Optimal Solutions for Constrained Bimatrix Games with Payoffs Represented by Single-Valued Trapezoidal Neutrosophic Numbers

. Single-valued neutrosophic set (SVNS) is considered as generalization and extension of fuzzy set, intuitionistic fuzzy set (IFS), and crisp set for expressing the imprecise, incomplete, and indeterminate information about real-life decision-oriented models. The theme of this research is to develop a solution approach to solve constrained bimatrix games with payoﬀs of single-valued trapezoidal neutrosophic numbers (SVTNNs). In this approach, the concepts and suitable ranking function of SVTNNs are deﬁned. Hereby, the equilibrium optimal strategies and equilibrium values for both players can be determined by solving the parameterized mathematical programming problems, which are obtained from two novel auxiliary SVTNNs programming problems based on the proposed ranking approach of SVTNNs. Moreover, an application example is examined to verify the eﬀectiveness and superiority of the developed algorithm. Finally, a comparison analysis between the proposed and the existing approaches is conducted to expose the advantages of our work.


Introduction
Constrained bimatrix games are nonzero-sum two-player noncooperative games which play a dominant role in many real-life applications such as in military, finance, economy, strategic welfares, cartel behaviour, management models, social problems or auctions, political voting systems, races, and development research [1,2]. Usually, the constrained bimatrix game makes the assumption that the payoff values are described with crisp elements and exactly known by each player. However, players are not able to evaluate the games outcomes exactly due to the unavailability and ambiguity of information. To handle that, Zadeh [3] introduced the fuzzy set concept and since then various researchers have extended it to the different sets such as interval intuitionistic fuzzy set, IFS, linguistic interval IFS, and cubic IFS. Many scholars have studied various kinds of noncooperative games under uncertainty. For instance, Li et al. [4] proposed a bilinear programming algorithm for solving bimatrix games with intuitionistic fuzzy (IF) payoffs. Figueroa et al. [5] studied group matrix games with interval-valued fuzzy numbers payoffs. Jana et al. [6] introduced novel similarity measure to solve matrix games with dual hesitant fuzzy payoffs. Singh et al. [7] established 2-tuple linguistic matrix games. Zhou et al. [8] constructed novel matrix game with generalized Dempster-Shafer payoffs. Seikh et al. [9] solved matrix games with payoffs of hesitant fuzzy numbers. Han et al. [10] described new matrix game with Maxitive Belief information. Roy et al. [11] discussed Stackelberg game with payoffs of type-2 fuzzy numbers. Bhaumik et al. [12] solved Prisoners' dilemma matrix game with hesitant intervalvalued intuitionistic fuzzy-linguistic payoffs elements. Ammar et al. [13] studied bimatrix games with rough interval payoffs. Brikaa et al. [14] developed fuzzy multiobjective programming technique to solve fuzzy rough constrained matrix games. Bhaumik et al. [15] introduced multiobjective linguistic-neutrosophic matrix game with applications to tourism management. Brikaa et al. [16] applied resolving indeterminacy technique to find optimal solutions of multicriteria matrix games with IF goals. So far, as the authors are aware, there are only four articles that studied constraint bimatrix games. Jing-Jing et al. [17] proposed linear programming method for solving constrained bimatrix games with IF payoffs. Koorosh et al. [18] presented constrained bimatrix games and their application in wireless communications. Fanyong et al. [19] applied two approaches to solve the classical constrained bimatrix games. Bigdeli et al. [20] discussed constrained bimatrix games with fuzzy goals.
However, the IFS and fuzzy set theories are unable to deal with inconsistent and indeterminate data correctly. To consider that, Smarandache [21] introduced the theory of neutrosophic set (NS), defining the three components of indeterminacy, falsity, and truth; all lie in ]0 − , 1 + [ and are independent. As NS is difficult to implement on realistic applications, Wang et al. [22] developed the single-valued neutrosophic set (SVNS) concept, which is an extension of the NS. Due to its importance, many scholars have applied the SVNS theory in various disciplines. For example, Garg [23] studied the analysis of decision-making based on sine trigonometric operational laws for SVNSs. Murugappan [24] presented neutrosophic inventory problem with immediate return for deficient items. Garg [25] proposed new neutrality aggregation operators with multiattribute decision-making (MADM) approach for single-valued neutrosophic numbers (SVNNs). Abdel-Basset et al. [26] investigated resource levelling model in construction projects with neutrosophic information. Garai et al. [27] discussed variance, standard deviation, and possibility mean of SVNNs with applications to MADM models. Broumi et al. [28] solved neutrosophic shortest path model by applying Bellman technique. Garg [29] proposed TOPSIS and clustering approaches to solve SVNNs decision-making model. Mullai et al. [30] presented inventory backorder model with neutrosophic environment. Garg et al. [31] studied MADM based on Frank Choquet Heronian mean operator for SVNSs. Leyva et al. [32] introduced a new problem of information technology project with neutrosophic information. Garg [33] presented nonlinear programming approach for solving MADM model with interval neutrosophic parameters. Sun et al. [34] developed new SVNN decisionmaking algorithms based on the theory of prospect. Garg [35] introduced biparametric distance measures on SVNSs and their applications in medical diagnosis and pattern recognition.
In the imprecise data game, players may encounter some assessment data that cannot be represented as real numbers when estimating the utility functions or uncertain subjects. Since SVNS has great superiority and flexibility in describing many uncertainties with complex environments, it is effective and convenient to represent the constrained bimatrix games with neutrosophic data. Due to decision-making growing requirements of expressing their judgments in a human friendly and neatly manner, it is important to extend the IF or fuzzy constrained bimatrix games into neutrosophic environment. e SVNS is an effective tool to satisfy the increasing requirement of higher uncertain and complicated constrained bimatrix game models. Probably, this is the first attempt of solving constrained bimatrix game with SVTNNs payoffs. e fundamental targets of this article are listed as follows: (1) To propose a novel constrained bimatrix games model with SVTNNs payoffs (2) To develop an effective algorithm for SVTNN constrained bimatrix games to obtain the optimal strategies for such games (3) To formulate crisp linear optimization problems from the neutrosophic models based on the defined ambiguity and value indexes of SVTNN (4) To present an application example to demonstrate the effectiveness and applicability of the proposed method (5) To compare our results with other existing approaches e remainder of the manuscript is summarized as follows. Section 2 introduces the concept, cut sets, and arithmetic operations of SVTNNs. Section 3 gives the concept of ambiguity and value indexes of SVTNNs and the ranking technique of SVTNNs. Section 4 formulates constrained bimatrix games with SVTNNs payoffs and the solution approach to solve such games. e illustrative example with comparative analysis is discussed in Section 5. Lastly, a short conclusion is given in Section 6.

