ECON 159: Game Theory

Lecture 6

 - Nash Equilibrium: Dating and Cournot


We apply the notion of Nash Equilibrium, first, to some more coordination games; in particular, the Battle of the Sexes. Then we analyze the classic Cournot model of imperfect competition between firms. We consider the difficulties in colluding in such settings, and we discuss the welfare consequences of the Cournot equilibrium as compared to monopoly and perfect competition.

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Game Theory

ECON 159 - Lecture 6 - Nash Equilibrium: Dating and Cournot

Chapter 1. Coordination Games: Scope for Leadership and Strategic Complements [00:00:00]

Professor Ben Polak: All right, so last time we were talking about The Investor Game and this was a coordination game, and we learned some things. I just want to recall some of the things we learned so I can highlight them a bit. One thing we learned was that communication can help in a coordination game. So I forget who it was, but someone down here who has disappeared, was our Jimmy Stewart character and helped coordinate you on a better equilibrium simply by suggesting what you should do.

There are two lessons here. One is this is very different from the Prisoner’s Dilemma. So we learned the very first time that in the Prisoner’s Dilemma communication per se won’t help, but in the coordination problem, which could be just as serious socially as a Prisoner’s Dilemma, in a coordination problem it may well help. The reason it helps is you’re trying to coordinate onto a Nash Equilibrium. One way to think about Nash Equilibria is that they are self-enforcing agreements, so provided that everyone believes that everyone is going to go along with this agreement, then everyone in fact will.

I want to make another, slightly more philosophical remark associated with this and it’s to do with the idea of “leadership.” So leadership is kind of a big word that you see written probably too often these days, in too many newspaper articles, and it probably comes up in too many Yale classes, and I don’t claim to know anything about leadership. And I don’t think Game Theory is going to contribute to anything to understanding about leadership. But one thing we can do is tell you where leadership may help. In the coordination game, where the idea is to try and get people to coordinate on a particular equilibria rather than on another equilibria, or worse still, to be uncoordinated entirely. In those kind of games, leadership can help tremendously. A little bit of leadership can help tremendously. So these games, these coordination games, are games where there is a “scope for leadership.”

Just to see that in a very simple example, again, we don’t use such a complicated example as the one we looked at last time, you could imagine a game, a really trivial coordination game, which looked like this (1,1) (0,0) (0,0) (1,1). And clearly in this game what matters is coordinating. You either want to coordinate on up left or you want to coordinate on down right. You don’t want to end up uncoordinated on down left or up right. Everyone see that? So in this game if you just played it, it’s quite likely you’re going to end up uncoordinated, but if you have a little bit of leadership can say okay let’s make sure this is where we coordinate, or let’s make sure this is where we coordinate. So this matters a lot. And I don’t want to overplay the social importance of this, but go back a couple of years to what was happening in the aftermath of Katrina and realize how important–how bad things get when things fail to be coordinated.

One other remark before we leave this, in the game we played last time, in the investment game, one feature of that game was that the more you thought other people were going to invest, the more you wanted to invest. The more you thought other people were going to invest, the more likely that you were going to invest. If you go back to the time before that, we talked about this partnership game. In the partnership game you were contributing effort to a joint project. It could have been a law firm; it could have been working in the study group on a homework assignment.

In that game, if you remember what the best responses looked like, they looked like this where this was the effort of Player 1; this was the effort of Player 2. This was the best response of Player 1 and this was the best response of Player 2. This game also has the feature that the more effort the other person–the more the other person does–the more you want to do. The more effort your partner provides into this project the more effort you want to provide in this project. I just want to introduce a bit of jargon here. These games in which the more the other person does the more I want to do, these are called games of strategic complements. These are games of strategic complements. So both the investment game and the game with–the partnership firm game are games with strategic complements. We could call the strategies strategic complements. We’ll come back to this later on today.

Chapter 2. Coordination Games: The Battle of the Sexes [00:04:59]

So before we leave coordination games, I want to look at another one, a little bit more complicated one perhaps, that we mentioned briefly last time. So we’ll look at and play another game, and we’ll call this game, “Going to the Movies.” So I always regard one purpose of this class to help hapless Yale students in their dating strategies. That seems like a good thing to do. How many of you are Econ majors? There’s a lot of people who probably need a lot of help with their dating strategies, right? Okay, so the idea of this game is a couple is going to meet up at the movies. And let’s have a look at the movies concerned here and we’ll draw–there’s three possible movies. The movies we’re going to look at are The Bourne Ultimatum, and the movie called Good Shepherd, and a movie called Snow White and I’ll explain the game in more detail in a second.

Now, it used to be in the old days before I had kids, I could list off 15 current movies and help you a bit more with your dating strategy by giving you instant movie reviews, but now I have kids, I get to see precisely two movies a year, and the two movies I got to see this year were The Bourne Ultimatum and the Good Shepherd. And now I think about it, both of those have Matt Damon in it. My wife chose both so maybe I need to work out some more. So just to give you some idea about these movies, how many of you saw Bourne Ultimatum? Quite a lot of you; this is a movie, it has pretty good action, virtually no plot. Basic lesson of this movie is–if you take home a lesson–it’s that all spies are psychos or something like that.

