) and (MOD-11) are same. Let us consider that ′ justifies the constraints of (MOD-11), then we get Proof. We have ≤ ≤ for all and . Therefore, the proof is straightforward. Algorithm 4.6. Now we develop an algorithm to find the optimal strategies of player I and player II. Also we find the expected score of each alternative corresponding to the attributes. In this procedure, player I decides the -cut based on the chosen value .
-Step 1. Player I selects the -cut to obtain the optimal solutions ′ * and ′ * of (MOD-5) and (MOD-6), respectively consequently determines the mixed strategies * = 1 ′ * and * = 2 ′ * . Definition 4.1 and Corollary 4.2 show that these strategies are the optimal solutions of the two-person zero-sum game.
-Step 2. Player I considers the weight ∈ [0, 1], and then determines the desired score of every alternative according to equation (4.1). -Step 3. At last, we present the preference order of each alternative according as first (1), second (2), third (3) and fourth (4).

The proposed HHFOWA-TOPSIS approach
In this subsection, we propose the TOPSIS approach based on HHFOWA operator. The basic principle of the proposed TOPSIS approach is the chosen alternative should have the maximal and minimal hybrid hesitant fuzzy normalized Euclidean distance from PIS and NIS respectively. Suppose that an MADM problem has alternatives ( , = 1, 2, . . . , ) corresponding to attributes ( , = 1, 2, . . . , ). All the values are assigned to the alternatives with hesitant fuzzy attribute can be expressed in a decision matrix. So the procedure of HHFOWA-TOPSIS approach is discussed in Algorithm 4.7. -Step 4. Then rank the alternatives according to their scores.

Application example
An investment company is wishing to invest a lump-sum amount of money in one of possible four alternatives which are as: the first alternative ( 1 ) is a TV company, the second alternative ( 2 ) is a computer company, the third alternative ( 3 ) is a food company and the fourth alternative ( 4 ) is a car company. The investment company also suggests five attributes based on the alternatives and they are: production risk ( 1 ), financial risk ( 2 ), management risk ( 3 ), environment risk ( 4 ) and market risk (  Here we consider the weight = 0.9 so that the center value ( ) lies between and .  Table 7. Score of each hesitant fuzzy alternative with its priority.

Results and discussion
To solve this problem, we apply the LPP and HHFOWA-TOPSIS approach to obtain the optimal ranking order.

Applying the LPP
We impose the proposed Algorithm 4.6 to validate the MADM game. First, the -cut is fixed for different values within the interval [0, 1], and then calculate the optimal strategies * and * which are shown in Table 6 (by Step 1 of Algorithm 4.6). Secondly, player I sets the value of . Then determine the expected score of every alternative (by Step 2 of Algorithm 4.6) corresponding to various -cuts and different . Thereafter, the accepted outcomes are displayed in Table 7. Now for this example, we say that the ranking of each alternative remains adjustable for various values of . Here, we choose: poor (0.3), medium (0.5), good (0.7) and excellent (0.9). At last, we give the preference order of each alternative by Step 3: 4 (Car company) is considered as the best alternative with largest score and 1 is a computer company, likes to be the worst alternative with the smallest score in this MADM game. So, the ranking order of this MADM game will be 4 > 2 > 3 > 1 (> means "is preferred to"). So this example displays that the MADM game process can be applied for the investment selection problem.

Applying HHFOWA-TOPSIS aproach
The example concerning investment company is employed to compare the effectiveness of proposed HHFWOA-TOPSIS approach with the LPP. Suppose that the weight vector of the criteria is = (0.1, 0.2, 0.2, 0, 2, 0.3). Now we employ Algorithm 4.7 to obtain the optimal ranking order.
-Step 1. Using the score function, calculate the PIS ( + ) and NIS ( − ) respectively as:  -Step 4. Then the ranking order of the alternatives is 4 > 1 > 2 > 3 . So the alternative 4 is the best according to their scores.

