Okay, so let me summarise what we have learnt so far. So we have learnt that any social problem can be formulated as a mathematical model of a game. And the next question is to find a single solution concept which can be applied to all those social problems. Okay? So this question was posed by the fathers of game theory, von Neumann and Os, Oskar Morgenstern, back in 1944. But even in 1950s, six years later, the central problem was still open. Okay? So the central problem is this. Any social interaction can be formulated as a mathematical, mathematical model of a game which specified players, strategies and payoffs. The question is if we can find a unified general theory to find the solution to all of those social problems. Is there any single solution concept that can be applied to all those social interactions? The answer was discovered by a mathematical genius, John Nash. John Nash is a, John Nash is a very smart young person and he entered the graduate school of mathematics of Princeton University when he was age 19. And the way he came to Princeton, he came with recommended, recommendation letter coming from his teacher. And usually recommendation letter is very long. And it has a detailed description about why this guy i, is a good guy, and what he did, and so on. But surprisingly, his recommendation letter was very short. It only had a single sentence, saying that this man is a genius and he was accepted to Princeton University and he encountered this intriguing question posed by von Neumann and Oskar Morgenstern, and within a few years he found an answer. And then he had a problem of schizophrenia. And he couldn't continue his research, but sooner or later, a columnist discovered that his unified principle was very useful in addressing lots of economics questions. And in 1994 he received Nobel Prize in economics, okay? So his life was so interesting that it was made into a movie called A Beautiful Mind. Okay, so if you are interested, you can see this movie performed by Russell Crowe. Okay, so let me just explain what the genius John Nash discovered. The problem is to find a unifying solution concept, which can be applied to all those social problems. Okay. And surprisingly, Nash find, Nash found the answer in a cup of coffee. If you stir the surface of coffee. And then you get a vortex, a point that doesn't move. So a vortex and human behavior, why are they similar? This is what I'm going to explain now. Okay, so this is the picture of the surface of coffee. And this point here moves to another point after one second or something, okay? But the vortex point doesn't move, okay? So if you stir coffee surface, you always get a vortex, and vortex point doesn't move. Okay, so suppose the surface of coffee represents the set of all possible human behavior in a social problem. This was really an ingenious idea of John Nash. I'm going to explain how this analogy works in the second week possibly. But if you buy this analogy between coffee surfaces and the set of possible human behavior in a social problem you can realize that the following is true. So a point in this fixture now represents original behavior of people in a social situation. Okay? And the destination, this point here, represents another social situation where players are moving towards best replies. Originally, people are not taking best replies, best strategies against others. But now they can move to better strategies. And then social situation changes from this point to that point. Okay. Coffee surface has a vortex. What about a social situation? Okay? The vortex represents a case where people already are taking best replies. Okay? So at the vortex point, all players are doing their best against the others. Okay? So this is a stable situation and a social problem. Okay? And just as coffee surface has a vortex point, any social problem has such a stable point. Okay? This is what is called Nash equilibrium. So by using advanced mathematics called topology, Nash discovered or found that any social problem has a natural equilibrium point. Nash discovered that every social problem has a stable point where all individuals are doing their best against others. And this is called Nash equilibrium. Now, you can apply the idea of Nash to any of those social situations. So first, you formulate a social situation as a game, a mathematical model of a game, by specifying players, strategies, and payoffs. And then you use your mathematics to find out something like a vortex point, okay, where all individuals are doing their best against others. And then you get a prediction in every single social problem. That's called Nash equilibrium.