Microeconomics

Learn the toughest concepts covered in Microeconomics with step-by-step video tutorials and practice problems by world-class tutors

Oligopoly

One-Time Games and the Prisoner's Dilemma

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Game Theory, One-Time Games, and the Prisoner's Dilemma

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All right, you guys ready to have some fun? We're about to learn about games. So in an oligopoly, what we're going to see is that they have to work strategically to make their pricing decisions because there's only a few firms and it makes a big difference what your competitors are doing in these markets, In other markets. We were dealing with that idea of marginal revenue equals marginal cost to find profit maximizing quantity. Well, that's why I said this one's kind of like an odd one out. We're gonna be dealing with it through this game theory. All right, let's check it out. So game theory, this is when we're making decisions and the outcomes depend on the interaction with others. Okay, so we're gonna be thinking about what other people are doing when we make our decisions. Okay, so the first type of game we're gonna talk about is a one time game. Alright. A one time game is a game that gets played one time. Who would have thought? Right, so let's go right into our example. This is the prisoner's dilemma. This is one of the most famous examples of a one time game and it's pretty much always used to introduce students to game theory. Okay, so you're gonna see what happens here in the prisoner's dilemma. So let's scroll down a little bit and let's see what we get. We've got Bad Boy Benny and Evil Eddie were recently arrested after some casual B and E. If you weren't raised by the streets like me B and E. That's a little breaking and entering. Just a little friday night. Fun. Right? The police do not have enough evidence to make a strong case against them but can nail them for smaller crimes. So after being separated into separate into different cells, that's that's key. Right? There separated into different cells. They're not able to communicate with each other. Okay? The police make each prisoner the same offer. Okay so this is important right here, this next sentence. These are gonna be the payoffs of the game, right? This is where all of our information pretty much comes from. Right now we can lock you up for a year. That's what they tell the prisoner. Right? Right now we can lock you up for a year. If you confess we will let you go free and your partner will get 20 years. But if you both confess then you're each gonna get an eight year sentence, right? So this is a pretty tricky case, right? Because now they have to think oh is my partner gonna confess? Are they not gonna confess? Right, how am I gonna get the best sentence that I can get? Alright, so to funnel all this information and make it easy to take in, we build what's called a payoff matrix. Okay? And it's gonna look something like this, We're gonna have bad boy Benny's decisions going down, he can either confess or not confess and then we've got Evil Eddie's decisions, he can either confess or not confess. Right? So this is a payoff Matrix and it's still empty. Let's go ahead and fill it in. Alright, this is our payoff Matrix. So let's go one by one here. Let's start with this first sentence. Right? Now, we can lock you up for a year. So this doesn't even matter if you don't confess, we're gonna lock you up for a year, Right? So if both partners don't confess, if I don't confess or if Bad Boy Benny doesn't confess and Evil Eddie doesn't confess, they're gonna each get one year, right? So what we're gonna do is we're gonna go to this box where it's don't confess and don't confess for both of them and we're gonna put on their payoffs. So I'm gonna put B for Bad Boy Benny, he would get one year if they both don't confess, right? How about Evil Eddie? If they both don't confess, they're gonna nail him for the smaller crimes and they are gonna get one year each. Right? Excuse me. Alright, so how about our next sentence? If you confess, we will let you go free and your partner will get 20 years. Okay? So this is a situation where one person confesses and the other one doesn't confess. Okay, So let's say that Bad Boy Benny doesn't confess and Evil Eddie does confess. Well if Evil Eddie is the confessor right? Bad Boy Benny will get 20 years in prison, right? Evil Eddie ratted him out and Bad Boy Benny gets 20 years in prison and Evil Eddie right here, Evil Eddie will get zero years right? He's gonna go scot free. Um If he confesses while Bad Boy Benny doesn't confess. Now it's the same thing the other way around, right? If Bad Boy Benny is the confessor and Evil Eddie doesn't confess. Well, in that case, Bad Boy Benny is the confessor and Bad Boy Benny right down here, right where Bad Boy Benny confesses in this column and Evil Eddie doesn't confess in the bottom row. Well, Bad Boy Benny for confessing, he's gonna get zero years right? And Evil Eddie will end up spending 20 years in prison there. Wow, that sounds awful. Alright, how about the last case? Right, in this last sentence, they say if you both confess, you each get an eight year sentence, so, if they both decide to confess, they're gonna each get eight years right? So, Bad Boy Benny, if they confess and confess, would get eight years and Evil Eddie will also get eight years. Alright, So that's how we set up our payoff matrix. Now we have all our information in one place, the next thing to do is decide what is each person's best decision. All right? So let's pause here and in the next video we'll continue by solving this prisoner's dilemma. Alright, let's do that now
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Dominant Strategy and the Nash Equilibrium

