Fuzzy Payoffs Research Articles

Correlated Equilibrium in Bimatrix Games with Fuzzy Payoffs

Correlated Equilibrium in Bimatrix Games with Fuzzy Payoffs

Interactive Fuzzy Approaches for Solving Multiobjective Two-Person Zero-Sum Games

In this paper, we consider multiobjective two-person zero-sum games with vector payoffs and vector fuzzy payoffs. We translate such games into the corresponding multiobjective programming problems and introduce the pessimistic Pareto optimal solution concept by assuming that a player supposes the opponent adopts the most disadvantage strategy for the self. It is shown that any pessimistic Pareto optimal solution can be obtained on the basis of linear programming techniques even if the membership functions for the objective functions are nonlinear. Moreover, we propose interactive algorithms based on the bisection method to obtain a pessimistic compromise solution from among the set of all pessimistic Pareto optimal solutions. In order to show the efficiency of the proposed method, we illustrate interactive processes of an application to a vegetable shipment problem.

Matrix Games with Intuitionistic Fuzzy Pay-offs

In this paper two person zero sum matrix games are considered in which elements of payoff matrix are expressed with triangular intuitionistic fuzzy numbers (TIFNs). The purpose of this paper is to develop a solution methodology for solving such types of matrix games. For this, a pair of auxiliary intuitionistic fuzzy programming (IFP) models for each player are established. Then these two IFP problems are transformed into conventional crisp linear programming problems by defining a suitable defuzzification function to determine optimal strategies for each player and value of the game. The proposed method in this paper is illustrated with a numerical example to show the validity and applicability.

On Nash Equilibrium Strategy of Two-person Zero-sum Games with Trapezoidal Fuzzy Payoffs

In this paper, we investigate Nash equilibrium strategy of two-person zero-sum games with fuzzy payoffs. Based on fuzzy max order, Maeda and Cunlin constructed several models in symmetric triangular and asymmetric triangular fuzzy environment, respectively. We extended their models in trapezoidal fuzzy environment and proposed the existence of equilibrium strategies for these models. We also established the relation between Pareto Nash equilibrium strategy and parametric bi-matrix game. In addition, numerical examples are presented to find Pareto Nash equilibrium strategy and weak Pareto Nash equilibrium strategy from bi-matrix game.

Solving Matrix Games with I-fuzzy Payoffs: Pareto-optimal Security Strategies Approach

A recent research work of Clemente et al. [12] on Pareto-optimal security strategies (POSS) in matrix games with fuzzy payoffs is extended to I-fuzzy scenario. Besides, the membership and the non-membership functions of the I-fuzzy values for both players are obtained by employing the technique of multiobjective optimization. The presented approach provides an efficient solution to a class of I-fuzzy matrix games with piecewise linear membership and non-membership functions. This class also includes I-fuzzy matrix games with triangular and trapezoidal I-fuzzy numbers as special cases. Further, POSS approach also provides an approximate solution to I-fuzzy matrix games with payoffs as general I-fuzzy numbers.

A methodology for matrix games with payoffs of triangular intuitionistic fuzzy number

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.

A fast approach to bimatrix games with intuitionistic fuzzy payoffs.

The aim of this paper is to develop an effective method for solving bimatrix games with payoffs of intuitionistic fuzzy value. Firstly, bimatrix game model with intuitionistic fuzzy payoffs (IFPBiG) was put forward. Secondly, two kinds of nonlinear programming algorithms were discussed with the Nash equilibrium of IFPBiG. Thirdly, Nash equilibrium of the algorithm was proved by the fixed point theory and the algorithm was simplified by linear programming methods. Finally, an example was solved through Matlab; it showed the validity, applicability, and superiority.

