#### Abstract

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.

#### 1. 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 Prisonersâ€™ dilemma matrix game with hesitant interval-valued 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 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 decision-making 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. The 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. The 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

The 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. The illustrative example with comparative analysis is discussed in Section 5. Lastly, a short conclusion is given in Section 6.

#### 2. Preliminaries

In the following, we introduce the basic concepts of fuzzy sets, IFSs, NSs, SVNSs, and SVNNs.

*Definition 1. *(see [36]). A fuzzy number is said to be a trapezoidal fuzzy number (TFN), if its membership function is given by

*Definition 2. *(see [37]). Suppose that is a universal set. An IFS is defined as follows:where and are the nonmembership degree and the membership degree of to the set , such that .

*Definition 3. *(see [22]). An SVNS in a universe *Y* is defined bywhere , , and such that . The values, respectively, express the falsity membership, indeterminacy membership, and truth membership degree of *y* to .

*Definition 4. *(see [22]). An -cut set of SVNS , a crisp subset of , is given bywhere , , , and .

*Definition 5. *(see [22]). An SVNS is called neutrosophic normal, if there exist at least three points such that .

*Definition 6. *(see [22]). An SVNS is said to be neutrosophic convex, if, and , the following conditions are satisfied:(i)(ii)(iii)

*Definition 7. *(see [22]). An SVNS , is said to be single-valued neutrosophic number when(1) is neutrosophic normal(2) is neutrosophic convex(3) is upper semicontinuous, is lower semicontinuous, and is lower semicontinuous(4)The support of , that is, , is bounded

*Definition 8. *(see [38]). An SVTNN is a special neutrosophic set on the set of real numbers , whose truth membership, indeterminacy membership, and falsity membership are represented asrespectively.

*Definition 9. *(see [38]). Let and be two SVTNNs and let be any real number. Then,(1)(2)(3)

*Definition 10. *(see [38]). Let be an SVTNN. Then, -cut set of the SVTNN , represented by , is given aswhich satisfies the following conditions:Obviously, any -cut set of an SVTNN is a crisp subset over the set of real numbers .

*Definition 11. *(see [38]). Let be an SVTNN. Then, -cut set of the SVTNN , represented by , is given aswhere .

Obviously, any -cut set of an SVTNN is a crisp subset over the set of real numbers .

Here, any -cut set of an SVTNN for the truth membership function is a closed interval, represented by *.*

*Definition 12. *(see [38]). Let be an SVTNN. Then, -cut set of the SVTNN , represented by , is given aswhere .

Obviously, any -cut set of an SVTNN is a crisp subset over the set of real numbers .

Here, any -cut set of an SVTNN for the indeterminacy membership function is a closed interval, represented by *.*

*Definition 13. *(see [38]). Let be an SVTNN. Then, -cut set of the SVTNN , represented by , is given aswhere .

Obviously, any -cut set of an SVTNN is a crisp subset over the set of real numbers .

Here, any -cut set of an SVTNN for the falsity membership function is a closed interval, represented by *.*

#### 3. Characteristics and the Ranking Approach for SVTNNs

##### 3.1. Value and Ambiguity of SVTNNs

Here, we introduce the basic definitions of value and ambiguity indices of SVTNN.

*Definition 14. *(see [38]). Let be an SVTNN and let , , and be any -cut set, -cut set, and -cut set of the SVTNN , respectively. Then, we have the following.(1)The value of the SVTNN for -cut set, represented by , is given asâ€‰where , , and is nondecreasing and monotonic of .(2)The value of the SVTNN for -cut set, represented by , is given asâ€‰where , , and is nondecreasing and monotonic of .(3)The value of the SVTNN for -cut set, represented by , is given asâ€‰â€‰where , , and is nondecreasing and monotonic of .

*Definition 15. *(see [38]). Let be an SVTNN and let , , and be any -cut set, -cut set, and -cut set of the SVNN , respectively. Then, we have the following.(1)The ambiguities of the SVTNN for -cut set, represented by , are given asâ€‰where , , and is nondecreasing and monotonic of .(2)The ambiguities of the SVTNN for -cut set, represented by , are given asâ€‰where , , and is nondecreasing and monotonic of .(3)The ambiguities of the SVTNN for -cut set, represented by , are given asâ€‰where , a, and is nondecreasing and monotonic of .Here, the weighting functions , , and can be supposed according to the decision-making model nature. Suppose that , , and *.*

Let be an SVTNN. Then the value and ambiguity indices, using the above descriptions, are constructed as

##### 3.2. A Ranking Approach of an SVTNN Based on Value and Ambiguity Indices

This 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].

*Definition 16. *Let be an SVTNN. The weighted value ambiguity index for an SVTNN is given aswith .

*Definition 17. *Let and be two SVTNNs and let . For the weighted value ambiguity index of the SVTNNs and , the ranking order of and is given as follows:(1)if , then (2)if , then (3)if , then where â€śâ€ť and â€śâ€ť are neutrosophic versions of the order relations â€śâ€ť and â€śâ€ť in the real line, respectively.

