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The purpose of this article is to present a novel biobjective biform game framework for optimizing strategy selection of nodes (agents) in bilateral link network formation under dual objectives of profit and link security. First, in the first part of the noncooperative game, we construct various possible strategy combinations (situations) of all players (nodes) in the network, which define the competitive environment in the second part of the cooperative game. Second, in the second part, we derive coalitions’ payoffs under the two objectives of profit and link security and propose the biobjective Banzhaf values to allocate the corresponding cooperative games in all strategy situations. Then, a biobjective noncooperative game is formed on the basis of the obtained biobjective Banzhaf values. Third, we redefine Pareto–Nash equilibriums (in the sense of pure strategies) and obtain the Pareto–Nash equilibriums of the biobjective noncooperative game in the first part. Subsequently, the choice of the obtained Pareto–Nash equilibriums is regarded as a biobjective group decision-making problem, which is solved by employing the redefined Borda choice function method with objective weights. Hence, we can obtain the optimal strategy choices of all the nodes (players) for the entire bilateral link network formation. Finally, numerical examples and comparative analysis are given to verify the validity and applicability of the proposed models and methods. [ABSTRACT FROM AUTHOR]

Uncertainty provides an importance in decision making problems like game theory. Several types of uncertainties occur in the literature as fuzzy, soft, rough, interval, etc., and game theory is treated by these types of uncertainties and vagueness by game theorists and others in different aspects. Neutrosophic set and logic are emerged nowadays as another type of uncertainty. Single-valued triangular neutrosophic numbers liberally assume the indeterminacy in choice of elements based upon decision makers' intuition, assumption, judgement, behaviour, evaluation and decision. Here, a new ranking approach is based on the (α , β , γ) -cut of single-valued triangular neutrosophic number and is applied on bi-matrix game theory. We compare our derived results with some previously defined score and accuracy functions and put some interesting and comparatively good results without any plausible fiasco. In this paper, our principal purpose is to validate and approve the proposed contemplations by applying it on illuminating real-life problems in neutrosophic realm through bi-matrix game theory. [ABSTRACT FROM AUTHOR]

Through adding a discounted factor, Shapley values of cooperative games are extended to interval-valued (IV) cooperative games. The main purpose of this paper is to propose the concept of IV discounted Shapley values and hereby develop a simplified method for computing IV discounted Shapley values for a class of IV cooperative games. In this method, we introduce the concept of discounted Shapley values of -cooperative games associated with IV cooperative games for any attitude factor and find a weaker coalition monotonicity-like condition so that the defined discounted Shapley values possess the monotonicity. Hence we can obtain the analytical formula of IV discounted Shapley value of any IV cooperative game with the coalition monotonicity-like through utilizing only the lower and upper bounds of coalitions’ IV values. The proposed method utilizes the weaker coalition monotonicity-like condition rather than the irreversible interval subtraction and hereby can effectively overcome some irrational issues. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a new methodology for solving matrix games with payoffs of Atanassov's intuitionistic fuzzy (I-fuzzy) numbers. In this methodology, we define the concepts of I-fuzzy numbers and the value-index and ambiguity-index and develop a difference-index-based ranking method, which is proven to be a total order. By doing this, the parameterized nonlinear programming models are derived from a pair of auxiliary I-fuzzy mathematical programming models, which are used to determine solutions of matrix games with payoffs of I-fuzzy numbers. The validity and applicability of the models and method proposed in this paper are illustrated with a practical example. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to develop a new ranking methodology for solving matrix games with intuitionistic fuzzy payoffs in terms of a value and ambiguity of triangular intuitionistic fuzzy numbers (TIFNs). In order to obtain an equilibrium solution of this matrix game, we construct two auxiliary TIFN programming models that are transformed into bi-objective linear programming models which are solved by a lexicographic methodology. The validity and applicability of the proposed methodology in this paper are illustrated with a numerical example. [ABSTRACT FROM AUTHOR]

Abstract: Constrained matrix games with payoffs of triangular fuzzy numbers (TFNs) are a type of matrix games with payoffs expressed by TFNs and sets of players’ strategies which are constrained. So far as we know, no study have yet been attempted for constrained matrix games with payoffs of TFNs since there is no effective way to simultaneously incorporate the payoffs’ fuzziness and strategies’ constraints into classical and/or fuzzy matrix game methods. The aim of this paper is to develop an effective methodology for solving constrained matrix games with payoffs of TFNs. In this methodology, we introduce the concepts of Alpha-constrained matrix games for constrained matrix games with payoffs of TFNs and the values. By the duality theorem of linear programming, it is proven that players’ gain-floor and loss-ceiling always have a common interval-type value and hereby any Alpha-constrained matrix game has an interval-type value. Moreover, using the representation theorem for the fuzzy set, it is proven that any constrained matrix game with payoffs of TFNs always has a TFN-type fuzzy value. The auxiliary linear programming models are derived to compute the lower and upper bounds of the interval-type value and optimal strategies of players for any Alpha-constrained matrix game. In particular, the mean and the lower and upper limits of the TFN-type fuzzy value of any constrained matrix game with payoffs of TFNs can be directly obtained through solving the derived three linear programming models with data taken from only 1-cut and 0-cut of payoffs. Hereby the TFN-type fuzzy value of any constrained matrix game with payoffs of TFNs are easily computed. The proposed method in this paper is compared with other methods and its validity and applicability are illustrated with a numerical example. [Copyright &y& Elsevier]

The aim of this paper is to study how to solve a type of matrix game with interval payoffs. In this paper, the interval inequality relations and the concept of solutions of the matrix game with interval payoffs are defined. Based on the fuzzy ranking index defined, the solution of the matrix game with interval payoffs can be obtained through solving a pair of bi-objective linear programming models derived from the constructed auxiliary interval programming models. It is shown that the models proposed in this paper extend those of the classical matrix game. The validity and applicability of the proposed methodology are illustrated with a numerical example. [ABSTRACT FROM PUBLISHER]

The aim of this note is to point out and correct some vital mistakes in the paper by P K Nayak and M Pal, "Linear programming technique to solve two person matrix (games with interval pay-offs). Asia-Pacific Journal of Operational Research, 26(2), 285-305". Lots of serious mistakes on the definitions, conclusions, models, methods, proofs and computing results have been corrected and modified in this note. We also indicate inappropriate formulations regarding their proposed linear programming models for solving generic matrix games with interval pay-offs and suggest a pair of linear programming models with any minimal acceptance degree of the interval inequality constraints which may be allowed to violate. The lexicographic method is suggested so that a rational and credible solution of the generic matrix game with interval pay-offs can be achieved. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to study how to solve a type of matrix games with payoffs of triangular fuzzy numbers. In this paper, the value of a matrix game with payoffs of triangular fuzzy numbers has been considered as a variable of the triangular fuzzy number. First, based on two auxiliary linear programming models of a classical matrix game and the operations of triangular fuzzy numbers, fuzzy optimization problems are established for two players. Then, based on the order relation of triangular fuzzy numbers the fuzzy optimization problems for players are decomposed into three-objective linear programming models. Finally, using the lexicographic method maximin and minimax strategies for players and the fuzzy value of the matrix game with payoffs of triangular fuzzy numbers can be obtained through solving two corresponding auxiliary linear programming problems, which are easily computed using the existing Simplex method for the linear programming problem. It has been shown that the models proposed in this paper extend the classical matrix game models. A numerical example is provided to illustrate the methodology. [ABSTRACT FROM AUTHOR]