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Alright, so before we check what our best situation is, let's go down here and think about our dominant strategy? So to make our best decision, what we wanna do is we want to think what would we do in response to each of our opponents decisions. Right? So so what would Bad Boy Benny wanna do if Evil Eddie decides to confess or what would Bad Boy Benny want to do if Evil Eddie decides not to confess? Right? We want to see what's our best scenario based on our opponent's decisions, right? This is the interdependence that we see happening. Okay. And we're gonna say that a dominant strategy is your best strategy regardless of what the other player does. So you could have a dominant strategy would be a situation where whether Eddie confesses or doesn't confess, it's gonna make more sense for me to confess, no matter what right, that would be a dominant strategy where you would do the same thing regardless of the other person's decisions. I just want to make sure that you understand that not every game has a dominant strategy and also not every player, sometimes one player will have a dominant strategy and the other one doesn't. Sometimes they both do. Sometimes nobody does. Right? So dominant strategy is only when you have one best best option overall. Okay, so let's go now to our payoff matrix and let's think what each person's best decision could be. In each case, let's start with bad boy Benny's decisions. Okay, so for bad boy Benny to decide what's his best choices? He has to think what Evil Eddie would do. So the first hypothetical would be for Bad Boy Benny to think, Okay, what would I do if Evil Eddie were to confess? Okay, so if Evil Eddie were to confess, I have the option to confess and get eight years right up here. If Evil Eddie is for sure confessing, I can either confess and get eight years or not confess and get 20 years, right? So if the situation is that Evil Eddie is gonna confess, it makes more sense for Bad Boy Benny to confess right? He's gonna get eight years instead of 20 years. So I'm gonna circle this right here. Okay, Bad Boy Benny would be best off confessing if Evil Eddie confesses. Now, what if Evil Eddie doesn't confess? Let's think what would be Bad Boy Benny's best decision. Well, if Evil Eddie's not gonna confess, Bad Boy Benny is either gonna get zero years right, he's gonna go free if he confesses or he'll get one year where they both don't confess, keep their mouth shut and they just get pegged for the one year each. Okay, so in this situation, it makes more sense for Bad Boy Benny to confess again, right? It makes more sense for him to confess because he's gonna go scot free, he spends less time in jail. So in his best interest it would be it would be better to confess, right? Zero years in prison would be his better choice. So you can already see that for Bad Boy Benny, his dominant strategy is going to be to confess, right? Because whether or not evil Eddie confesses or doesn't confess, Bad Boy Benny gets a lower sentence by confessing. Okay, let's do the same thing now with Evil Eddie's decisions. So first Evil Eddie has to think, what would I do if bad Boy Benny confesses? So bad Boy Benny confesses evil Eddie? Well, he has to choose between eight years by confessing. Where they both write each other out or not confessing and bad Boy Benny would go free, but he would get 20 years right. So bad Boy Benny confesses, it makes more sense for Evil Eddie to confess, right? He'd rather spend eight years than 20 years in prison. How about the other one if Bad Boy Benny decides not to confess? What's evil Eddie's best decision? Well, he has to choose between zero years and one year, right? By confessing he can go scot free, but by not confessing, he would spend one year in prison. So his best option again is to confess and this is the prisoner's dilemma because both of them have a dominant strategy to confess, right? So it's in their best interest personally to confess. But what we see is that that's a bad thing overall, right? Because they're both going to confess and they're both going to end up at eight years in prison, which is not ideal, right? We would probably think more ideal at least would be this box right? Where they both don't confess at least instead of eight years each, they're only getting one year each, right? So we would think that that box is where they would want to be, but since they can't talk to each other, they're not gonna be able to make that decision and they're both going to end up confessing at least that would be their best choice. Right? So what we call this situation where where they both end up in the same place, this is called the nash equilibrium, nash. So this guy nash, he's a famous economist who discovered this idea about game theory, about this nash equilibrium. They made this excellent movie. There was a book written about his life called A Beautiful Mind. If you haven't seen it, it's a great movie whether or not you love Economics or not. Um and I would suggest that he won a nobel prize for these ideas. So this is what he thought here, the nash equilibrium is gonna occur where all players are making their best choice, so they're all making their best choice which in this case was for each to confess right? They're making their best choice given their competitors options, right? So that nash equilibrium is where they're both making their best choice and that was confess and confess, right? So the nash equilibrium would be this box up here that I'm gonna circle in green. The confessed confessed box, right? The confessed confessed box is our nash equilibrium okay? But notice that it's not the best situation, right? We've been talking about equilibrium in this class and equilibrium is usually a good thing, right? Equilibrium was where we're being efficient, this and that. But in game theory, the equilibrium does not necessarily mean it's the best outcome for all the players, right? It's just where they're all making the best decisions and this is where we end up. Okay? So it's not necessarily the best outcome. It's where we're gonna end up if everyone's just making the best decisions. So that's unfortunate, right? They're both gonna end up spending eight years in prison. But they could possibly have had a different situation if they could cooperate. Alright, so let's pause here and in the next video, let's discuss cooperation.
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Collusion, Cartels, and Price Leadership