Then how many of you saw Good Shepherd? A lot of you saw that as well. It’s actually pretty good. It has a lot of plot and no action, so it’s the other way around. The basic lesson of this movie is–and you probably knew this already–everyone at Yale is a spy … and by the way also a psycho. The third movie is Snow White, which I haven’t gone out to see but my four-year old daughter has seen on 24 of the last 27 nights on video. And this movie, I don’t know if I would recommend it that much, it’s–perhaps I’m being too PC–but I’m not convinced that for the modern woman, hanging around waiting for your prince to come is really a good strategy. By the way, for those of you–those of you who are, if you are doing that strategy, got me in trouble. Never mind, if you are doing that strategy, take it from a Brit, most princes are as dumb as toast, not worth waiting for.

So let’s put the payoffs in here. So the idea here is that these people are going to meet at the movie, they’re going to go out to a movie. And they’ve decided to go to the Criterion [New Haven movie theatre], or the local movie house, and there were these three movies showing, and they’re all excited about going to this movie except being Economics majors and not very good at dating, they have forgotten to tell each other which movie they’re actually going to go to. They’re going to meet in there on the–in the back row probably and–but they’re not telling us which movie. So that’s a problem and let’s put in the payoffs, and we’ll put in the payoffs that roughly I think would correspond–so my preferences and we’ll talk about it–what these preferences mean in a second.

So here we go so … -1,( -1, 0), (-1, 0), (-2, -2). So here are the preferences for these movies of Player 1 and Player 2, and you can see from these preferences, from these payoffs, that the best thing for Player 1 is for both players, both people to meet and go to the Bourne Ultimatum; this is the action movie. So it’s not to be sexist–assume this is the she and this is the he, so she would like to go and see Matt Damon beat people up and coordinate on that, her favorite thing. And she gets nothing if they fail to coordinate and so on and so forth. Her second choice is to manage to coordinate at Matt Damon being a Yale spy, and then she really doesn’t want to go and see Snow White, and the worst thing of all, actually this is for both players, the worst thing of all is to coordinate with Snow White because then you’ve got to talk about it afterwards at Koffee [New Haven café].

Same thing with the other person. Same thing exactly, except that he would rather meet at the Good Shepherd. Since he’s a Yale guy, he probably thinks of himself as the Matt Damon character though perhaps not, and failing that, he’d like to coordinate at the Bourne Ultimatum. But again, the disaster is to fail to coordinate and go to see different movies. So these are the preferences. What can we tell immediately, if you’re stuck in this situation? What can you tell immediately about this game? What’s an easy first step? Let’s get the guy in the red here.

Student: Snow White is dominated by both players.

Professor Ben Polak: Good, so for both players Snow White is a dominated strategy. So both players having taken this class–I told you it would help you with your dating. So both players will realize that they should at least not go to Snow White, is that right? Great animation but not perhaps a good date movie, so it’s gone. Everyone see that? So that leaves us just with these two options, the two Matt Damon movies, Bourne Ultimatum and Good Shepherd. And here are the remaining payoffs, and let’s see how we play this out. So let’s pick on some people. Ale you want to pick on somebody? I’ve got Kai here as well. Okay, so you’ve picked on somebody there. The person Ale picked on, you want to stand up? Okay, your name is?

Student: Nina.

Professor Ben Polak: Say again loudly.

Student: Nina.

Professor Ben Polak: Ana? Shout it out so everyone hears.

Student: Nina.

Professor Ben Polak: Nina, okay Nina I’m sorry. Nina and Kai get your guy to stand up there. And your name is?

Student: David.

Professor Ben Polak: David so–do you know each other?

Student: No.

Professor Ben Polak: No you’re–so you’re about to go on a date. So Nina’s preferences are Player 1’s preferences here. So she–her best thing now is to successfully meet David at The Bourne Ultimatum and David’s preferences are Player 2’s preferences, so his favorite thing is to successfully mette at The Good Shepherd. Why don’t you write down a second on the corner of your notepad what it is you’re going to choose to do with my T.A.’s watching so we can’t cheat. Both have it written down? Should we have a good look at them? Should we just embarrass them a bit? We looked at them already I guess. So Nina, shout out to the crowd what it is you’ve–where it is you’ve chosen to go.

Student: Bourne Ultimatum.

Professor Ben Polak: Bourne Ultimatum; and David where do you choose to go?

Student: I also put down Bourne Ultimatum.

Professor Ben Polak: They’re going to coordinate; that’s very good. I don’t know what this tells us about David and Nina, but don’t let them off the hook quite yet. So now, that was good, we managed to coordinate. I think we can all see that we might not have done. Let’s try some communication in this game. We said that coordination games are helped by communication. So let’s imagine we’re playing the game again, once again, you’re going to the movies, once again the first one didn’t happen so you didn’t–for some reason it was cancelled that night because someone had a bad cold or something. You are about to go off on Friday night to see these movies and let’s allow some communication ahead of time. So David, you’re about to go, you realize you’ve got this coordination problem, so you phone Nina up and you can say whatever you like, so what would you say to Nina?

