Greetings, we are thinking about understanding how to use game theory to model strategic interaction between firms in an oligopoly situation with a small number of players. Many of our videos, I've referenced how people like, for example, Toyota with its Camry and Honda with its Accord, are the big players in the mid-size automobile market, and they constantly think about what the other side doing.What will happen if I increase my output a little bit, how will they respond, how do we model that stuff and we're turning the game theory to see if we can figure this stuff out. So in this particular video, we're going to think about normal form games. Remember, normal form games, we talked about earlier. Normal form games are games that we call simultaneous play. So now, a little model Honda and Accord would announce their outputs simultaneously. We'll see how that works out. We're also going to look right now at single play games. Now what that means is, and you would say well, that's a single-player game means that they're only going to play the game once. Clearly, that's not true. Honda and Toyota will play repeated play. They know they're going to see each other in the market every four times, for different quarters depending on what your time for this. You want to talk about quarter, you want to talk about an annual, whatever. They're going to be playing with each other. They don't expect to quit at any point in time in the future. So there's going to be lots of repeated games. We will talk very little about repeated games. Repeated games are important, but we can get most of the groundwork done here by just thinking about a single play game and then I'll talk to you about what happens when you play a repeated play game going forward. Normal form games are typically the games, people are going to have some rules and there's going to be potential behaviors and on a real-world like the number of cars that they produce, it's almost a continuum. They'll be some functional form and you would have to use some pretty high level math to solve the equilibrium in that particular game. Out what we're going to use is, we're going to use more simplistic games where I'm going to limit the number of plays to just two. We're going to start. So the number of choices they have are just two, like high output or low output. That's an easy one, and we're going to start with something that we call the prisoner's dilemma. The prisoner's dilemma is a game that is probably the most studied single game in all of game theory. That's because it has lots of counter-intuitive outcomes, but yet it's also wildly successful in predicting how people behave with each other, how firms behave with each other, how nation-states behave with each other in terms of the arms races. All of these things are, essentially, they all boil down to the prisoner's dilemma model. But we're going to talk about the original prisoner's dilemma. In the prisoners dilemma, we need to set up something called a payoff matrix. We're going to construct something called a payoff matrix. This payoff matrix is going to look something like this. I'll show you a much better one on the next page, which will be smoother. It's going to say, look here, you've got this player, which we'll call by the original name of top player. You've got this player, which is side player, and we're going to give them two choices. The little example I was just doing, they could produce high output or they can produce low output. The top player could produce high output or low output. That's the only choices. In the real world, there would be a continuum choices and we'd have to do it with mathematical functions. Here, I've limited their choices to just two possible outcomes. That allows us to think about the drama that's going to happen in the scene. For the top player can say, I'm going to produce low output. That means that the outcome's are going to be somewhere in these two cells. Which cell we go depends on other side player does high, side player does high and top player does low. We're going to be up here. If side player does goes low and top players at low, we're going to be down here. So let's do this in a more formal matter like thinking about the prisoner's dilemma model. The prisoner's dilemma model is a pretty straightforward. Here's the story. The prisoner's dilemma is that two guys go in and they decide they're going to rob a bank, and they go on the bank. They got their ski masks on and they robbed the bank and they head out. Unfortunately for them, the banks already ran the silent alarm and cops are coming all over the place and they run to the alley, and the cops are coming up. They get the alley and they decided we're going to get caught and so dump their ski masks and their guns and the money into dumpster. They just stand around. The cops come by, "Did you see anybody come by? We had a couple of guys ran down there really fast." The cops go down and then come back and say, "What do you guys doing here?" They look around and the cops look in a dumpster and they see this stuff and they say, "Hey." They said, "We saw them dump something in there." I don't know what it to was." Well, the cops says, "We'll see about that. Come on, we're going to take you down." So when they get to the police station, the police station is got to put them in separate rooms. So we're going to put their names here as, instead of side player and top player or instead of Toyota versus Honda, we want to put them in separate rooms and the two players are Tim and Larry. So Larry's in one room, you've seen these on these television to police dramas. They've gotten in one room and Tim's another room. So we knew this was going to happen. We had a plan if we're caught. We have a plan, we we have an alibi. We're going to stick to the alibi. So what I want do is figure out what's the payoff for these guys. So when these payoff matrix is, what I've done to make it more, so that you can understand what's supposed to happen here, is I've added these diagonals, which just means that if we are fell to this cell, that this would be the payoff to Tim, and the lower part of the one would be the payoff to me. If we fell into this cell, this part would be the payoff to Tim and this part we be the payoff to Larry. So these this is just how we look at a typical path matrix and we're going to do this quite a bit. We're going to look at this going forward. But right now, we're not talking about Honda against Toyota. We're talking about these two guys and how the prisoner's dilemma model. That's why it's called the prisoner's dilemma. They have a dilemma. Their dilemma is that they had a plan that says that they are going to stick to an alibi. Their alibi says, we were just there and I'm sorry we're in the wrong place at the wrong time we didn't do it in. So if they stick to the alibi, no, by the way, what the payoff that we're going to put in here is what happened, what the district attorney's going to do to you if you do the situation. So what the district attorney says, is, "If you guys stick to the alibi, I've done a certain payoff for you, the payoff for Tim is in this cell and we're going to say the profit that Tim, would be that he's going to end up spending one year in jail." In the proper to Larry, was just going to spend one year in jail. That one year in jail is basically for loitering and other sorts of things, small things they picked up on. Now, on the other hand, he's talking to Larry and he says to Larry, "Why don't you just go ahead and confess." If you confess, what's going to end, and Tim sticks to the alibi, we're going to let you walk. That's called turning state's evidence. So the profit for Larry would be zero years, no time in jail. The profit for Tim, will be the maximum in the state of Illinois, 20 years in jail. On the other hand, in the other room, they are telling Tim the same thing. They're saying, "Come on, go ahead and sign this paper. If you turn state's evidence against Larry, he's down there lying like crazy and the judge will make him get 20 years." Instead of some months. But if you sign this document and confess, he tells Tim, "If you've signed this document and confess and Larry's still doing that lie, well, you're going to get off with nothing." Profit for Tim would be zero years. Now it turns out, if you guys both side, then what the judge will be as lenient as possible. You can't turn state's evidence against somebody who's already confessed. So in this case, you'll each get the minimum according to the law of the Illinois, profit for Tim, profit for Larry's five years. Now obviously, this is a situation where I'm thinking about these simple Pi as the payoff to them. In this case, it's not dollars and cents, it's years in jail, so they would like to minimize that number. They don't want to spend any time in jail, so we'd like to find the best possible outcome. Given that we put these out in that particular way to think this through, we're going to try and figure out what's the possible equilibrium for this model. So remember what we've done. We've taken this stylistic model, we have two players and those players each have only two choices. That means that there's only four possible cells that we can see the world ending up in. The world could end up here, where they both confess, the world could end up here, where they both do alibi, or it could end up here where Tim sticks to the alibi and Larry stabs him in the back. Or it could end up over here where Larry sticks to the alibi and Tim stabbed him in the back. Those are the four possible outcomes, and we have to then figure out what are they going do. We'll do that next. Thanks.