An Effective Methodology for Solving Matrix Games With Fuzzy Payoffs

Of the different types of games, the matrix games with fuzzy payoffs have been extensively discussed. Two major kinds of solution methods have been devised. One is the defuzzification approach based on ranking functions. Another is the two-level linear programming method which can obtain membership functions of players' fuzzy values (or gain floor and loss ceiling). These methods cannot always ensure that players' fuzzy/defuzzified values have a common value. The aim of this paper is to develop an effective methodology for solving matrix games with payoffs expressed by trapezoidal fuzzy numbers (TrFNs). In this methodology, we introduce the concept of Alpha-matrix games and prove that players' fuzzy values are always identical, and hereby, any matrix game with payoffs expressed by TrFNs has a fuzzy value, which is also a TrFN. The upper and lower bounds of any Alpha-cut of the fuzzy value and the players' optimal strategies are easily obtained through solving the derived four linear programming problems with the upper and lower bounds of Alpha-cuts of the fuzzy payoffs. In particular, the fuzzy value can be explicitly estimated through solving the auxiliary linear programming with data taken from the 1-cut and 0-cut of the fuzzy payoffs. The proposed method in this paper is illustrated with a real example and compared with other methods to show validity and applicability.

Application of Atanassov's I-fuzzy Set Theory to Matrix Games with Fuzzy Goals and Fuzzy Payoffs

We aim to extend some results in [6, 7, 8, 21] on two person zero sum matrix games (TPZSMG) with fuzzy goals and fuzzy payoffs to I-fuzzy scenario. Because the payoffs of the matrix game are fuzzy numbers, the aspiration levels of the players are fuzzy as well. It is reasonable to believe that there is some indeterminacy in estimating the aspiration levels of both players from their respective expected pay offs. This situation is modeled in the game using Atanassov's I-fuzzy set theory. A new solution concept is proposed for such games and a procedure is outlined to obtain the degrees of suitability of the aspiration levels for each of the two players.

Credibilistic extensive game with fuzzy payoffs

This paper considers a finite extensive game with fuzzy payoffs. Within the framework of credibility theory, expected value criterion, optimistic value criterion and pessimistic value criterion are introduced to define the behaviors of players in different decision situations. Accordingly, three types of credibilistic equilibria and credibilistic subgame perfect equilibria for the fuzzy extensive game are proposed, respectively. Moreover, theorems are given to confirm the existence of these credibilistic equilibria in the fuzzy extensive game. Finally, three examples are given to demonstrate the significance of these new concepts.

Set-valued cooperative games with fuzzy payoffs. The fuzzy assignment game

In this paper we study cooperative games with fuzzy payoffs. The main advantage of the approach presented is the incorporation into the analysis of the problem of ambiguity inherent in many real-world collective decision situations. We propose extensions of core concepts which maintain the fuzzy nature of allocations, and lead to a more satisfactory study of the problem within the fuzzy context. Finally, we illustrate the extended core concepts and the approach to obtain the corresponding allocations through the analysis of assignment games with uncertain profits.

Water Resources Allocation Using a Cooperative Game with Fuzzy Payoffs and Fuzzy Coalitions

In this paper, two fuzzy cooperative games are utilized for modeling equitable and efficient water allocation among water users in both inter-basin and intra-basin water allocation problems. The proposed all-inclusive water allocation approach consists of three main steps, following Sadegh et al. (Water Resour Manage 24(12):2991–2310, 2010). In the first step, an initial water allocation is carried out using an optimization model taking into account an equity criterion. In the second step, the water users form crisp coalitions with fuzzy characteristic functions to increase the total net benefit of the system and also their own benefits. In the methodology used in this step, the water users do not need to have exact information about their payoffs and they can evaluate their payoffs as fuzzy numbers. In the second step, based on the Hukuhara difference of fuzzy numbers, optimum water allocation strategies are determined using a game with fuzzy characteristic function. In the third step, we applied another methodology which considers a class of fuzzy games with fuzzy characteristic functions and also fuzzy coalitions for water allocation. The methodology of this step is on the basis of the Hukuhara difference and the Choquet integral. The usefulness of the mentioned methodologies is studied by applying them to three defined real life scenarios in a case study of water allocation in Iran. The results showed that the proposed methodologies are professionally appropriate to real-world uncertain problems of equitable and economic inter-basin and intra-basin water resources allocations.