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All right? So let's continue here on the bottom part of the page. So in this game, both players would have been better off, right? They could have been better off if they could cooperate if they could have discussed with each other. Hey, let's both keep our mouth shut and will only get one year in prison, right? If they could cooperate and both don't confess, they would have been in a better place, right? They would have had only one year each instead of eight years each. Right? So what do we call this? Right now, let's extend this to the idea of the oligopoly. Right here we were just talking about the prisoner's dilemma. Let's see how this relates in the big sense. Well, when we think about this with firms, this could be firms colluding together. We're gonna think of this idea of collusion where they work together to set prices, right? So it's an agreement between the firm to to set their quantities and price. Okay, so that's called collusion where they're working together and that's illegal in the United States, competing firms are not allowed to explicitly talk about pricing or quantity, right? It's not like Mcdonald's can call burger king and say, hey, let's let's raise our prices of burgers and everyone's gonna charge $3 for burgers or coke. Calling up Pepsi and saying, hey no longer are we gonna sell 20 ounces for a dollar. 20 ounces are gonna be $4 now. Okay, that's not allowed. That's illegal in the U. S. Because it's reducing competition. Okay. Um So what we call firms that are colluding right is called a cartel. Okay, a cartel is firms that are colluding together. The group colluding together is a cartel. A very famous cartel is Opec. So opec opec is a group of Middle Eastern nations that get together and set prices and quantities that they're gonna put for oil, right? They control a lot of the world's oil and they limit the quantity so that they can inflate the price. Okay. That would be illegal in the United States but they're not in the United States so they can do whatever they want over in their country. Okay, so opec is an existing cartel. But what we're gonna see in situations where people are colluding and where we have cartels is that is that there's an incentive to cheat to increase um their individual profits. I'm gonna put individual profit up here. Okay, so you could cheat when you know other people are going to cooperate, right? So the idea is this cartel, they all cooperate on what quantity they're gonna produce. But then you have that information, you know what everyone's gonna produce and you have that incentive to cheat and produce a little extra to make more money or something like that. So let's think about this idea of cartel in our situation up here, right? If they had gotten together and they were able to collude together to say, hey let's both not confess right? If we both don't confess, we only get one year instead of the eight year equilibrium. They're obviously both very smart at game theory and have been through this class before. Alright, So they think they could collude and they'll each get one year. But let's think about that incentive to cheat. Now, Bad Boy Benny, after this discussion with Evil Eddie knows that Evil Eddie is planning to not confess, right? So if Evil Eddie is planning to not confess, Bad Boy Benny could cheat and confess, right? He could confess and he could go free. We could get me in the situation where Evil Eddie doesn't confess and Bad Boy Benny takes advantage of it confesses. Bad Boy Benny's gone zero years in prison. Evil Eddie gets 20 years and Bad Boy Benny is better off, right? He increased his individual profit, right? His own profit, which was less time in prison by cheating. Cool. So there's that incentive to cheat when you know that other people are going to cooperate. So last thing here is an idea of implicit collusion. Okay, so price leadership is not illegal in the United States, it might be a bit unethical, but the idea here is where we're gonna have um one firm taking the role of the price leader. Okay, So it's a form of collusion where one firm announces a price. So let's say there's some new video game system, right? And it's like the PS 12 or whatever, and the PS 12 comes out and walmart announces we are selling PS twelve's for $500 Right? And then you see targets $500 toys r us, $500. Everybody's selling the PS 12 for two or for $500. Right? And that's because of this price leadership situation, right? The first firm announced the price and everyone followed suit because they knew that they could make more money if they all charge the same high Price and increase their profit. Okay, so this is not illegal because you're not really colluding, you're just saying, Hey, they're charging 500, I'm gonna charge 500. We didn't talk about this at all. It just seems like the right price to me, right? So you can see how there could be some gray areas there, right? Price leadership. That's a form of implicit collusion. Cool. Let's go ahead and move on to the next video now.
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One-Time Games:Finding Dominant Strategies and Nash Equilibrium with the "Check and X"Method

