A Study on Cooperative Continuous Static Games Without Differentiability Under Fuzzy Environment

1. Introduction

The possibility of competition among the system controllers, called “player”, and the optimization problem under consideration is therefore termed as “game”. Each player in the game controls a specified subset of the system parameters (called his control vector) and seeks to minimize his (her) own cost criterion, subject t specified constraints. Applications of game theory may be found in economics, engineering, biology, and in many other fields. Three major classes of games are matrix games, continuous static games, and differential games. In continuous static games, the decision possibilities need not be discrete, and the decisions and costs are related in a continuous rather than a discrete manner. The game is static in the sense that no time history is involved in the relationship between costs and decisions. Game Theory plays a vital role in Economics, Engineering, Biology and other computational cum mathematical sciences with wide range of applications in real world problems. The major classes of games are matrix games, continuous static games and differential games. Matrix games derive their name from a discrete relationship between a finite / countable number of possible decisions and the corresponding costs. The relationship is conveniently represented in term of a matrix (or two-player games) in which one player’s decision corresponds to the selection of a row and other player’s decision corresponding to the selection of a column, with the corresponding entries denoting the costs. It is vivid that decision probabilities are not mandatory in the cooperative games. In addition, there is no time in the relationship between costs and decisions in static games. Differential games are characterized by continuously varying costs along with a dynamic system governed by ordinary differential equations. For continuous static games, there are several solution concepts. How player uses these concepts depends not only on information concerning the nature of the other players, but on his/her own personality as well. A given player may or may not play rationally, cheat, cooperate, bargain, and so on. A player in making the ultimate choice of his/ her control vector must consider all of these factors. Thomas & Walter (1981) introduced different formulations in continuous static games. The three basic solution concepts for these games (Vincent & Grantham, 1981) are:

• 1.

  Nash Equilibrium Solution

• 2.

  Min-Max Solutions

• 3.

  Pareto Minimal Solutions