#### 4. Constrained Bimatrix Games with SVTNNs Payoffs and Solution Method

Let us consider the constrained bimatrix game with SVTNNs payoffs. Suppose that and are pure strategies sets for two players I and II, respectively. When player II selects pure strategy and player I selects pure strategy , at the situation , player II gains payoff and player I gains payoff, which are expressed with SVTNNs as and , where each and are SVTNNs defined as above. The mixed strategies vectors are represented as and , where and are probabilities for both players selecting their pure strategies and , respectively. The mixed strategies and are affiliated with the strategies sets (convex polyhedron) which are described by some inequalities and equations. Let represent the strategy constraint set of player I, where , , and *e* is a positive integer. Let express the strategy constraint set of player II, where , , and *b* is a positive integer. Note that contains , since is equivalent to and . Similarly, contains . In the sequel, the above SVTNN constrained bimatrix game is simply denoted by for short.

Without loss of generality, suppose that both players I and II, respectively, select mixed strategies and in order to maximize their own payoffs; then their expected payoffs can be obtained as follows:

*Definition 18. *(see [40]). If satisfies the following conditions:for any mixed strategies and , then and are called equilibrium strategies, and and are called equilibrium values of players I and II, respectively.

Theorem 1. *If and are the optimal solutions of the following linear programming problems:respectively, then and are equilibrium strategies of the SVTNN constrained bimatrix game , and and are equilibrium values of players I and II, respectively.*

*Proof:. *The 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 in (21) and in (22). The two players may allow some violations on the set of constraints and .

Therefore, the equilibrium strategies and and equilibrium values and of the SVTNN constrained bimatrix games are equal to the optimal values and optimal solutions of (23 and 24) as follows:respectively, where , , and all the vectors elements of and are SVTNNs that are approximately equal to zero, which represent the maximum violations that the two players may permit on the set of constraints. The parameter is a real number.

Applying the ranking approach of SVTNNs, as proposed in Subsection 3.2, the SVTNN mathematical programming problems (equations (23) and (24)) can be transformed into the following parameterized programming problems:respectively.

For given , solving equations (25) and (26), we can obtain the optimal game values and and the optimal solutions and , respectively.

Theorem 2. *If and are optimal solutions of equations (25) and (26), respectively, then and are equilibrium strategies, and and are equilibrium values of both players for SVTNN constrained bimatrix games, respectively.*

#### 5. 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.

##### 5.1. The Company Development Strategy Choice Model

â€śWe consider two companies and (i.e., players I and II). In order to improve the two companies competitiveness, both players have two strategies: introducing the advanced equipment or and introducing the senior talent or . When player I chooses pure strategies and , he wants to invest 7 million and 5 million dollars, respectively. Due to a lack of fund, player I can invest up to 6.5 million dollars, which means that player I has a constraint, , when selecting strategy. Likewise, player II wants to invest 4 million and 6.5 million dollars when he chooses pure strategies and , respectively. However, due to a lack of fund, player II can invest up to 5.5 million dollars. Namely, player II has a constraint, , when choosing strategies.â€ť This is a typical SVTN constrained bimatrix game. According to the previous description of the matrix game model, the two playersâ€™ constrained strategy sets are given as follows:respectively. The SVTNNs payoff matrices of the two players are given by

The vectors of the constraints and the coefficient matrices are given by

Let the two players select and , respectively.

##### 5.2. The Solution Procedure

Applying the ranking approach presented in Section 3 to the SVTN constrained bimatrix game, we have

According to equations (25) and (26), we can formulate the optimization problems with four parameters , and as follows:

For different values , and , the equilibrium strategies and the equilibrium values of both players can be obtained by solving equations (31) and (32), as depicted in Tables 1â€“12.

It can be easily seen from Table 1 that when , and , the equilibrium value and the equilibrium strategy for player I are and , respectively; and the equilibrium value and the equilibrium strategy for player II are and , respectively. The results indicate that different optimal solutions can be obtained for different values of , and . Thus, it is essential to take all the parameters into consideration.

##### 5.3. Comparison Analysis

In this subsection, the proposed ranking approach is compared with three other approaches that were introduced by Khalifa [41], Ye [42], and Garai et al. [43].

We compare our results with those of Khalifa [41], where a score function is described by

Here, expresses an SVTNN. Based on this score function, we obtain a set of linear optimization models as follows:

Using the Simplex technique, we can obtain that the equilibrium value and the equilibrium strategy for player I are and , respectively; and the equilibrium value and the equilibrium strategy for player II are and , respectively, although this approach provides the same optimal solutions as our results.

We compare our results with those of Jun Ye [42], where the score function is given by

Here, expresses an SVTNN. Based on this score function, we obtain the following mathematical programming models:

Using the Simplex technique, we can obtain the equilibrium value and the equilibrium strategy for player I as and , respectively; and the equilibrium value and the equilibrium strategy for player II are and , respectively, although this approach provides the same optimal solutions as our results.

Finally, we compare our results with those of Garai et al. [43], where the ranking function is described by

Here, represents an SVTNN. Based on this ranking function, we can get a set of optimization models as follows: