scholarly journals Solving Rectangular Fuzzy Games through

2017 ◽  
Vol 7 (1) ◽  
pp. 46-50 ◽  
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.

scholarly journals Solving Game Problems Involving Heptagonal and Hendecagonal Fuzzy Payoffs

2019 ◽  
Vol 8 (9) ◽  
pp. 2114-2120
Keyword(s):  
Political Parties ◽  
Fuzzy Numbers ◽  
Game Problem ◽  
Payoff Matrix ◽  
Matrix Game ◽  
The Matrix ◽  
Fuzzy Game ◽  
Fuzzy Payoff ◽  
Zero Sum ◽  
Ranking Of Fuzzy Numbers

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.

MONTE CARLO METHODS IN FUZZY GAME THEORY

2007 ◽  
Vol 03 (02) ◽  
pp. 259-269 ◽  
Author(s):  
AREEG ABDALLA ◽  
JAMES BUCKLEY
Keyword(s):  
Game Theory ◽  
Monte Carlo ◽  
Monte Carlo Method ◽  
Monte Carlo Methods ◽  
Mixed Strategy ◽  
Minimax Theorem ◽  
Approximate Solutions ◽  
Mixed Strategies ◽  
Fuzzy Game ◽  
Zero Sum

In this paper, we consider a two-person zero-sum game with fuzzy payoffs and fuzzy mixed strategies for both players. We define the fuzzy value of the game for both players [Formula: see text] and also define an optimal fuzzy mixed strategy for both players. We then employ our fuzzy Monte Carlo method to produce approximate solutions, to an example fuzzy game, for the fuzzy values [Formula: see text] for Player I and [Formula: see text] for Player II; and also approximate solutions for the optimal fuzzy mixed strategies for both players. We then look at [Formula: see text] and [Formula: see text] to see if there is a Minimax theorem [Formula: see text] for this fuzzy game.

Exponential Families and Game Dynamics

1982 ◽  
Vol 34 (2) ◽  
pp. 374-405 ◽  
Author(s):  
Ethan Akin
Keyword(s):  
Game Theory ◽  
Evolutionary Game ◽  
Payoff Matrix ◽  
Exponential Families ◽  
Von Neumann ◽  
Rational Player ◽  
Large Populations ◽  
Pure Strategies ◽  
Classical Game ◽  
Classical Game Theory

A symmetric game consists of a set of pure strategies indexed by {0, …, n} and a real payoff matrix (aij). When two players choose strategies i and j the payoffs are aij and aji to the i-player and j-player respectively. In classical game theory of Von Neumann and Morgenstern [16] the payoffs are measured in units of utility, i.e., desirability, or in units of some desirable good, e.g. money. The problem of game theory is that of a rational player who seeks to choose a strategy or mixture of strategies which will maximize his return. In evolutionary game theory of Maynard Smith and Price [13] we look at large populations of game players. Each player's opponents are selected randomly from the population, and no information about the opponent is available to the player. For each one the choice of strategy is a fixed inherited characteristic.

The behaviour of strategy-frequencies in Parker's model

1980 ◽  
Vol 12 (01) ◽  
pp. 5-7
Author(s):  
D. Gardiner
Keyword(s):  
Finite Number ◽  
Evolutionarily Stable Strategy ◽  
Pure Strategy ◽  
Payoff Matrix ◽  
Stable Strategy ◽  
Expected Payoff ◽  
Pure Strategies ◽  
Image Position ◽  
Evolutionarily Stable

Parker's model (or the Scotch Auction) for a contest between two competitors has been studied by Rose (1978). He considers a form of the model in which every pure strategy is playable, and shows that there is no evolutionarily stable strategy (ess). In this paper, in order to discover more about the behaviour of strategies under the model, we shall assume that there are only a finite number of playable pure strategies I 1, I 2, ···, I n where I j is the strategy ‘play value m j ′ and m 1 < m 2 < ··· < m n . The payoff matrix A for the contest is then given by where V is the reward for winning the contest, C is a constant added to ensure that each entry in A is non-negative (see Bishop and Cannings (1978)), and E[I i , I j ] is the expected payoff for playing I i against I j . We also assume that A is regular (Taylor and Jonker (1978)) i.e. that all its rows are independent.

Evaluation of Quantified Statements Using Gradual Numbers

2011 ◽  
pp. 246-269 ◽  
Author(s):  
Ludovic Liétard ◽  
Daniel Rocacher
Keyword(s):  
Decision Making ◽  
Expert Systems ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Relational Databases ◽  
Fuzzy Numbers ◽  
Theoretical Background ◽  
Definition Of ◽  
Flexible Querying

This chapter is devoted to the evaluation of quantified statements which can be found in many applications as decision making, expert systems, or flexible querying of relational databases using fuzzy set theory. Its contribution is to introduce the main techniques to evaluate such statements and to propose a new theoretical background for the evaluation of quantified statements of type “Q X are A” and “Q B X are A.” In this context, quantified statements are interpreted using an arithmetic on gradual numbers from Nf, Zf, and Qf. It is shown that the context of fuzzy numbers provides a framework to unify previous approaches and can be the base for the definition of new approaches.

scholarly journals Enhanced Two-Step Method via Relaxed Order ofα-Satisfactory Degrees for Fuzzy Multiobjective Optimization

