scholarly journals A Fictitious Play Algorithm for Matrix Games with Fuzzy Payoffs

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Emrah Akyar
Keyword(s):  
Order Relation ◽  
Fictitious Play ◽  
Matrix Games ◽  
Fuzzy Matrix ◽  
Zero Sum Games ◽  
Numerical Example ◽  
Value Of The Game ◽  
Zero Sum

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 WEAKENED FORM OF FICTITIOUS PLAY IN TWO-PERSON ZERO-SUM GAMES

2000 ◽  
Vol 02 (04) ◽  
pp. 307-328 ◽  
Author(s):  
BEN VAN DER GENUGTEN
Keyword(s):  
Optimal Strategies ◽  
Fictitious Play ◽  
Large Games ◽  
Zero Sum Games ◽  
Game Matrix ◽  
Numerical Iteration ◽  
Speed Up ◽  
New Form ◽  
Zero Sum ◽  
Weakened Form

Fictitious play can be seen as a numerical iteration procedure for determining the value of a game and corresponding optimal strategies. Although convergence is slow, it needs only a modest computer storage. Therefore it seems to be a good way for analysing large games. In this paper we introduce a weakened form of fictitious play, where players at each stage do not have to make the best choice against the total of past choices of the other player but only an increasingly better one. Theoretical bounds for convergence are derived. Furthermore, it is shown that this new form can speed up convergence considerably in practice. It is seen that weakened fictitious play can be extended to models in which the game matrix itself becomes better known as the number of stages increases.

scholarly journals Strategic Management in Marketing: A Game Theoretic Approach

2021 ◽  
Vol 12 (6) ◽  
pp. 2518-2524
Author(s):  
Vinod Jangid, Gaurav Sharma, Ganesh Kumar
Keyword(s):  
Fuzzy Numbers ◽  
Status Quo ◽  
Theoretic Approach ◽  
Ranking Methods ◽  
Matrix Games ◽  
Fuzzy Matrix ◽  
Game Theoretic ◽  
Zero Sum ◽  
Determining Factor ◽  
Game Theoretic Approach

In marketing, a real-world dilemma emerging between two rivals, McDonald's and Burger King, is investigated. Both firms use three strategies: discounted pricing, status quo, and aggressive commercial. In such cases, ambiguity is a determining factor. To deal with confusion in payoffs, octagonal fuzzy numbers are used. To rank fuzzy numbers, the average of odd positions, average of even positions, and quartile deviations are used. To solve the reduced modelled two competitors zero sum fuzzy matrix games, the proposed ranking methods are used. Finally, the findings are compared to current approaches that are quite similar to the proposed approach.

scholarly journals Zero-sum games for continuous-time jump Markov processes in Polish spaces: discounted payoffs

2007 ◽  
Vol 39 (03) ◽  
pp. 645-668 ◽  
Author(s):  
Xianping Guo ◽  
Onésimo Hernández-Lerma
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Infinitesimal Operator ◽  
Polish Spaces ◽  
Zero Sum Games ◽  
Stationary Strategies ◽  
Value Of The Game ◽  
Optimal Stationary Strategies ◽  
Zero Sum ◽  
Jump Markov Processes

This paper is devoted to the study of two-person zero-sum games for continuous-time jump Markov processes with a discounted payoff criterion. The state and action spaces are all Polish spaces, the transition rates are allowed to beunbounded, and the payoff rates may haveneither upper nor lower bounds. We give conditions on the game'sprimitive dataunder which the existence of a solution to the Shapley equation is ensured. Then, from the Shapley equation, we obtain the existence of the value of the game and of a pair of optimal stationary strategies using theextended infinitesimal operatorassociated with the transition function of a possibly nonhomogeneous continuous-time jump Markov process. We also provide arecursiveway of computing (or at least approximating) the value of the game. Moreover, we present a ‘martingale characterization’ of a pair of optimal stationary strategies. Finally, we apply our results to a controlled birth and death system and a Schlögl first model, and then we use controlled Potlach processes to illustrate our conditions.