Student: Looks like you want to go to Bourne Ultimatum. I’d rather go and–if you’re going to be stubborn I’d rather go see Bourne Ultimatum with you than not go on a date at all.

Professor Ben Polak: So here’s another clue. Stubborn is not a good first date word, but okay. So Nina what would you say in response to that?

Student: I’ll go to Bourne Ultimatum.

Professor Ben Polak: All right, so write down what you’re going to do this time. Nina what did you do?

Student: Bourne Ultimatum.

Professor Ben Polak: David?

Student: Bourne Ultimatum.

Professor Ben Polak: So they’re still managing to coordinate but you–okay, so thank you for this couple, let’s give them a round of applause. We may come back and pick on you later on in the course, but we’ll leave it for now. So in this case, the communication worked but am I right in thinking that the communication isn’t such an instant solution as it was in the game we saw last time? This game’s a little bit harder to get communication to work. Why? Is that right first of all? So what’s the problem here? Why is this a slightly more difficult game? Let’s keep the mikes handy a second. Why is this a more difficult game to attain coordination in?

Student: Can I say the best response is the Nash Equilibrium?

Professor Ben Polak: Well no, hang on a second. Just slow down a second, that’s a good one. Let’s start with that. What are the Nash Equilibria in this game? Somebody, let’s just pick on somebody. Just pick on anybody okay. Yeah, what are the Nash Equilibria in this game?

Student: Both players doing Bourne Ultimatum and both players doing Good Shepherd.

Professor Ben Polak: All right, so in fact, if we checked the Nash Equilibria in this game, the Nash Equilibria, and you can check that they are in fact best responses, are both people doing The Bourne Ultimatum or both people doing The Good Shepherd. And the reason for that is if the other person’s going to go to Bourne Ultimatum I want to do that, and if the other person’s going to go Good Shepherd I want to do that. So these are both Nash Equilibria.; That isn’t so much the problem. That was true last time as well. There were two well coordinated Nash Equilibria although one was better than the other. What’s the tricky thing here? What’s the extra trick here? Is there a mike on this side? There was a hand up there, yeah. Stand up. Great.

Student: Each player prefers a different Nash Equilibrium.

Professor Ben Polak: All right, so unlike the other game, the two examples of coordination games we saw so far were really pure coordination problems. Is that right? They were pure coordination problems. There was no conflict at all. In the case of the Investor Game last week, every player preferred one equilibrium to the other. The game I just put up, that’s a trivial example, it didn’t really matter which equilibrium we played here, we just wanted to play an equilibrium, is that right? But here, there’s a potential source of conflict here. Both people would rather be at an equilibrium than to be mal-coordinated or uncoordinated, but Player 1 wants to go to Bourne Ultimatum and Player 2 wants to go to Good Shepherd, and actually I thought Nina’s strategy there was pretty good. Nina’s version of communication was, “I’m going to Bourne Ultimatum.” That basically fixed things.

But you could imagine that communication could break down here. There could be a negotiation going on. So think about–I don’t want to push this too hard but think about other games where people are communicating to fail–to avoid a failure of coordination. There’s a strike negotiation going on as we speak. In fact, there was a critical moment this morning up in Detroit between General Motors and the united autoworkers and I don’t want to say that strikes are only about mis-coordination, but clearly, everyone’s better off if they come to some agreement rather than strike. There’s at least some agreement that’s going to be better for everybody than a strike, and yet, because they’re conflicting interests there, basically in that case conflicting interest about health and pension payments, it could well be that you end failing to coordinate.

So this game has a name, and actually somebody mentioned it last time. Who was the person who mentioned this last time? Somebody mentioned this. What was this game called? Shout it out! Battle of the Sexes. That’s good. So this game’s called The Battle of the Sexes and we’ll see it in various forms over the course of the semester. It’s actually a very interesting game. Games like it, they are coordination games but different people disagree about where you’d like to coordinate.

Chapter 3. Cournot Duopoly: Math [00:18:37]

So much for talking about coordination games and helping you with your dating strategy. This being a Game Theory class, there’s at least one game we have to discuss and we’ve come to it now. So we’re going to spend most of the rest of today talking about Cournot Duopoly. Before I do that, let me just check. How many of you have seen Cournot Duopoly before? Just raise your hands just for me. All right, so maybe about half of you. So those of you who have, don’t worry, I mean this will be a bit review, but we’ll see it more through the eyes of Game Theory this time, and for those of you who haven’t, don’t worry we’re going to go through it.

So this is a classic game, perhaps it’s one of the most famous games, and therefore worth studying in the class. Now, as a purely Game Theory exercise, one reason for studying Cournot Duopoly, is that so far we’ve discussed how to find Nash Equilibrium when there are few players, each of whom has few strategies, and we’ve discussed how to find Nash Equilibrium where there are many players, each of whom has few strategies, and this is a game where there are few players, maybe two, but they have many strategies. They actually have a continuum of strategies. For those people who aren’t familiar with this and are worried about the economics of it in particular, it’s gone over in considerable detail in chapter 6 of the textbook.