A fast approach to compute fuzzy values of matrix games with payoffs of triangular fuzzy numbers

The aim of this paper is to develop an effective method for solving matrix games with payoffs of triangular fuzzy numbers (TFNs) which are arbitrary. In this method, it always assures that players’ gain-floor and loss-ceiling have a common TFN-type fuzzy value and hereby any matrix game with payoffs of TFNs has a TFN-type fuzzy value. Based on duality theorem of linear programming (LP) and the representation theorem for fuzzy sets, the mean and the lower and upper limits of the TFN-type fuzzy value are easily computed through solving the derived LP models with data taken from 1-cut set and 0-cut set of fuzzy payoffs. Hereby the TFN-type fuzzy value of any matrix game with payoffs of TFNs can be explicitly obtained. Moreover, we can easily compute the upper and lower bounds of any Alfa-cut set of the TFN-type fuzzy value for any matrix game with payoffs of TFNs and players’ optimal mixed strategies through solving the derived LP models at any specified confidence level Alfa. The proposed method in this paper is demonstrated with a numerical example and compared with other methods to show the validity, applicability and superiority.

The Banzhaf Value for Fuzzy Games with Fuzzy Payoffs

The Banzhaf Value for Fuzzy Games with Fuzzy Payoffs

A Fictitious Play Algorithm for Matrix Games with Fuzzy Payoffs

Fuzzy matrix games, specifically two‐person zero‐sum games with fuzzy payoffs, are considered. In view of the parametric fuzzy max order relation, a fictitious play algorithm for finding the value of the game is presented. A numerical example to demonstrate the presented algorithm is also given.

A Novel Algorithm For Finding Equilibrium Strategy In Two Person Zero Sum Game With Fuzzy Strategy Sets And Payoffs

This paper investigates a two person zero sum matrix game in which the payoffs and strategy are characterized as random fuzzy variables. Using the operations of triangular fuzzy numbers, the fuzzy payoffs for all synthetic outcomes are calculated. Based on random fuzzy expected value operator, a random fuzzy expected minimax equilibrium strategy to the game is defined. After that, based on the constraints, the feasible strategy string sets of the players for multi conflict situations are constructed. Then an iterative algorithm based on random fuzzy simulation is designed to seek the minimax equilibrium strategy. Using a linear ranking function, the aggregation model can be solved by transforming it into a crisp bimatrix game. Then, the fuzzy synthetic aggregation model is established and solved by transforming it into a crisp bimatrix game. Finally, a military example is provided to illustrate the practicality and effectively of the model.

Pareto-optimal security strategies in matrix games with fuzzy payoffs

We present a new methodology for the analysis of fuzzy payoff matrix games. The main difficulty that appears in the study of these games is the comparison between the payoff values associated to the strategies of the players because these payoffs are fuzzy quantities. Our approach does not transform the fuzzy payoffs to crisp numbers via standard defuzzyfication but we use standard fuzzy orders which allows us to find solutions within the same space of fuzzy numbers. Moreover, we provide a method to solve these games finding equivalent fuzzy linear programs whose maximal solutions give the solutions of the games.

Solutions for fuzzy matrix games

In this paper, we deal with two-person zero-sum games with fuzzy payoffs and fuzzy goals. We have presented two models for studying two-person zero-sum matrix games with fuzzy payoffs and fuzzy goals. We assume that each player has a fuzzy goal for each of the payoffs. We obtained that the fuzzy relation approach and the max–min solution are equivalent.

Coalitional game with fuzzy payoffs and credibilistic core

As an important branch of game theory, coalitional game deals with situations that involve cooperation among the players. This paper deals with this topic further by incorporating the fuzzy payoff information. Based on the credibility theory, we introduce two decision criteria to define the preferences of players, which leads to two definitions of credibilistic cores--the solution of coalitional game with fuzzy transferable payoffs. Meanwhile, we give a sufficient and necessary condition to ensure non-emptiness of the credibilistic cores. Finally, an example is provided for illustrating the usefulness of the theory developed in this paper.

Two-Person Non Zero-Sum Bimatrix Game with Fuzzy Payoffs and Its Equilibrium Strategy

In this paper, we consider fuzzy bimatrix games with fuzzy payoffs. Based on fuzzy max order, for such games,we define three kinds of concepts of minim ax equilibrium strategies. Some basic results obtained.