Preliminaries
In the following, we introduce the basic concepts of fuzzy sets, IFSs, NSs, SVNSs, and SVNNs.
Definition 1 (see [36]). A fuzzy number B � (b 1 , b 2 , b 3 , b 4 ) is said to be a trapezoidal fuzzy number (TFN), if its membership function δ B (y) is given by

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Definition 2 (see [37]). Suppose that Y is a universal set. An IFS C is defined as follows: where c C : Y ⟶ [0, 1] and δ C : Y ⟶ [0, 1] are the nonmembership degree and the membership degree of y ∈ Y to the set C⊆Y, such that 0 ≤ δ C (y) + c C (y) ≤ 1, ∀y ∈ Y.
Definition 3 (see [22]). An SVNS B in a universe Y is defined by where e valuesF B (y), I B (y) and T B (y), respectively, express the falsity membership, indeterminacy membership, and truth membership degree of y to B.
which satisfies the following conditions: Obviously, any 〈α, β, c〉-cut set b 〈α,β,c〉 of an SVTNN b is a crisp subset over the set of real numbers R.
where α ∈ [0, u b ]. Obviously, any α-cut set b α of an SVTNN b is a crisp subset over the set of real numbers R.
Here, any α-cut set of an SVTNN b for the truth membership function is a closed interval, represented by Definition 12 (see [38]). Let b � 〈((k 1 , l 1 , m 1 , n 1 ), where β ∈ [v b , 1]. Obviously, any β-cut set b β of an SVTNN b is a crisp subset over the set of real numbers R.
Here, any β-cut set of an SVTNN b for the indeterminacy membership function is a closed interval, represented by Definition 13 (see [38] where c ∈ [w b , 1]. Obviously, any c-cut set b c of an SVTNN b is a crisp subset over the set of real numbers R.
Here, any c-cut set of an SVTNN b for the falsity membership function is a closed interval, represented by