So just to motivate this a little bit, so there’s two reasons why this game is interesting. This is a game in which there are two firms who are competing in the same market, and we’ll give a bit more detail in a second. One reason this game is interesting from the point of view of economics, is this game lies between the two extreme cases that you learn about in your Intro Economics class. One extreme case is perfect competition and the other extreme case is monopoly. So this is really the first attempt, way back in the nineteenth century to study a market that’s somewhere in the middle, where it happens most markets are–there are two firms.

We’re interested in two things here. We’re interested in what’s going to happen in these markets. And then from a welfare point of view, from a policy point of view, we’re interested about whether this is good for consumers or good for producers or what. How this relates to profits and consumer surplus. So with that in mind, let’s just set the game up.

So the players in this game are two firms and the strategies in this game for the firms–and this is going to turn out to be important–the strategies are the quantities that they produce of an identical product. So they are the quantities they produce, each of them produces, of an identical product. So as far as the consumers are concerned, these two products are perfect substitutes. You could think of these as two companies producing bottled water–and now we’re going to get hundreds of letters saying not all bottled waters are the same, especially from Italians and the French, but never mind. Let’s just pretend that they are. So just to emphasize that the strategies are quantities, rather than using S let me use q today to be the strategies. So qi and q-i or q1 and q2 will be the strategies.

We need to give a little bit more structure on the payoff before I get to the payoffs. So in particular, I need to tell you what is the cost of production. And the cost of production in this game is simply going to be cq. So if I produce one unit it costs me c; if I produce two units it costs me 2c; if I produce 100 units, it’ll cost me 100c and if I produce .735 units that costs me .735c. So for those people who took Intro Economics, which is most of you, this is a game–this a setting in which we have constant marginal costs. The constant marginal cost is c.

I’ll need to tell you about how prices are determined in this market. So prices are determined as follows: prices depend on some parameters I’m just going to call a and b, and let me write the equation and then we’ll see what it looks like. So basically the idea is that the more these firms produce, so the more the total quantity produced q1 + q2, the lower is the price in the marketplace for this product. Let’s just draw a picture of that; we’ll come back to this in a minute. Let’s draw a picture which we can keep for later. Actually, let’s save myself some time and bring this down. So this equation I’ve just written, if you can imagine q1 + q2, total quantity on the horizontal axis, and you could imagine price on the vertical axis, and basically what we’re saying is that prices depend on total quantities as follows: where the slope of this line is -b.

Now, for those of you who took Economics 115 or equivalent, what is the name of this line I’ve just drawn? That was way fewer responses than I wanted. Let’s try again; what’s the name of this downward sloping (hint-hint) line that I’ve just drawn? It’s a demand curve. Thank you; it’s a demand curve. That was the demand. It tells me–the other way around–to look at how prices correspond to quantities, it tells me the quantity demanded at any given price. We’ll come back to that in a second. Meanwhile, let’s just finish up what we’re doing here and put in payoffs.

So payoffs for these firms are going to depend on profits, the payoffs. The firms’ aim to maximize profit and profit is going to be given by P times q minus–sorry, let’s do this for Firm 1. Let’s be careful here; let’s do this for Firm 1 and we’ll do Firm 2 in a second. So the payoff for Firm 1, as it depends on the quantity that she produces and the quantity that the other firm produces, is going to be prices times the quantity that Firm 1 produces minus costs times the quantity that she produces. So this is–this term here is revenues and this term here is total costs. Revenues minus costs makes profits, and again, for those of you who are less familiar with economics, I’m hoping this is not really too hard but you can read up on it.

What I’m going to do here is I’m going to substitute in for prices. So I’ve got this expression for prices here, here it is, and I want to plug it into that P there. And I’m going to rewrite it now with that P expanded out. So I’m going to get–I’m going to multiply it out at the same time, so I’m going to have aq1 - bq1² - bq1q2 - cq1. All I’m doing here, nothing particularly exciting, I’m simply plugging this expression in for P and then I multiplied the whole thing out because it was all multiplied by q. I should warn you again that I’m very likely to make mistakes when I’m doing this kind of thing, so please catch me I do. So here I have a new expression for Firm 1’s profit and I could do the same for Firm 2, but I’m not going too because it’s–I’m getting bored. So this is just Firm 1’s profit.

Now, with this in hand, what I want to do is, and in Firm 2 I have a similar expression, what I want to do is, I want to figure out the Nash Equilibrium of this game. Now, I will need the other board; I want to figure out the Nash Equilibrium on this game. Both firms are producing quantities; both firms are trying to maximize profit, and I want to find out the Nash Equilibrium. How do I go about that exercise? What’s the–what do I need to do find each firm’s Nash Equilibrium? How do we always find Nash Equilibrium when we’re not guessing? What do we do? Anybody? Somebody way over there who’s going to be way off camera but we can actually get a mike in there. Can we get a–he’s way in the corner there. Come this way, a little closer to the aisle, there you go.

Student: Well, we would just delete the–we would iteratively delete dominated strategies.

Professor Ben Polak: Okay, we could do that. We could do that; we could try iteratively deleting dominated strategies and see if that process converged. I had something else in mind here, something else more straightforward. What else? The guy in red.

Student: Identify the best responses of each player as a function of the others and find out where they intersect.