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Na Wang ◽  
Chaofang Hu ◽  
Wuxi Shi ◽  
Chunbo Xiu ◽  
Yimei Chen
Keyword(s):  
Multiobjective Optimization ◽  
Set Theory ◽  
Optimization Problem ◽  
Fuzzy Numbers ◽  
Fuzzy Relations ◽  
Preemptive Priority ◽  
Step Method ◽  
Satisfactory Solution ◽  
Order Constraint ◽  
Fuzzy Parameters

An enhanced two-step method via relaxed order of satisfactory degrees for fuzzy multiobjective optimization is proposed in this paper. By introducing the concept of fuzzy numbers andα-level set theory, fuzzy parameters are taken as variables, and all the objectives are transformed into fuzzy goals involving three fuzzy relations. The order ofα-satisfactory degrees which means the objectives with higher priority achieving higher satisfactory degree is applied to model preemptive priority requirement. This strict order constraint is relaxed by priority variable to find the preferred solution satisfying optimization and priority. The original optimization problem is divided into two steps to be solved iteratively. The M-α-Pareto optimality of the solution is ensured, and the satisfactory solution can be acquired by regulating the slack parameterΔδor changingα. The numerical examples demonstrate the power of the proposed method.

scholarly journals Fuzzy Insurance

1990 ◽  
Vol 20 (1) ◽  
pp. 33-55 ◽  
Author(s):  
Jean Lemaire
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Fuzzy Set ◽  
Life Insurance ◽  
Fuzzy Set Theory ◽  
Fuzzy Numbers ◽  
Fuzzy Decision Making ◽  
Fuzzy Decision ◽  
Basic Concepts ◽  
Definition Of

AbstractFuzzy set theory is a recently developed field of mathematics, that introduces sets of objects whose boundaries are not sharply defined. Whereas in ordinary Boolean algebra an element is either contained or not contained in a given set, in fuzzy set theory the transition between membership and non-membership is gradual. The theory aims at modelizing situations described in vague or imprecise terms, or situations that are too complex or ill-defined to be analysed by conventional methods. This paper aims at presenting the basic concepts of the theory in an insurance framework. First the basic definitions of fuzzy logic are presented, and applied to provide a flexible definition of a “preferred policyholder” in life insurance. Next, fuzzy decision-making procedures are illustrated by a reinsurance application, and the theory of fuzzy numbers is extended to define fuzzy insurance premiums.

On the Foundations of Nash Equilibrium

1996 ◽  
Vol 12 (1) ◽  
pp. 67-88 ◽  
Author(s):  
Hans Jørgen Jacobsen
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Cooperative Game Theory ◽  
Mixed Strategy ◽  
Pure Strategy ◽  
Mixed Strategies ◽  
Positive Theory ◽  
Equilibrium Concept ◽  
Strategy Equilibrium ◽  
Pure Strategies

The most important analytical tool in non-cooperative game theory is the concept of a Nash equilibrium, which is a collection of possibly mixed strategies, one for each player, with the property that each player's strategy is a best reply to the strategies of the other players. If we do not go into normative game theory, which concerns itself with the recommendation of strategies, and focus instead entirely on the positive theory of prediction, two alternative interpretations of the Nash equilibrium concept are predominantly available.In the more traditional one, a Nash equilibrium is a prediction of actual play. A game may not have a Nash equilibrium in pure strategies, and a mixed strategy equilibrium may be difficult to incorporate into this interpretation if it involves the idea of actual randomization over equally good pure strategies. In another interpretation originating from Harsanyi (1973a), see also Rubinstein (1991), and Aumann and Brandenburger (1991), a Nash equilibrium is a ‘consistent’ collection of probabilistic expectations, conjectures, on the players. It is consistent in the sense that for each player each pure strategy, which has positive probability according to the conjecture about that player, is indeed a best reply to the conjectures about others.

Research on Fuzzy Fault Diagnosis of the Equipments

2013 ◽  
Vol 411-414 ◽  
pp. 1484-1487
Author(s):  
Ji Yang Qi ◽  
Li Na Ren ◽  
Shan Ping Ning ◽  
Yu Fu
Keyword(s):  
Fault Diagnosis ◽  
Set Theory ◽  
Fuzzy Set Theory ◽  
Fuzzy Numbers ◽  
Triangular Fuzzy Numbers ◽  
Comparison Matrix ◽  
Fuzzy Diagnosis ◽  
The Given ◽  
Pair Wise Comparison ◽  
Pair Wise Comparison Matrix

The paper introduces a method of fault diagnosis using fuzzy set theory. In the paper, the principle that a fault symptom either exists or doesnt exist is abandoned. A crisp number between 0 and 1 is used to denote the degree of fault symptom, by which the fault symptom vector is constructed. For every kind of fault symptom, a fuzzy pair-wise comparison matrix is constructed. The elements of the pair-wise comparison matrix are triangular fuzzy numbers which denote the qualitative comparisons between the membership values of the given fault symptom with the reference to a pair of possible faults respectively. The least logarithm squares method is applied to determine the membership of the fault symptom with respect to each fault, and then the fuzzy diagnosis matrix is constructed. A simple weighted addition is used to calculate the fault vector based on the fuzzy diagnosis matrix and the fault symptom vector. Center of area is used to determine the best non-fuzzy performance value of the fuzzy number, according to which the fuzzy numbers can be ranked. The ordering of all the possible faults based on the fault symptoms is determined. At the end of the paper, an example is used to demonstrate the procedure of fuzzy fault diagnosis.

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