Value and Ambiguity of SVTNNs.
Here, we introduce the basic definitions of value and ambiguity indices of SVTNN.
be any α-cut set, β-cut set, and c-cut set of the SVTNN b, respectively. en, we have the following.
(1) e value of the SVTNN b for α-cut set, represented by V μ (b), is given as where , is given as where Definition 15 (see [38]).
be any α-cut set, β-cut set, and c-cut set of the SVNN b, respectively. en, we have the following.
(1) e ambiguities of the SVTNN b for α-cut set, represented by A μ (b), are given as where (2) e ambiguities of the SVTNN b for β-cut set, represented by A δ (b), are given as (3) e ambiguities of the SVTNN b for c-cut set, represented by A η (b), are given as where g(c) Here, the weighting functions h(α), f(β), and g(c) can be supposed according to the decision-making model nature. Suppose that h(α) � α, f(β) � 1 − β, and g(c) � 1 − c.
Let b � 〈(k, l, m, n); u b , v b , w b 〉 be an SVTNN. en the value and ambiguity indices, using the above descriptions, are constructed as

A Ranking Approach of an SVTNN Based on Value and Ambiguity Indices.
is section provides a ranking approach of SVTNNs based on the ambiguity and value indices of SVTNNs in a similar way to those of SVNNs introduced by A. Bhaumik et al. [39].
〉 be an SVTNN. e weighted value ambiguity index for an SVTNN b is given as Definition 17. Let c and d be two SVTNNs and let λ 1 , λ 2 , λ 3 ∈ [0, 1]. For the weighted value ambiguity index of the SVTNNs c and d, the ranking order of c and d is given as follows: where " > N " and " < N " are neutrosophic versions of the order relations " > " and " < " in the real line, respectively.

