So John Nash has found a unified principle which can be applied to all social problems, Nash equilibrium. Just as coffee surface has a vortex, by the same mathematical reasoning, any social problem has a point where all players are doing their best against others. Okay. So in this lecture, I'm going to give you the formal definition of Nash equilibrium and then I'm going to apply the idea of Nash equilibrium to the second question I posed in the first lecture. That’s the traffic problem, okay? So let's just, let me just give you the definition, the formal definition of Nash equilibrium. So for simplicity, let's confine our attention to two player case. Okay? So we consider our game which has two players, player one and two. Okay? Nash equilibrium. Is a pair of strategies. One for each player. So A-1-star is the Nash equilibrium action or strategy for player 1. And A-2-star is the Nash equilibrium strategy for player 2. If the pair of strategies, A-1-star and a-2-star satisfy the following condition- We call it a Nash equilibrium. Okay so let's confine our attention to player number one. Okay, if everybody chooses the Nash equilibrium strategy, a1 star and a2 star, this is Mr. One's payoff. Okay? What happens if he changes his behavior from equilibrium to something else? Well, the answer is given by g1 of a1 which may not be equal to the Nash equilibrium strategy and a2 star. Okay? So originally everybody was sticking to Nash equilibrium. But now Mr. One is changing his action from equilibrium to something else. Okay? Well Nash equilibrium requires that player one cannot increase his payoff by deviating. Okay? So this should be true for all other strategies. Other than the equilibrium. Okay? And a similar condition applies to player two. Okay? So what is Nash equilibrium? Well, this condition says that Nash equilibrium has the following property. No one can gain by deviating by himself or by herself. If you are the only one person deviating from equilibrium, you cannot really gain. Okay? That's the definition of Nash equilibrium. In other words, if player, if players one and two. Are taking Nash equilibrium strategies, a1 star and a2 stars. They are taking mutually best reply. Best reply. Okay? So, Mr. One cannot increase his payoff by deviating from the equilibrium strategy. Given that other guy is using equilibrium strategy a2 star. So that means a1star is the best reply to player two's equilibrium strategy. Okay? So, a1 is doing his best against a2 star. Okay? And likewise, player 2 is also taking the best reply against mister 1’s strategy. Okay? This is the nature of Nash equilibrium. Okay? Now, I'd like to apply this basic solution concept. To the second problem I posed in the first lecture, the traffic problem. So let's recall question number two I posed during the first lecture. Question number two says, what happens to the traffic flows if we construct a new bypass from city x to city y? Okay? You can answer this question by calculating Nash equilibrium before construction- constructing of the new bypass, and after the construction of a new bypass. If you compare those two Nash equilibria, you can see the answer. Okay. So,. Let me just present a very much simplified version of, of this question. Okay so let's suppose 150 cars are commuting from city x to city y, it's a very simplified traffic problem. And there are three routes. Okay? And the number in parentheses represents the lengths of each route. So this route here is very long, length is 350. The second route, is, is. The length of the second route is 250 and the new bypass, the length of the new bypass is 200. Okay? Okay, the traveling time on each route. So to simplify the analysis, let's suppose that traveling time is equal to the route length. If the route is long it takes a long time to go to the destination. Plus the number of cars on the road. So if the route is congested it takes more time to get to the destination. Okay? Later I'm going to show you the real example, real estimation- real life estimation of the relationship between Traffic on the traveling time, but that's coming later. So now- Let's stick to this very simple formulation to solve for Nash equilibrium. Okay. the payoff is the negative of traveling time. So each player is trying to minimize- The time to destination. So this new bypass from City X to Y. is the shortest one. The length is 200. But if all drivers, 150 cars, jump into this. Jump on to this new route then the traveling time is equal to. The route length 200 plus 150. Okay? It's 350 and it's better to choose the route in between. Okay. So, what is the Nash equilibrium? Okay. Nash equilibrium has a property, saying that no single driver can save his or her traveling time by deviating to another route. And the answer is given by the following allocation of traffic. Okay, at the Nash equilibrium, you have zero cars traveling from x to y. On this route. Very long route. And 50 cars are allocated to the second route. And 100 cars are allocated to the new bypass. Okay? So let's try to examine the traveling time for this one. For this route and that route. Okay, so what's the traveling time of the route in between? Okay. The length is 250 and the amount of cars, number of cars on this route is 50. So if you sum up the traveling time on this route, it's 300, okay? The new bypass has more cars, 100, right? But it's shorter. So if you sum up the length and number of cars, the traveling time is again 300, okay? So traveling time is equal on those routes, and you can see that. This allocation satisfies the Nash condition. The Nash condition says that no single player can increase his payoff by deviating by himself or by herself, okay? This allocation has a Nash property that no one can save traveling time by changing his or her route, okay? So if you are driving on this new bypass coming from x to y, your traveling time is 300. And if you switch to the second route, your traveling time is 300 plus 1. Plus 1 comes from you, okay? So it takes longer. And if you switch to the third way, route, it's very long and your travelling time is 351. So no one can increase his payoff by deviating by himself and herself. So by using this kind of reasoning, you can see the answer to the original traffic question, okay? So you can calculate the Nash equilibrium before construction of the new bypass. And you can calculate the Nash equilibrium of traffic allocation after the construction of new bypass. And you can see how the traffic on the existing road is affected. And how much saving you can get about the traveling time from city x to city y by constructing the bypass.