Okay. So we have learned the basic solution concept which can be applied to any social problem. That's the concept of Nash equilibrium. So to really understand the nature of Nash equilibrium, the best way to really understand the nature of Nash equilibrium, is to see lots of examples of Nash equilibrium. So let me just present one more example of Nash equilibrium in a simple game, okay? The game I'm going to explain is called the location game, okay? So this location game has two players, player A and B and they are selling ice cream so we have vendor A and vendor B selling ice cream on a street. And the street is represented by this segment, or line, from here to here, and every morning simultaneously, vendors A and B choose their locations. Okay, so that's the nature of this location game. 'Kay? The customers are uniformly distributed over this road, okay? And each customer goes to the nearest vendor. So if you are here, you are going to vendor B to buy ice cream. And customers are uniformly distributed over this street. Okay, so what happens if A and B are located at exactly the same place. Well, I assume that they split the customers. Half of the customers are going to place A and half, the other half goes to place B. Okay, what about vendors payoff? 'Kay, I'm going to assume that a vendors payoff is equal to the number of customers it gets. And so, this clearly defines a game played by two ice cream vendors, A and B on a street. Okay, so let's try to understand the nature of this game and suppose vendor A is here and vendor B is located here, okay? Okay, so this is a point in between A and B And all the customers located here, all the customers located here goes to vendor A. And all the customers walking here are going to buy ice cream from vendor B. This is the nature of this location game. And the number of customers here is Mr. A's payoff, and the number of customers here is Mr. B's payoff, okay. So this game was invented by an economic theorist, Harold Hotelling, and, therefore, it is sometimes called Hotelling's location game. Okay, so suppose this is the locations of- this picture depicts the locations of A and B. And A is getting customers here, and B is getting customers here, is this a Nash equilibrium? Well, it is not a Nash equilibrium. Okay? What was the definition of Nash equilibrium? The Nash equilibrium has the property saying that, no single player can increase its payoff, by deviating by himself, or herself. Okay? So Nash equilibrium conditions says that no one should increase, should- Nash equilibrium condition says that no one can increase his or her payoff by deviating by himself, or by herself. So let's examine A's payoff when he deviates. ' Kay? Well, vendor A can steal customers from B if he moves closer to Mister B.' kay, like this one. Okay? And this is not Nash equilibrium again because if A move further closer to B, his customer, the volume of customers coming to A increases. Okay. So, finally, maybe A is located at the same place as B, is this the Nash equilibrium? Well, I would say no, this is not a Nash equilibrium. Okay, I would argue that this is not a Nash equilibrium because originally Mr. A is getting half of the customers. because remember that this segment here is larger than this segment, okay? So if A moves, towards a larger segment, like this. Then he get more than half customers. So originally he was getting half of the customers but by deviating by himself, Mr. A is getting more than half customers. So therefore, the original configuration doesn't satisfy Nash equilibrium condition. Okay, by deviating slightly to the left, A can get more customers. Therefore, the original configuration is not a Nash equilibrium, okay? So, the only Nash equilibrium in this location game is this one, okay? A and B are located exactly at the same point, exactly in the middle of the road. Well, this is very unfortunate for the customers if A is located here and B is here, it's more convenient, but Nash equilibrium predict that they choose the same location. Okay, so let's examine if this configuration satisfies the Nash condition, okay? So, in the original situation, A and B are splitting customer, they are getting half of the customer. And even if A moves to the left or right Mr. A cannot really increase the number of customers. Therefore, no player can de-, can increase his or her payoff by deviating by himself. This configuration satisfies the Nash equilibrium condition. And this is the unique Nash equilibrium in the location game.