Friday, October 7, 2016
4. The Nash Equilibrium
Each type of genius is unique to itself. Gödel took two seemingly unrelated things, improved they were related to each other. He did this even though two geniuses of field tried to do things the opposite way – and Whitehead and Russell were geniuses. Turing was, for all intensive purposes, a mystic – seeing things in a completely different fashion, even though the objects that he reached with were right there. They just thought of them as machines, whereas he understood them to be computers. Finally, we reach Nash, whose gift was to see things in place site. Remember that Gödel himself see this though he was a resident at Princeton at the time, but did recognize for what was. It can easily be said that genius has the gift of a complex mind, but is trumped by a genius with a simple mind. So let us get to Nash simple genius, which is often overlooked next to the complex geniuses that surround him.
Though I would like to mention that in 1951 he was hired by this institution, to be a C. L. E. Moore instructor in the mathematics faculty, and eventually a full professorship.5
So what exactly does he prove with his famous Nash Equilibrium? Is only a few simple steps, and it proves two things: first that the group that is a Nash equilibrium is convex rather than concave, and second one does not prove it forward but backwards. Everything else in the group is devoted to one of these two concepts. You might think that anyone can prove this, and you would be right. Anyone can do this once knows that it has been proved, but no one can prove this if he does not know that the proof is sitting right there in front of them. Only one person could figure that out, even Gödel did not figure that out.
Now we have we have to ask: why is this important?
Let us examine the proof, looking for the key texts, which are important.
First he finds an equilibrium point such that if and only if for every individual high there is only one option, for every strategy that the others choose on. That is, whatever your opponents choose, there is only one choice for you, as opposed to one choice in that they choose one way, and a different choice if they choose a different way. This is powerful, because you do not have to concern yourself with what they are deciding. It means that in best case for doing well, and in worst case you are losing least.
Then says that an equilibrium point it can be expressed as pairs of use functions. By this he equates a single equilibrium point to its group on a line. So an equilibrium point stands in for a set of values, which is hard to grasp at first, but settles in your mind once have grasped. Again I will remind you that this will be grasped only with time. That a single point can stand in a curve is not to grasp.
Then he makes a leap into two unknowns, based on Kakutani generalized fixed point theorem. Though cleans it up by making reference to Brouwer. Then he proceeds to show that all of the equilibrium points are connected.
First he shows that every finite game has an equilibrium point. This is obvious, until you actually try and prove it. Then a web comes over you. What Nash proves is that the top part of the equation is the s part with pi as the numerator, over 1 plus the whole amount. Which is hardly obvious.
What he then must show is that the fixed points of the mapping are also equilibrium points, which looks non-trivial, but has a solution. He shows that under Brouwer, the cell must have at least one fixed point which also means it is an equilibrium point. This does not show that all equilibrium points are accessible from one strategy, however. It just proves that there is an equilibrium point for each strategy, but what he needs to prove is that all of the equilibrium points will yield to the same step.6
This makes it provable that any finite game has a symmetric equilibrium point. Which means that it is not random, and it is not a sphere of points that are related: but a curve of points that have a symmetric relationship. This is one half of the problem, because instead of a random series of points, he has shown that they lie on a curve. What he now needs to show is that there is one curve that dominates all of the others, if it exists. Remember it does not have to; there are unsolvable games, but if they are solvable then, they are solvable on a plane. This means that there are no strategies which have a myriad of unrelated points. Either they are solvable on a curve or they are not solvable at all.
This means if you have a position which is insoluble in any way, you do not have to look for any other examples. Similarly, if you find a solution to a problem, that means that the solution exists on any curve. This is unexpected, and the proof, while simple, is hard to get through.
But it has repercussions, because it means that to solve any problem which has a solution, one has to start at the end and work backwards. This is a general plan: start with the end.
But then proves that for any game which is soluble, regardless of number of players or their strategies, the same rule applies: if it is solvable, work backwards. He called this the “Geometrical Form Of Solutions”. And they had the property of Dominance and Contradiction Methods”.
Or More colloquially, the prisoners dilemma. This again, is well known, and you can look it up any place. Imagine that two criminals are caught, and they are separated. Each one is given the same choice: either blame the other one, or be quiet. If both of them are quiet, the police will get them on a minor charge, which will give them about one year. If they both blab, they will get at least 5 years. But if one blabs, while the other stays quiet, then the talkative one goes free, and the quiet one gets 10 years. The core of the situation is that neither one knows what the other one has planned, and looks down his route, and sees that whatever the other one is planning, the best route for him is to talk. So they both get five years. All of the solutions that do not end in this fashion, require some means of communication before, and that is the prisoners dilemma. How do they know? They do not. One can elaborate this with side payouts, and other things – there is an entire science of game theory.