Constrained Bimatrix Games with SVTNNs Payoffs and Solution Method
Let us consider the constrained bimatrix game with SVTNNs payoffs. Suppose that T 1 � ξ 1 , ξ 2 , . . . , ξ κ and T 2 � η 1 , η 2 , . . . , η ℓ are pure strategies sets for two players I and II, respectively. When player II selects pure strategy η j ∈ T 2 and player I selects pure strategy ξ i ∈ T 1 , at the situation (ξ i , η j ), player II gains payoff and player I e mixed strategies vectors are represented as r � (r 1 , r 2 , . . . , r κ ) T and s � (s 1 , s 2 , . . . , s ℓ ) T , where r i (i � 1, 2, . . . , κ) and s j (j � 1, 2, . . . , ℓ) are probabilities for both players selecting their pure strategies ξ i ∈ T 1 and η j ∈ T 2 , respectively. e mixed strategies r i and s j are affiliated with the strategies sets (convex polyhedron) which are described by some inequalities and equations. Let R � r: G T r ≥ p, r ≥ 0 represent the strategy constraint set of player I, where p � (p 1 , p 2 , . . . , p e ) T , G � (g in ) κ×e , and e is a positive integer. Let S � s: Hs ≥ q, s ≥ 0 express the strategy constraint set of player II, where q � (q 1 , q 2 , . . . , q b ) , H � (h mj ) b×ℓ , and b is a positive integer. Note that G T r ≥ p contains κ i�1 r i � 1, since κ i�1 r i � 1 is equivalent to κ i�1 r i ≥ 1 and − κ i�1 r i ≥ − 1. Similarly, Hs ≥ q contains ℓ j�1 s j � 1. In the sequel, the above SVTNN constrained bimatrix game is simply denoted by (C, D) for short.
Without loss of generality, suppose that both players I and II, respectively, select mixed strategies r ∈ R and s ∈ S in Journal of Mathematics 5 order to maximize their own payoffs; then their expected payoffs can be obtained as follows: Definition 18 (see [40]). If (r * , s * ) ∈ R × S satisfies the following conditions: for any mixed strategies r ∈ R and s ∈ S, then r * and s * are called equilibrium strategies, and U * � r * T Cs * and W * � r * T Ds * are called equilibrium values of players I and II, respectively. Gz ≤ N Ds, Hs ≥ q, respectively, then r * and s * are equilibrium strategies of the SVTNN constrained bimatrix game (C, D), and U * � q T y * � r * T Cs * and W * � p T z * � r * T Ds * are equilibrium values of players I and II, respectively.
Proof:. e proof of this theorem is similar to the proof given by Jing-Jing et al. [17].
It is obvious that the two players often cannot calculate the payoffs accurately in each situation, and the game values of the SVTNN constrained bimatrix games are not equal to q T y in (21) and p T z in (22). e two players may allow some violations on the set of constraints H T y ≤ N C T r and Gz ≤ N Ds. erefore, the equilibrium strategies r * and s * and equilibrium values U * and W * of the SVTNN constrained bimatrix games are equal to the optimal values and optimal solutions of (23 and 24) as follows: max q T y , Hs ≥ q, s ≥ 0, respectively, where m � (m 1 , m 2 , . . . , m ℓ ) T , n � (n 1 , n 2 , . . . , n k ) T , and all the vectors elements of m and n are SVTNNs that are approximately equal to zero, which represent the maximum violations that the two players may permit on the set of constraints. e parameter ρ (0 ≤ ρ ≤ 1) is a real number.

Application Example
In this section, an example of the company development strategy choice model adapted from Jing-Jing et al. [17] is used to illustrate the solution procedure of a constrained bimatrix game with payoffs of SVTNNs.

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For different values λ 1 , λ 2 , λ 3 , and ρ, the equilibrium strategies and the equilibrium values of both players can be obtained by solving equations (31) and (32), as depicted in Tables 1-12. It can be easily seen from Table 1 that when λ 1 � 0.2, λ 2 � 0.3, λ 3 � 0.5, and ρ � 0, the equilibrium value and the equilibrium strategy for player I are U * � q T y * � 4.868 and r * � (0.75, 0.25) T , respectively; and the equilibrium value and the equilibrium strategy for player II are W * � p T z * � 4.855 and s * � (0.4, 0.6) T , respectively.
e results indicate that different optimal solutions can be obtained for different values of λ 1 , λ 2 , λ 3 , and ρ.
us, it is essential to take all the parameters into consideration. max − 5.5y 1 + y 2 − y 3 s.t.

Conclusion
e constrained bimatrix games with payoffs of SVTNNs are studied and constructed in this article. e ranking order relation, important theorems, and arithmetic operations of SVTNNs are outlined. Novel neutrosophic optimization problems for both players are established from the arithmetic operations of SVTNNs and solution method for SVTNNs constrained bimatrix games. Based on the ranking approach of SVTNNs presented in this article, the neutrosophic optimization problems for both players are converted into crisp parameterized problems, which are solved to obtain the equilibrium optimal strategies and equilibrium values for two players. Moreover, the ranking approach proposed in this article is demonstrated with a numerical simulation. Finally, our article is the first to study the constrained bimatrix games under neutrosophic environment and provide algorithm and practicable application for SVTNNs constrained bimatrix games.
In the future, we will study game theory under other types of uncertain environment such as linguistic neutrosophic, interval neutrosophic, linguistic interval neutrosophic, and linguistic interval intuitionistic neutrosophic. Furthermore, we will apply the proposed ranking approach to other areas such as pattern recognition, supply chain, risk evaluation, teacher selection, and optimization models.

Data Availability
e data used to support the findings of this research are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest.