Solving Game Problems Involving Heptagonal and Hendecagonal Fuzzy Payoffs

The main aim of this paper is to deal with a two person zero sum game involving fuzzy payoff matrix comprising of heptagonal and hendecagonal fuzzy numbers. Ranking of fuzzy numbers is a hard task. Many methods have been proposed to rank different fuzzy numbers such as triangular, trapezoidal, hexagonal, octagonal etc. In this paper, a matrix game is considered whose payoffs are heptagonal and hendecagonal fuzzy numbers and ranking method is used to solve the matrix game. By using this proposed approach the fuzzy game problem is converted into crisp problem and then solved by applying the usual game problem techniques. The validity of proposed method is illustrated with the help of two different practical examples; one where the two companies are venturing into online restaurant business and the other where the two political parties with conflicting interests during elections are competing with each other.

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A Fast Approach to Solve Matrix Games with Payoffs of Trapezoidal Fuzzy Numbers

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595916500470 ◽

2016 ◽

Vol 33 (06) ◽

pp. 1650047 ◽

Cited By ~ 2

Author(s):

Sanjiv Kumar ◽

Ritika Chopra ◽

Ratnesh R. Saxena

Keyword(s):

Linear Programming ◽

Linear Programming Problem ◽

Fuzzy Numbers ◽

Optimization Problems ◽

Fuzzy Linear Programming ◽

Matrix Game ◽

Matrix Games ◽

Trapezoidal Fuzzy Numbers ◽

Fast Approach ◽

The Matrix

The aim of this paper is to develop an effective method for solving matrix game with payoffs of trapezoidal fuzzy numbers (TrFNs). The method always assures that players’ gain-floor and loss-ceiling have a common TrFN-type fuzzy value and hereby any matrix game with payoffs of TrFNs has a TrFN-type fuzzy value. The matrix game is first converted to a fuzzy linear programming problem, which is converted to three different optimization problems, which are then solved to get the optimum value of the game. The proposed method has an edge over other method as this focuses only on matrix games with payoff element as symmetric trapezoidal fuzzy number, which might not always be the case. A numerical example is given to illustrate the method.

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Games in Extensive Form

Noncooperative Game Theory ◽

10.23943/princeton/9780691175218.003.0007 ◽

2017 ◽

Author(s):

João P. Hespanha

Keyword(s):

Saddle Point ◽

Matrix Game ◽

Extensive Form ◽

Multi Stage ◽

Form Representation ◽

Other Information ◽

The Matrix ◽

Recursive Computation ◽

Zero Sum ◽

Computation Of Equilibria

This chapter discusses a number of key concepts for extensive form game representation. It first considers a matrix that defines a zero-sum matrix game for which the minimizer has two actions and the maximizer has three actions and shows that the matrix description, by itself, does not capture the information structure of the game and, in fact, other information structures are possible. It then describes an extensive form representation of a zero-sum two-person game, which is a decision tree, the extensive form representation of multi-stage games, and the notions of security policy, security level, and saddle-point equilibrium for a game in extensive form. It also explores the matrix form for games in extensive form, recursive computation of equilibria for single-stage games, feedback games, feedback saddle-point for multi-stage games, and recursive computation of equilibria for multi-stage games. It concludes with a practice exercise with the corresponding solution, along with additional exercises.

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An Algorithmic Procedure for Finding Nash Equilibrium

GANIT Journal of Bangladesh Mathematical Society ◽

10.3329/ganit.v40i1.48196 ◽

2020 ◽

Vol 40 (1) ◽

pp. 71-85

Author(s):

HK Das ◽

T Saha

Keyword(s):