Comparisons
In comparison with the related papers, the proposed study has the following advantages: -It ought to be noted that our primary problem is a hesitant fuzzy MADM problem. In our content, -cuts are taken to defuzzify the hesitant fuzzy MADM problem for obtaining the crisp two-person zero-sum MADM game. Thereafter, LPP and HHFOWA-TOPSIS approach are applied to solve the hesitant fuzzy MADM game for deriving the optimal strategy and the expected score of each alternative corresponding to the attributes are different than Chen and Larbani [11]. -Sakawa and Yano [45] discussed the -cuts for solving multi-objective non-linear programming problems with fuzzy information. Again Sakawa and Yano [45] considered the minimax principle i.e., they chosen the decision variable that accepted values in the -cut but we consider maximin principle. Here we formulate the problem as a game versus Nature, and another procedure is introduced for solving hesitant fuzzy MADM game. -Nishizaki and Sakawa [36] and Sakawa and Nishizaki [46] proposed a two-person zero-sum fuzzy and multiobjective games for conflict solution. The elements of the payoff matrix are taken as L-R fuzzy numbers. A degree of acquirement of each fuzzy goal is defined due to the degree of satisfication of a decision. Therefore, a maximin solution with connection to a degree of acquirement of the aggregated fuzzy goal is discussed. In Nishizaki and Sakawa [36] and Sakawa and Nishizaki [46], the authors considered the maximin rule when conduct with the satisfaction degree of goals but in this paper, it is in the crisp game. Also, for obtaining maximin strategy of any player, the authors of Nishizaki and Sakawa [36] and Sakawa and Nishizaki [46] determined a fractional LPP, but here LPP and HHFOWA-TOPSIS approach are used to solve the hesitant fuzzy MADM game. -Campos [9] converted a two-person zero-sum fuzzy matrix game into a crisp LPP by using different approaches for ranking fuzzy numbers. Here we consider the hesitant fuzzy MADM game, and the solution procedure is different. The procedure is applied by Bector and Chandra ([4]) is same with Campos [9]. Table 8. Comparison among our approach and existing approaches.

Existing papers Solution procedure
Bector and Chandra ( [4]) Using the concept of duality in LPP, fuzzy matrix game reduces to the pair of crisp primal and dual LPPs Campos [9] Using ranking function to convert a zero-sum fuzzy matrix game into a crisp LPP Chen and Larbani [11] Using -cut and maximin principle to formulate fuzzy MADM problem into two-person zero-sum game with uncertain payoff matrix Sakawa and Yano [45] Using -cut for solving multi-objective non-linear programming problems with fuzzy information Nishizaki and Sakawa [36] and Sakawa and Nishizaki [46] Using maximin approach for solving a fractional LPP

This paper
Using -cut and maximin principle to convert hesitant fuzzy MADM problem into crisp matrix game, which is solved by LPP and analyzed the result with the solution of hesitant fuzzy MADM game by HHFOWAapproach Actually, Bector and Chandra ( [4]) discussed an unfamiliar duality idea in LPPs with fuzzy payoffs and due to defuzzification they converted a fuzzy matrix game to the pair of crisp primal and dual LPPs. The comparison between our approach and existing approaches is shown in Table 8.
Moreover, in all mentioned approaches, the elements of the payoff matrix are fuzzy number but in our work, elements are hesitant fuzzy numbers which depend on the value of . Thus our proposed approaches can be applied to analyze not only a hesitant fuzzy MADM problem but also a two-person zero-sum game with the hesitant fuzzy payoffs.

Pros and Cons of the Proposed approaches
-The LPP and HHFOWA-TOPSIS approach based on the hybrid hesitant fuzzy normalized Euclidean distance measure are incorporated in this paper. Based on the context of the problem, HHFOWA-TOPSIS is very easy than the LPP to apply in real-life problem. -It is less computational effort for solving hesitant fuzzy MADM game.
-The LPP and HHFOWA-TOPSIS approach are not only solved the hesitant fuzzy MADM game problem, but also the other decision making problem under uncertain environment. -In this paper, the weights of each attribute are given for HHFOWA-TOPSIS. But in some real-life problems, the weights are always not known. So it is a questionable to select the proper weight for the decision making problem, can be treated as one of the drawbacks of the approach.

Conclusion
In conventional MADM game problem, the weight of every attribute identifies the substances of such type of difficulty. Also numerous methods are available for obtaining the weights. Many researchers has investigated the chance of applicability of two-person zero-sum game in MADM process. In this content, we have considered a hesitant fuzzy MADM problem; and then we have formulated it as a crisp MADM game by considering the -cut and maximin principle. The proposed approaches has been conveyed us to initiate a new way to define the membership function for triangular hesitant fuzzy numbers. Here we have applied the LPP and HHFOWA-TOPSIS approach to solve the hesitant fuzzy MADM game problem. Two algorithms have been introduced to check the efficiency of application in this paper. An example of investment company has been shown the applicability and feasibility of using such MADM game. Based on the discussion, we have concluded that the HHFOWA-TOPSIS approach has been allowed more suitable to obtain the optimal ranking order of the alternatives. In practical situation, a large number of MADM games can be analyzed frequently and successfully by using the proposed approaches.
Future studies can be extended by considering the proposed approaches in the following directions: -In this work, we have discussed the proposed approaches to the hesitant fuzzy MADM game. Further, the notion of proposed approaches can be utilized for analyzing the interval-valued hesitant, dual hesitant, Pythagorean hesitant fuzzy MADM problem, Lotfi et al. [31], Lotfi et al. [32], Lotfi et al. [29] and Lotfi et al. [30]. -Also aggregation operator is an effective tool to make the decision. Then the proposed TOPSIS can be extended for any other aggregation operator.