Greetings. In the last video we introduced to the advertising game where we looked at sort of a model where the defensive advertising turned out to be the dominant strategy. They just keep throwing lots of commercials on the market, okay? Even though the joint profit, let's recall what that looked like, even though the money that the industry had made in total would have been greater, 10, if they could just stick to low. But they each have such a strong incentive to start throwing more on the market. That's better. More on the market. That's better for me. And Pepsi said I'm always better, always better by running bigger campaigns. And Coke says I'm always better by running big campaigns. Look at that. And so, they both end up running big campaigns and they end up making very little money because they'd spend so much money at creative shops making those funny commercials, as well as paying networks to run those 30 second spots or whatever it is that they want to put out there. And then, in that game, I said yeah, but I wonder what would happen if we had just changed one cell. And I just changed that one cell. And now, we're in sort of a dilemma. Because looks like Pepsi always wants to do one thing, but Coke is really going to depend on what Cokes thinks Pepsi was. Now, I could have changed two or three or more and don it, but this was the simplest one of all. And this does not lend itself well to a type of solution concept. So we need a new solution concept. Now, economists have been writing game theory papers for 40 years, okay? And over those 40 years, many economists and mathematics professors have offered their own candidate solution concept, okay? So you can have Shetlon's equilibrium. You can have minimax models. You can have maximin models. There's just tons of different authors whose names have been attached, papers have been published and names have been attached to those new algorithms to solve this problem. But one of them turns out to be the one that is most frequently used. It's called the Nash equilibrium. You may remember, there was this movie called A Beautiful Mind about John Nash. Russell Crowe played John Nash as the troubled mathematics professor turned economist who had developed this thing called the Nash equilibrium. John Nash won a Nobel prize for the Nash equilibrium. And most people who win Nobel prizes in economics have just unbelievably monstrous resumes. I mean, they published hundreds and hundreds and hundreds of articles and dozens of books, each one. They're really giants of the profession. Not so much John Nash. But what John Nash did produce was the Nash equilibrium. The paper about the Nash equilibrium was a paper that has gotten more citations than any other paper in economic. In fact, they don't even count it anymore. Because the Nash equilibrium became so dominant. So remember, there's probably 100 different algorithms that we could apply to try to solve this problem, okay? There's probably 100 different ones, different economist's names on there, different mathematicians on there. But the Nash equilibrium is what everybody has been using. And the reason is that Nash works. The Nash equilibrium, whether you look at the banking industry or the steel industry or the maritime shipping industry or the airline industry, where you have like Airbus and Boeing going head-to-head, and you say look at these guys. I'm going to look at their cost curves. I'm going to look at demand functions. I'm going to see how they play. They play Nash. The outcomes that we observe in the market is exactly the outcome that Nash would predict, okay? Not the other ones. The other ones are sometimes good at predicting 20, 30, 40% of the market. And Nash just kills it. And so, John Nash and the Nash equilibrium, John Nash won the Nobel Prize on the strength of the fact that this is just a remarkably, remarkably important part of economics today. Not just economics. If you were to go over to the library and grab a copy of the of the leading journal in political science, and there's 15 articles in that journal today, in the most recent one, all of them somewhere are going to have John Nash's name in there. because Nash is what people use in political science. Nash what people use and psychology. Nash is what people use in sociology. Nash is what people use in economics. And it's not because it's easy. [LAUGH] The Nash equilibrium can be profoundly difficult to crank out. But when you get there, it's amazing how accurate it is of predicting how people behave. So let's go back and think about the Nash equilibrium, because I want you to be able to understand what it is. The Nash equilibrium says do the best you can given what your rival is doing. Do the best you can given what your rival is doing. Now, that's a little bit of a simplification, but I hope that if you're looking at that, you'd say what the heck, I'm paying all this money to get a to get a degree, and the professor says this is the