please provide every step Nashs Problem in R2 If X 0 and V

please provide every step

Nash\'s Problem in R^2: If X = {0} and V is the set of vector fields F on R^2\\X which satisfy curl(F) = 0 and W is the set of vector fields F which are gradient fields, then the dimension of V/W is one. The vector field F(x, y) = -y/x^2 + y^@ i + x/x^2 + y^2 j has vanishing curl in IR^2\\X but the line integral around the origin is 2pi. It is not a gradient field. Now: What set X in R^3 would you have to take to get the dimension of V/W to be eight?

Solution

The theory constructs a notion of \"equilibrium,\" to which the complex chain of thinking about thinking could converge. Then the strategies of all players would be mutually consistent in the sense that each would be choosing his or her best response to the choices of the others. For such a theory to be useful, the equilibrium it posits should exist. Nash used novel mathematical techniques to prove the existence of equilibrium in a very general class of games. This paved the way for applications. Biologists have even used the notion of Nash equilibrium to formulate the idea of evolutionary stability. Here are a few examples to convey some ideas of game theory and the breadth of its scope.

The Prisoner\'s Dilemma
In Joseph Heller\'s novel Catch-22, allied victory in World War II is a foregone conclusion, and Yossarian does not want to be among the last ones to die. His commanding officer points out, \"But suppose everyone on our side felt that way?\" Yossarian replies, \"Then I\'d certainly be a damned fool to feel any other way, wouldn\'t I?\"

Every general reader has heard of the prisoner\'s dilemma. The police interrogate two suspects separately, and suggest to each that he or she should fink on the other and turn state\'s evidence. \"If the other does not fink, then you can cut a good deal for yourself by giving evidence against the other; if the other finks and you hold out, the court will treat you especially harshly. Thus no matter what the other does, it is better for you to fink than not to fink -- finking is your uniformly best or \'dominant\' strategy.\" This is the case whether the two are actually guilty, as in some episodes of NYPD Blue, or innocent, as in the film LA Confidential. Of course, when both fink, they both fare worse than they would have if both had held out; but that outcome, though jointly desirable for them, collapses in the face of their separate temptations to fink.