Bimatrix game example

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. 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. Garai et al. Broumi et al. Mullai et al. Garg et al. Leyva et al. Garg [ 33 ] presented nonlinear programming approach for solving MADM model with interval neutrosophic parameters.

Sun et al. 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. 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 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. Definition 1. A fuzzy number is said to be a trapezoidal fuzzy number TFN , if its membership function is given by.

Definition 2. 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. The values , respectively, express the falsity membership, indeterminacy membership, and truth membership degree of y to. Definition 4. An - cut set of SVNS , a crisp subset of , is given by where , , , and. Definition 5. An SVNS is called neutrosophic normal, if there exist at least three points such that.

Definition 6. An SVNS is said to be neutrosophic convex, if, and , the following conditions are satisfied: i ii iii. Definition 7. 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. An SVTNN is a special neutrosophic set on the set of real numbers , whose truth membership, indeterminacy membership, and falsity membership are represented as respectively. Definition 9. Then, 1 2 3. Now, just pick any other spots for player 2's best responses choosing from the options top, middle, and bottom , the only key is that there can be no intersections. Or, anything else, as long as the two have no intersections.

Then just fill in any numbers that satisfy the 1s the above matrices being the best responses. Is a 3x3 bimatrix game with no pure strategy equilibria. But this method could easily generate infinitely many. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams?

Learn more. Example of a completely mixed bimatrix game. Ask Question. Asked 4 years, 9 months ago. Active 4 years ago. Viewed times. Perhaps you have in mind some additional constraints. Anbalagan, S. Norin, R. Savani, and A. Fearnley, T.

Igwe, and R. Goldberg, P. Krysta, and C. Hortala-Vallve and A. Sandholm, A. Gilpin, and V.