Greetings. We're still in the process of looking at how markets operate. We started this whole process by thinking about this number line, about the number of firms in an industry. But even though we know a lot about the cost for example within, because we did a lot of understanding of production functions and costs. We understand that there's just one firm like this, we call this a monopoly. And we did monopoly, and we understood monopoly equilibrium. We understand how to analyze a monopoly. And this end, we have N, where N is very large. And we looked at that too, we called this the perfectly competitive model. Perfectly meant, we had some really tough conditions that we had to meet. And we understood that in fact, in many markets, most of those conditions are met but maybe not all of them. And so the market, we might call it competitive, but we might not be able to go all the way and say it's perfectly competitive. And then we have this sort of middle zone here, where 2, or 3, or 4, or dot, dot, dot, dot certain number of firms, and we call that oligopoly. And we looked at two different forms and we said, oligopoly is a hard problem. And so what you said, well, sometimes oligopolies equilibrium happens because firms act sort of like cartels, collusive arrangements. So we've looked at those, but then we have another approach to think about this. And what we're going to do is we're going to introduce this idea called game theory, okay? Now, game theory is a branch. It was basically created by mathematicians, mathematicians have, if you've taken some calculus courses, that mathematics you have the calculus exams. They say well, I want you to figure out how to maximize this function, that's easy. But then, you get it at a deeper level of calculus they say, figure out how to maximize this function subject to the following constraints, okay? And those constraints might say, maximize your revenue, price times quantity, subject to the cost function, okay? And you could go ahead and solve that, and that would be perfectly happy for an individual firm and that's a nice calculus problem. But the mathematicians, they're not really worried about helping economics out in terms of showing you how firms might solve their own particular profit function. But what they were interested in is saying, I wonder if we can figure out how to find the optimal strategic play in a game. In other words, they say, maybe we can take the rules of a game and write those rules out in algebraic notation. Things like, alpha to the x power, plus beta to the y power, and all those sorts of rules. Or maybe we can take the rules of a game and write it out in algebraic notation, and then we can figure out how would you, given the rules of the game, maximize the probability that you're going to win. And so they did this, they played around, they understood, they had success, okay? But let's talk a bit about the types of general class of games that we could talk about. There's two real types of games, cooperative games and non-cooperative games. And we're not really going to do cooperative games. Cooperative games, it's not so important to us as economists. Believe me, cooperative games are very important. Cooperative games, the way to think about this is cooperative games, or like think about, they're like games where you can sign. Basically where you can sign a binding commitment. You can sign a binding contract. So, cooperative games are sometimes relevant to economics, think about labor-management negotiations. Labor and management get into a negotiation, the unions and the management now butt heads about what they ought to get. Well, we want this type of wages, we want this type of benefits, we want this type of vacation and they'll have all these arguments. But in the end, they can come to an agreement where they sign a binding agreement, okay? That game is what we would call a cooperative game, okay? And so, it's relevant to some branches of economics. But what we're talking about now is we're talking about, how could we use this game theory to talk about how firms compete against each other? And when firms compete against each other, okay, we call that non-cooperative games. And non-cooperative games are games where firms cannot sign binding contracts. So again, in a nutshell, cooperative games, very important, but cooperative games are games where people can have binding contracts. And so it's a small subset of things that happen in the economic sphere, by the way. There's lots of social sciences who study, psychologists and sociologists will study all sorts of cooperative games because there are arrangements where people can come to agreement with each other. How are you going to divide up the refrigerator space between you and your your roommate in an apartment, these sorts of things. How does that type of behavior happened to the right a little contractile each other that might hold up and judge duties court of law or something like that. But non-cooperative games, that's really where the action is. And so I want to think more about two types of non-cooperative games and we're going to write them out like this. The first type is what we we'll call extensive, Extensive form games. And the second type is something we'll call normal form games. Now, distinction here between these two is that extensive form games are essentially, the easiest way to think about this is they're sequential play. They're sequential play games, okay? And so, Their games were one firm makes a move, the next firm response. The first firm remix another move, the other one response, okay? Normal form games are what will say are simultaneous play games. So the two firms make simultaneously the announced number of cars that Honda is going to occur. The number of Accords that Honda is going to produce and the number of Camrys that Toyota is going to produce are basically announced at the same time, all right? Those are what we will call simultaneous play games, people are making decisions at the same time. And then we see what happens when the dust settles. What's the outcome on the market when they put different amounts of product onto the market? And what we're going to do next is we want to stop and think about how to model these two and how you would think about this process. So, just to summarize, game theory was started by a bunch of mathematicians in mathematics departments and they made it work and then they moved on. So they looked at how to solve certain games, certain things like chess and and other games, some games are too hard. There's too many complicated rules in say, playing a board game like risk to be able to solve it. They moved on but by that time they had established this idea of thinking about strategic responses between two different players in what might can we model what the outcomes going to be? That's what we're going to do going forward. Thanks.