Professor Ben Polak: Say that loud–more–that was correct, say it more loudly, shout.

Student: Identify the best responses of each player as a function of the others and find out where they intersect.

Professor Ben Polak: Okay good. So what we’re going to do is we’re going to figure out Player 1’s best response quantity to each possible choice of Player 2, and then we’re going to flip it around and figure out Player 2’s best response quantity to each possible choice of Player 1, and then we’re going to see where they–where those coincide, where they cross. So, just like we’ve done several times before. So what I need to do then is I need to figure out what is Player 1’s best response for each possible choice q2 of Player 2. So how do I do that? I want to find the best response. How do I do that? Everyone is looking like this is–I bet you all know really. Anybody? Cold call somebody? Athletic day for you. Somebody way at the back there; stand up and shout.

Student: Find the best response function by differentiating this and maximizing the function for every q2, so we could just–

Professor Ben Polak: Good, so what we’re going to do is–all right good, we’re going to–we’re trying to maximize–the best response is going to maximize these profits. So, in particular, what we’re going to do to find out what quantity q1maximizes this profits for each choice of q2. We’re going to differentiate this with respect to q1 and then what? And then set it equal to 0. So again, let me just take a look out there. How many of you remember this? We did this once before in this class. How many of you remember this from when you were–high school calculus days or from 112? Quite a number of you actually. Okay, good, so what we’re going to do is we’re going to differentiate this thing to find a first order condition. We’re going to differentiate with respect to q1, the thing we’re trying–our control variable, the thing we’re trying to maximize the thing with respect to and set it equal to 0.

So what are we going to get? So this term here aq1 is going to become an a and this term here -bq² is going to become a 2bq1 and this term here, -b q1q2 is going to become a -bq2, and the last term -cq1 is going to become a -c. Everyone happy with that? What I did was I differentiated this fairly simple function with respect to q1 and since I want to find a maximum, what I’m going to do is I’m going to set this thing equal to 0. At my maximum, I’ll put a hat over it to indicate this is the argmax; at my maximum I’m going to set this thing equal to 0. Now, since we’re being nerdy in this class, despite our attempts to go dating, we’re being nerdy, let’s actually be a little bit careful. This was a first order condition or a first order necessary condition. I actually need to check the second order condition. So how do I check the second order condition? I differentiate again, right? I differentiate a second time and check the sign, so the second order condition, I differentiate this expression again with respect to q1. The only place q1 appears here is here, so when I differentiate again I’m going to get -2b and sure enough that’s negative, which is what I wanted to know, just to check that when I’m finding this thing, I’m finding a maximum and not a minimum.

Good, so this is our key expression and I’m going to use this expression; I’m going to solve it for q1. So the best response for Player 1, as a function of what Player 2 chooses, q2, is just equal to the q1 hat in this expression and if I solve that out carefully, I will no doubt make a mistake, but let’s try it. I got to have a - c over 2b - qover 2. So try that at home, I think I did that right. So I took this to the other side and I divided through by 2b and the b here cancels. Everyone just stare at that a second, make sure I got it right. I’m very capable of getting these things wrong. Good, so what I have here is an equation that tells me Player 1’s best response for each possible choice of Player 2. I could do the same thing for Player 2 to find Player 2’s best response for every possible choice of Player 1.

On the other hand, we know it’s–the math is going to be the same, so let’s cheat. I know that Player 2’s best response for every possible choice of Player 1, which if we had done it would be q2 hat is going to a - c over 2b–q1 over 2, right? I’m just flipping around the 2’s and the 1’s. So at this point I’ve found Player 1’s best response as a function of q2. I’ve found Player 2’s best response as a function of q1, and the way I did it, just remember, the way I did it was I applied a little bit of 112 and/or high school calculus, so that’s your single variable calculus. Now, we could just solve at this point but let’s not. Let’s draw these.

So what I’m going to do is I’m going to go back to my picture and draw these functions. I’ll just remind us of what these functions are; it’s a - c over 2b - q2 over 2 and accordingly for Player 2. That’s the thing; it was hidden by the board. What I want to do is I want to draw a picture, a little bit like we did for the partnership game. So in the partnership game we put efforts on these axis and now I’m going to put quantities on these axis. So this is going to be the choice of Player 1, and this is going to be the choice of Player 2, and what I want to do is I want to figure out what this looks like. For each q2 that you give me or that Player 2 chooses, I want to find out and draw what is Player 1’s best response. Everyone happy with what I’m doing here?

So let’s start somewhere; what we’re going to do is we’ll refer back to this other picture, which is why I left it here. So, in particular, what would be Player 1’s best response if Player 2 didn’t produce at all? How do we find that? Without worrying about the economics of it a second, how would we find that just as a math exercise? So suppose Player 2 doesn’t produce at all. What is Player 1’s best response? Someone read it off for me. Why don’t we cold call somebody out, you can–

Student: a - c over 2b.

Professor Ben Polak: All right, so shout that out.

Student: a - c over 2b.