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Alright guys, now I want to show you an easy method and this is the method I use when solving payoff matrices and using game theory. Okay, So what this helps us do is it helps us find our dominant strategies and it helps us find nash equilibrium. Okay. So what I call this method, it's something like the check and X method, right? Kind of a weird name. But the check and X method is what it is. And these are the steps we're gonna do first. We're gonna put a check next to each of player one's best strategies and put an X for each of player two's best strategies. Okay, So that's that's all we have to do. Just the checks and the excess and then we're gonna analyze our solutions. Okay, So let's go ahead and do the checks and excess and then we'll see how our solutions work. Okay, so let's go down to this example notice we didn't even need to be given a story in this case. Right? They didn't have to tell us this whole back story. They just threw payoffs at us. Right? They told us in this case you're gonna get $300 in this case. 500 right? They just through numbers we don't need a whole backstory. The payoff matrix can just be filled in like this. Okay, so let's go ahead and start with player one's best decisions, right? We want to see what's player one's best strategy. Should he choose option a or choose option? Okay, So let's go ahead and start here, remember that to pick your best strategy, you have to decide what you would do in each case of your competitors choices? All right, So for player one's first choice, they have to decide what would I do if player two chooses a right? Player two is going to choose A. What's my best strategy? Well, player one, if player two chooses a player one can choose A. And get 300. Or player one can choose B. And get 400. Right? So you can see that if player two chooses a player one will want to choose B. So we're gonna put a check down here in this box because that is player uh Player one's best strategy. In that case. Now what if player two chooses be if player choo chooses be. Player one can choose A and get 100. Or they can choose B. And get 200. So again B is gonna be their best strategy. Okay, So before we make our conclusions, let's do the other thing. Let's do the exes with player two's decisions. All right, so the opposite. Now, Player two has to think what would be best if player one chooses a. What's my best choice. So player one's gonna choose A. So we have to look in this row right here, Player two can either choose A. For 500 or player two can choose B. For 400. So player two is gonna choose a right A. Gets them more more money. So they choose A. When player one chooses A. Now what if player one chooses B, what is player choose best strategy? What player two? Uh Could either get 100 If if player one chooses A or player two can get 200 right? That's behind me. Right? Player two can either get the 100 down here or the 200. So they're gonna pick the 200 right? So what we're gonna do is we're gonna put an X in this box as well. So notice that this box in the bottom right has a check and an X in it. And then there's a check in this box and an X. In that box up there. And then the box, player one choosing a player to choosing B has nothing in it. Okay, So now let's go to our analyze our solution up here. Step three. So, the first thing we wanna do is check if there's any row or any column that has both, checks, checks or both. Exes. All right. So what we see is that we have checks and checks in this row, right? There's a row with checks. But the exes are not in a column, right? The exes are diagonal. So this is not Okay, this diagonal with the X. Is that that does not give us our answer. Okay? So what we do, What we do know is that player one has a dominant strategy, right? Because the checks are for player one and player one has a row with both of the checks. Right? So both of the checks are in the be ro so for player one the dominant strategy is to choose B. Okay, So let's go down here. Player one's dominant strategy is be right. And that's because those checks are both in that row. What about player two's dominant strategy? Well, player two, we saw that the exes are not in the same column at all. Right. There's an X. In column A. And an X. In column B. So they have no dominant strategy, right? So I told you that there could be situations where you might not have a dominant strategy. Well, here's one, right? He's gonna choose a in one case and be in the other case. So last but not least. Let's discuss nash equilibrium and that's here in B. Remember that a nash equilibrium is where both players are making their best decisions based on their opponents decisions. Okay. So what we're gonna see is that any box that has both a check and an X. So if there's a box with a check and an X in it, that is a nash equilibrium. Okay. And what we have right here behind me, we've got that bottom right box with both a check and an X. Okay, so that is going to be our nash equilibrium right there. And that would be basically the situation where we're probably gonna end up if everyone's acting rationally, we're gonna end up in a situation where both player one and player to get $200. Okay, So nash equilibrium is, I'm gonna put it in a bracket here, be comma be right? Where they both choose B. I don't think that's like proper uh you know, writing etiquette with it, but that's just the idea, right? They're both choosing B. And that is the nash equilibrium. Alright, So let's see, is there anything else here? No, that's that's about it for this lesson, right? I just wanted to teach you how to use that check and X. Method. Okay, so let's go ahead and move on to the next video. Let's get some practice using this method.
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Problem

Based on the information in the payoff matrix, which of the following is true? 

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The game above has: 

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Problem

In a cartel, the incentive to cheat is significant because

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