most important thing ever, and it just says do the best you can? Well, that's kind of pretty silly. Well, it's not really. That's kind of just a distillation of the maximization process that happens when you're doing Nash, okay? Sometimes that maximization process is very difficult. You have to slog through a lot of math. Sometimes it's pretty straightforward. I'm going to use that rule right now to show you the Nash equilibrium in that cell that we just did. We're going to think about how to solve this Nash equilibrium, okay? We're going to say which one of these, if any, is a Nash, okay? And the way we're going to do it is this rule. See, I put this little thought bubble, solving Nash. You solve Nash by dropping yourself into that cell, and then ask both players if they want to stay there given what their rival's doing? In other words, do the best you can, do you want to stay here or do you want to go someplace else given what your rival's doing, okay? I'm going to ask each player. If both players say yeah, I'll stay here, that's a nice equilibrium. If either one of them says no, I'm going to get out of here, I don't like this cell after all, then it's not a Nash equilibrium, okay? And it's very simple in our little two-by-two game. Not simple when you think about the top ten banks in the United States competing against each other. It gets to be very, very much more difficult. Well, we can do it. Nash still wins. Okay, so let's ask ourselves how to do that. So we drop ourselves in this cell. We say okay, in this cell, Pepsi's doing low. Coke is doing low. We turn to Coke, we say hey, Coke, Pepsi is doing low. Do you want to stay here at low or do you want to go up and do high? Coke says I'll stay, I'm fine. 5 is better than 4, I'll stay. Okay, maybe we got it. So we turn over and ask Pepsi. Hey, Pepsi, Coke is going low. Do you want to stay here at low, or do you want to go high? Pepsi says no, I want to get out of here, I'd like that 8, it's much better. So at this cell, if we were to ask Pepsi are you satisfied to stay here, if Coke's going low, Pepsi's going to say no, I'm getting outta here. If I know Coke's going low, I'm going to go over here. So this cannot be a Nash equilibrium. So there's strike one, okay? Not a Nash equilibrium. Let's go up here and try this one. Pepsi's going low, and Coke's going high. So once again, we'll put us in the cell. And we say okay, Pepsi's doing low. Do you want to stay here at high? And Coke immediately says no, I don't want to stay there at high. If Pepsi's doing low, I'm getting 4, but I could go 5, I'm going away. Well, we don't even need to ask Pepsi. All we have to do is find one person that says I'm not happy here, and we already shown that this is an out of equilibrium situation. They both have to agree to stay there. Coke says I'm not staying. if Pepsi's going low, I'm definitely getting away from this 4 and going to this 5. I'm going to go low. So we're going to go mark that off. That's not a Nash equilibrium. Let's go over and try this one. It's got different numbers. Pepsi's doing high, and Coke's doing low. You turn to Coke and you say hey, Coke Pepsi's doing high, you're doing low. Do you want to stay here at low, Pepsi's doing high? No, I don't like that. I want to go up here, okay? So Coke wants out. We don't even need to ask Pepsi. We've got an out of equilibrium situation. At least one of the players wants out, so that can't be a Nash equilibrium. Let's hope this one works. We'll go up and try this cell. And in this cell, you say Pepsi, Coke's going high. You're going high. Do you want to stay high or you want to go to low? Pepsi says well, if Coke's high, I'll stay here. You turn over to Coke and you say Pepsi's doing high, you're doing high. Do you want to stay here or do you want to go over there? Coke says well, 1's better than 0, I'll stay here. Both players are satisfied to stay there, there's the definition, given what their rival is doing. Given that a rival has put me in that hole, do I want to stay in that hole or do I want to move someplace else? If Pepsi's going high, you can't say well, I want to go over here. If Pepsi's going high, you only have these two choices, this or this. And you're going to stick at the high. So this cell up here at that corner is the Nash equilibrium. And again, it's not that difficult to solve this in a two move game. A two layer, two move game, there's only four cells to check. Like I said, I could be nasty and say two players, three moves. Then, there would be nine cells to check, and you'd have to go. So a little bit harder, but I'm not going to do that either. You just need to understand that the Nash equilibrium is wildly successful, very popular. Here, it's pretty simplistic to work it out. But you do need to figure out how to do it, because you can be pretty confident that I'm going to be going down that road, seeing can you solve for a Nash equilibrium in a simple two-by-two matrix. Thanks.