Professor Ben Polak: Good, so if I plug q2 = 0 into here, this term disappears and I just get a - c over 2b. What’s that telling me? Let me put the 45º line in here. What’s that telling me is that if Player 2 chooses not to produce then Player 1’s best response is a - c over 2b. Now, I claim that that quantity, a - c over 2b actually has another name. What’s the name of that quantity? What’s the name of the quantity a - c over 2b? Once we escape the algebra just think about it in terms of economics. Let’s think about it a bit, let’s go back to the previous picture.

If Player 2 is producing nothing, then what is Player 1 effectively? He’s a monopoly; so she’s a monopolist, so we know how to figure out monopoly quantities from what we did in 115 or equivalent courses. Let me finish the picture here; here’s my demand curve, here’s my constant marginal cost at c, and I want to use this picture on the left now to figure out what the monopoly quantity is. Can anyone tell me–let’s cold call somebody again. Raise your hand if you took 115, 110 or 150. No, no, I know more of you took it than that. It’s a pre-req for the class, right? Raise your hands if you didn’t take any of those things. Ale, keep your hands up. Cold call somebody who hasn’t got their hand up. Anybody, go ahead. Do you want to tell me what the–where’s the monopoly quantity on this picture? The woman–where you are–the woman here.

Student: Oh no I really don’t remember.

Professor Ben Polak: Don’t remember.

Student: I’m sorry.

Professor Ben Polak: Who else doesn’t remember? Somebody must–don’t remember is a good reply if you’re auditioning to be the Attorney General, but it’s not such a good reply–anybody? Somebody help me out here, where’s the monopoly quantity on this picture? It’s a good review exercise, yeah.

Student: It’s where the marginal revenue is equal to the marginal costs.

Professor Ben Polak: It’s where marginal revenue equals marginal cost. I should have picked on some of the SOM people by the way. It’s where marginal revenue equals marginal cost, right? The reason I should pick on the SOM people, is their ambition in life is to run a monopoly, right. Trouble is I haven’t drawn the marginal revenue curve yet. Right, you can’t actually see the marginal revenue here. What does the marginal revenue look like in this picture? Why don’t we go to the same guy, the guy in brown. Where’s the marginal revenue–what’s the marginal revenue look like in this picture on the left?

Student: It’s half the slope of the price.

Professor Ben Polak: Yeah, or it’s–I guess it’s twice the slope but that’s fair enough. I know what you mean. So the marginal revenue here, let’s put it in a different color, the marginal revenue looks something like this and as–I don’t know what your name is–as the guy said, the monopoly quantity is when marginal revenue equals marginal costs. As we know, this line here has twice the slope of the original line of the demand curve, so the slope of this thing is -2b.

Now, I’m going to claim that this monopoly quantity I claimed before is going to–our intuition says if the other firm isn’t producing, my best response must be to produce monopoly quantity which I can see on this picture, and I also claim that the math is telling me, just a kind of nerdy math is telling me, that that quantity is a - c over 2b and I claim that they’re the same thing. How can I see they’re the same thing? Well look, here I have a line of slope -2. How far down does it have to go? It has to a - c down. This pink line has to drop off from a all the way to c. How far along do I have to go drop off a - c when I have slope -2b? Answer a - c over 2b. Not even calculus - all you need is high school.

So this monopoly quantity is indeed a - c over 2b. I’m going to ask again, is some of this being–coming out of the fog? I mean, you may not understand it with algebra before but some of these sorts of pictures have you seen before? It’s good review exercise for some of you who are–who have taken 115 or are about to take 150 or are there now. So we found precisely one point on this best response picture, and there’s a lot of points to find, and it’s 20 past 12, so we better get going. So let’s try and find another point. Let’s ask a different question. How much quantity would Firm 2 have to produce in order to induce Firm 1 not to produce at all? Again, how much quantity would Firm 2 have to produce in order to induce Firm 1’s best response to be 0? Help me out–it’s Katie. Is that right? The woman in green, let’s get a mike to Katie. Or, there’s a mike close by.

Student: a - c over 2b.

Professor Ben Polak: All right, so that’s just solving out the algebra, so we’re saying what’s–what q solves, a - c over 2b - q2 over 2 = 0. What q2 makes this equal to 0 and Katie’s answer is solving out the algebra here is that q2 that solves this must be a - c over b. That’s right; where can I see that one on the picture though? Let’s go back to the economics. So I’m claiming that there’s some quantity up here. I claim this quantity is a - c over b such that if Firm 2 produces that much quantity, Firm 1 will just produce nothing. But that’s just kind of algebra and math. Where can I see it on this picture? What quantity, if produced by Firm 2, would cause Firm 1 to shut down on this picture? Let’s get a mike in here.

Student: That’s where the marginal cost and the demand thing, demand lines intersect.

Professor Ben Polak: So it’s here right, it’s here. It’s where marginal–it’s where the marginal cost and the demand curve intersects. What’s the intuition? Let’s work it out. Suppose Firm 2 has produced all of this quantity up to here. So Firm 2 is already producing all of this quantity. So already, just from Firm 2’s production the price has been driven all the way down to costs. So if I–I’m Firm 1–if I produce any more quantity, what’s going to happen to the price? I produce any quantity at all in addition to what’s out there already, what’s going to happen to the price? It’s going to be pushed below costs. So I’ll be producing this stuff, this water at cost c and only getting p, which is not only less than c in return. So I’ll be losing money on any product that I produce. Does everyone see that?