Nash Equilibrium ◽

Nash Equilibria ◽

Payoff Matrix ◽

Matrix Game ◽

Perfect Equilibrium ◽

Repeated Use ◽

Definition Of ◽

Zero Sum ◽

Mathematical Construction ◽

Algorithmic Procedure

This paper proposes a heuristic algorithm for the computation of Nash equilibrium of a bi-matrix game, which extends the idea of a single payoff matrix of two-person zero-sum game problems. As for auxiliary but making the comparison, we also introduce here the well-known definition of Nash equilibrium and a mathematical construction via a set-valued map for finding the Nash equilibrium and illustrates them. An important feature of our algorithm is that it finds a perfect equilibrium when at the start of all actions are played. Furthermore, we can find all Nash equilibria of repeated use of this algorithm. It is found from our illustrative examples and extensive experiment on the current phenomenon that some games have a single Nash equilibrium, some possess no Nash equilibrium, and others had many Nash equilibria. These suggest that our proposed algorithm is capable of solving all types of problems. Finally, we explore the economic behaviour of game theory and its social implications to draw a conclusion stating the privilege of our algorithm. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 71-85

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Matrix Game with Payoffs Represented by Triangular Dual Hesitant Fuzzy Numbers

International Journal of Computers Communications & Control ◽

10.15837/ijccc.2020.3.3854 ◽

2020 ◽

Vol 15 (3) ◽

Author(s):

Zhihui Yang ◽

Yang Song

Keyword(s):

Decision Making ◽

Fuzzy Numbers ◽

Decision Makers ◽

Programming Models ◽

Decision Making Process ◽

Matrix Game ◽

Fuzzy Matrix ◽

The Matrix ◽

Definition Of

Matrix Game with Payoffs RepresentedDue to the complexity of information or the inaccuracy of decision-makers’ cognition, it is difficult for experts to quantify the information accurately in the decision-making process. However, the integration of the fuzzy set and game theory provides a way to help decision makers solve the problem. This research aims to develop a methodology for solving matrix game with payoffs represented by triangular dual hesitant fuzzy numbers (TDHFNs). First, the definition of TDHFNs with their cut sets are presented. The inequality relations between two TDHFNs are also introduced. Second, the matrix game with payoffs represented by TDHFNs is investigated. Moreover, two TDHFNs programming models are transformed into two linear programming models to obtain the numerical solution of the proposed fuzzy matrix game. Furthermore, a case study is given to to illustrate the efficiency and applicability of the proposed methodology. Our results also demonstrate the advantage of the proposed concept of TDHFNs.

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Benchmarking of Problems and Solvers: a Game-Theoretic Approach

Foundations of Computing and Decision Sciences ◽

10.2478/fcds-2019-0008 ◽

2019 ◽

Vol 44 (2) ◽

pp. 137-150

Author(s):

Joseph Gogodze

Keyword(s):

Payoff Matrix ◽

Matrix Game ◽

Theoretic Approach ◽

Mixed Strategies ◽

Game Theoretic ◽

Assessment Matrix ◽

Zero Sum ◽

Game Theoretic Approach

Abstract In this note, we propose a game-theoretic approach for benchmarking computational problems and their solvers. The approach takes an assessment matrix as a payoff matrix for some zero-sum matrix game in which the first player chooses a problem and the second player chooses a solver. The solution in mixed strategies of this game is used to construct a notionally objective ranking of the problems and solvers under consideration. The proposed approach is illustrated in terms of an example to demonstrate its viability and its suitability for applications.

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Solving Rectangular Fuzzy Games through

Open Computer Science ◽

10.1515/comp-2017-0006 ◽

2017 ◽

Vol 7 (1) ◽

pp. 46-50 ◽

Cited By ~ 1

Author(s):

Arindam Chaudhuri

Keyword(s):

Set Theory ◽

Operations Research ◽

Fuzzy Numbers ◽

Algebraic Method ◽

Payoff Matrix ◽

Mixed Strategies ◽

Maximin Principle ◽

Fuzzy Games ◽

Fuzzy Game ◽

Pure Strategies

Abstract Fuzzy set theory has been applied in many fields such as operations research, control theory and decision sciences. In particular, an application of this theory in decision making problems has a remarkable significance. In this paper, we consider a solution of rectangular fuzzy game with pay-off as imprecise numbers instead of crisp numbers viz., interval and LR-type trapezoidal fuzzy numbers. The solution of such fuzzy games with pure strategies by minimax-maximin principle is discussed. The algebraic method to solve 2 × 2 fuzzy games without saddle point by using mixed strategies is also illustrated. Here m × n payoff matrix is reduced to 2 × 2 pay-off matrix by dominance method. This fact is illustrated by means of numerical example.

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