Learning in Two-Person, Zero-Sum Games

1974 ◽  
Vol 34 (2) ◽  
pp. 503-510 ◽  
Author(s):  
James L. Pate ◽  
Elizabeth D. Broughton ◽  
Lorraine K. Hallman ◽  
N. Lynn Letterman
Keyword(s):  
Choice Behavior ◽  
External Control ◽  
Personality Variables ◽  
Zero Sum Games ◽  
Value Of The Game ◽  
Computer Confidence ◽  
Zero Sum

In a series of three studies, Ss were required to play a two-person zero-sum game. Ss low in dogmatism tended to approach the optimum or exploiting strategy more closely than highly dogmatic Ss, but two other personality variables (internal-external control and computer confidence, a measure of willingness to have decisions made by a computer) were unrelated to gaming strategies. The value of the game and the required strategy affected significantly Ss' choice behavior.

scholarly journals Zero-sum games for continuous-time jump Markov processes in Polish spaces: discounted payoffs

2007 ◽  
Vol 39 (3) ◽  
pp. 645-668 ◽  
Author(s):  
Xianping Guo ◽  
Onésimo Hernández-Lerma
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Infinitesimal Operator ◽  
Polish Spaces ◽  
Zero Sum Games ◽  
Stationary Strategies ◽  
Value Of The Game ◽  
Optimal Stationary Strategies ◽  
Zero Sum ◽  
Jump Markov Processes

This paper is devoted to the study of two-person zero-sum games for continuous-time jump Markov processes with a discounted payoff criterion. The state and action spaces are all Polish spaces, the transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. We give conditions on the game's primitive data under which the existence of a solution to the Shapley equation is ensured. Then, from the Shapley equation, we obtain the existence of the value of the game and of a pair of optimal stationary strategies using the extended infinitesimal operator associated with the transition function of a possibly nonhomogeneous continuous-time jump Markov process. We also provide a recursive way of computing (or at least approximating) the value of the game. Moreover, we present a ‘martingale characterization’ of a pair of optimal stationary strategies. Finally, we apply our results to a controlled birth and death system and a Schlögl first model, and then we use controlled Potlach processes to illustrate our conditions.

scholarly journals A Class of Two-Person Zero-Sum Matrix Games with Rough Payoffs

2010 ◽  
Vol 2010 ◽  
pp. 1-22 ◽  
Author(s):  
Jiuping Xu ◽  
Liming Yao
Keyword(s):  
Genetic Algorithm ◽  
Expected Value ◽  
Equilibrium Strategy ◽  
Matrix Games ◽  
Zero Sum Games ◽  
Equilibrium Strategies ◽  
Expected Value Operator ◽  
Maximin Equilibrium ◽  
Trust Measure ◽  
Zero Sum

We concentrate on discussing a class of two-person zero-sum games with rough payoffs. Based on the expected value operator and the trust measure of rough variables, the expected equilibrium strategy andr-trust maximin equilibrium strategy are defined. Five cases whether the game existsr-trust maximin equilibrium strategy are discussed, and the technique of genetic algorithm is applied to find the equilibrium strategies. Finally, a numerical example is provided to illustrate the practicality and effectiveness of the proposed technique.

FUZZY MULTIOBJECTIVE PROGRAMMING METHODS FOR FUZZY CONSTRAINED MATRIX GAMES WITH FUZZY NUMBERS

2002 ◽  
Vol 10 (04) ◽  
pp. 385-400 ◽  
Author(s):  
DENGFENG LI ◽  
CHUNTIAN CHENG
Keyword(s):  
Linear Programming ◽  
Multiobjective Programming ◽  
Fuzzy Numbers ◽  
Fuzzy Linear Programming ◽  
Computational Method ◽  
Matrix Games ◽  
Fuzzy Matrix ◽  
Numerical Example ◽  
New Type

The purpose of the paper is to introduce a new type of fuzzy matrix games: fuzzy constrained matrix games. A computational method for its solution based on establishment of the auxiliary fuzzy linear programming for each player is proposed. The approach based on the multiobjective programming is establisched to solve these fuzzy linear programming. Effectiveness is illustrated with a numerical example.

Sign in / Sign up

Export Citation Format

Share Document