Let me say it again. So Firm 2–if Firm 2 produces all the way up to here then any product produced by Firm 1 is going to push prices even lower. In particular, they’ll be lower here than costs and so you’ll make losses on that product. What’s the name–going back to Economics 115–what’s the name for this quantity? The quantity where demand and marginal costs equal–that’s the competitive quantity. So this is the perfect competition quantity. In a perfectly competitive market that’s exactly where prices are going to end up. This is not a perfectly competitive market, but if it were, that’s where price is going to end up.

So we have the monopoly quantity here, and we have the competitive here, and in between what does this best response curve look like? Somebody? It’s a straight line, thank you. So in between–this is just a straight line. So here is the best response of Firm 1 to each possible choice of Firm 2. Everyone’s looking really like this was hard, but it can’t be that hard. Most of you have seen this before. If you haven’t seen it before don’t panic. All I did was, I did a little bit of calculus, a little bit of algebra, and then I drew the thing. Okay, so this is Firm 1’s best response as a function of q2. What is Firm 2’s best response as a function of q1? Let’s just–we’ll get there, just to remind you, the way we read this is you give me a quantity of Firm 2, I find Firm 1’s best response by going across to the pink line and dropping down. So if Firm 2 produces this amount then Firm 1’s best response is this amount. If Firm 2 produces this amount then Firm 1’s best response is this amount. Now conversely, how do we find Firm 2’s best response as a function of q1? What does that look like? Those two guys in front of you.

Student: It’s

Professor Ben Polak: Shout it out.

Student: It’s symmetrical across the 45º axis.

Professor Ben Polak: All right, it’s reflected across the 45º line. So let’s reflect these two points. This will be here; this will be–I’ve got to be a bit more careful. Missed by a mile, let me try again. I missed by a mile, which is a problem in drawing my lines, but never mind. So this will be the monopoly quantity for Firm 2 and this is the competitive quantity of Firm 1. So if I had drawn that better it would look as if this was half of this distance, but I didn’t draw it well. Try and draw it better in your notes. So this is a - c over b and this is a - c over 2b so this distance and this distance are meant to be the same. The way I’ve drawn it they’re not the same at all. The way we read this green graph is you give me a choice of Firm 1, q1. I go up to the green line and go across and this tells me the best response for Firm 2. So at this point, how many of you have seen this picture before? A good many of you have seen it before. So I think we can really cold call somebody for this one. So somebody tell me–pick somebody to cold–why don’t you call a person, anybody that goes by. There we go, okay. So here’s the tough question. What’s the Nash Equilibrium on this picture?

Student: Where the green and the pink line intersect.

Professor Ben Polak: All right, where the green and the pink line intersect. So this must be the Nash Equilibrium, that wasn’t hard, right? Okay now–so why–let’s go back again. Why is this the Nash Equilibrium? Because at this point, as in the partnership game, which there was a similar thing, as in the partnership game where the best responses intersect is where Player 1 is playing a best response to Player 2, and Player 2 is playing a best response to Player 1. So this is the Nash Equilibrium in this Cournot game.

Let’s do a little bit of algebra just to make sure we can find this thing. So what we’re going to do is set these two things equal to each other, put in stars here. So at the Nash Equilibrium quantities I’m going to have q1* is equal to a - c over 2bq* and q2* is equal to a - c 2bq1* over 2 and now we’re going to solve this out by setting q1 = q2 since I know the game is symmetric and they were on the 45º line. So what I have to do here to do this algebraically is to solve out these equations. So let’s try and do that. So putting q1 in here–if I substitute q1 in here, what will that give me? It’ll give me q1* is equal to a - c over 2b - q1* over 2. Let’s multiply both sides by 2, I’ll get 2q1* is equal to a - c over b - q1*. Take that to the other side. I have 3 q1* is equal to a - c over b; and finally divide by 3 q1* is equal to a - c over 3b. So that crossing point actually occurs at a - c over 3b. I’m going to look desperately at my T.A.’s to make sure I didn’t screw that up. So this is something that’s called the Cournot quantity.

Now, this game, this game of imperfect competition between two firms competing in quantities, was thought up and studied by a French economist called Cournot almost a hundred years before Nash. So a hundred years really before the invention of Game Theory, someone had figured out this answer for this game. Okay, so what do we know so far? I’ll leave the algebra there, transfer it up here, q1* = q2* = a - c over 3b.

Chapter 4. Cournot Duopoly: Real World Examples [00:53:28]

So, so far, we’ve been working pretty hard and we haven’t, I guess, learned a lot, we’ve just kind of solved the thing out. Now, we get to draw some lessons out of this thing, so everybody who’s feeling a little bit shell shocked from having been doing algebra and calculus and drawing pictures and feeling like they’ve been cheated into taking a class that looks far too much like economics, calm down we’re going to actually talk right now.

So, one thing to remark about this game, one thing that we’ve learned immediately, is that this game is different in a significant way. It’s different from the partnership game. I mean, obviously it’s different because it’s about something different, but I mean in terms of the Game Theory it’s different. It’s different from the partnership game, it’s different from the investment game. What is it that’s different about this game from the–what’s the obvious thing just looking at this picture that makes it different from the partnership game? In the partnership game the best response lines sloped up. The more I did of my strategy, the more the other player did as a best response. In the investment game, the more likely I was to invest, the more likely you–the more you wanted to invest.

But in this game we have the exact opposites. The more Player 1 produces, the less Player 2 wants to produce and the more Player 2 produces, the less Player 1 wants to produce. So this game is a game, not of strategic complements, but of strategic substitutes. I want to be careful here. It’s not that these goods are substitutes. I mean, clearly that’s also the case, right? If both these firms are producing bottled water and it’s identical, then the goods themselves are substitutes. That’s not the point I’m making here. Strategic substitutes is a strategy–is a statement about the nature of the game. So strategic substitutes is telling me that my strategy is a strategic substitute of your strategy if the more I do at my strategy, the less you want to do of yours and conversely, the more you want to do of your strategy, the less I want to do of mine.

Now, let’s come back to the question we started with, which has to do with profits and so on in society. We know that if these players play these games, at least if we believe in Nash Equilibrium, then they’re going to produce these quantities, they’re going to produce here. But let’s ask a different question.; Each of these firms is trying to maximize profits, but how about the total profit in the industry? Each of these firms–we know the fact that they’re playing best response–is maximizing their profits taking as giving what the other firm is doing. But how about total industry profits? Are they being maximized here? Who thinks total industry profits are being maximized? Who thinks total industry profits are not being maximized? All right, good; shout out. Total industry profits are not being maximized. Where on the picture–let me do a bit more cold calling here–where on the picture maximizes total industry profits? We had him before? No idea? You’re taking the fifth here? The guy in there, yeah.

Student: Is it at the monopoly?

Professor Ben Polak: Monopoly, exactly. So clearly, if we produce the monopoly quantity, by definition, the monopoly quantity maximizes total industry profits. For example, if Firm 2 shut down and Firm 1 produced its monopoly profits, that is, a monopoly quantity, that would maximize firm profits. Conversely, if Firm 1 shut down and Firm 2 produced its monopoly quantity that would maximize industry profits. Where else in this picture maximizes industry profits? So this point maximized industry profits and this point maximizes industry profits. Where else maximizes industry profits? Anybody?

Well, how about all the points in between? Here, Firm 1 produced nothing and Firm 2 produced a monopoly quantity; and here Firm 2 produced nothing and Firm 1 produced a monopoly quantity. But you could also just split the monopoly quantity, for example, half half at this point here. So if the firms wanted to make more money, the only thing they could do is they could sign an agreement saying, why don’t each of us produce not our Cournot quantity, but produce half monopoly profits. I’m sorry, half the monopoly quantity and that would produce this much each. So Firm 1 would be producing half its monopoly quantity and Firm 2 would be producing half its monopoly quantity. So what’s wrong with that agreement? What are they? They’re two water companies, so they’re Poland Spring and Coca-Cola I guess these days, sign this agreement saying that each one’s going to produce half monopoly quantity and what’s going to go wrong with that agreement? The guy in red.

Student: First of all it’s illegal.

Professor Ben Polak: It’s illegal. So even without there being an Attorney General, somebody might be awake in the justice department and notice that that’s actually illegal. It’s illegal to sign an agreement–to sign a contract saying you’re going to restrict quantities. It’s illegal, so they can’t sign that as a contract, but they they can just agree to do it. So Coca-Cola and Poland Spring could just agree to produce half the quantity of water each. What’s going to go–what might go wrong in that agreement? Let’s just suppose that that game’s just been played once for now. Suppose that this is Coca-Cola’s quantity and this is Poland Spring’s quantity, and suppose you’re the manager of Poland Spring, so water is coming out of the ground in Maine for you and you know that this Coca-Cola guy is going to produce this quantity here, so this is qM over 2. So let’s pick on our two managers; let’s have two people who have spoken before, so the guy in the red shirt’s name is?

Student: Steve.

Professor Ben Polak: Steve, and I guess in red he should be the Coca-Cola guy, and let’s pick on our Jimmy Stewart from last week, whose name is Patrick. Can I get a mike in on Patrick? So Steve and Patrick are respectively the managers of Coca-Cola and Poland Spring and Patrick believes that the Coca-Cola manager is going to–may be we should just make this Coke and Pepsi since two identical–they’re near enough and identical, right? So these are Coke and Pepsi, that will confuse me less. So Patrick, our manager of Pepsi believes that Coca-Cola is going to produce this quantity and he’s agreed to produce this quantity. What is Patrick actually going to do? Patrick, what are you going to do?

Student: So I should cheat and pick the quantity on my best response line that’s much farther out so I’ll produce more than my quantity.

Professor Ben Polak: All right, so Patrick (being Jimmy Stewart or not, actually) when he’s in the–when he’s playing the manager of Pepsi he’s going to produce more of this undrinkable liquid and produce this quantity here. So what is this? This is Patrick’s best response to the other guy producing the monopoly quantity. So Patrick’s response in Pepsi is to overproduce relative to the monopoly quantity, actually overproduce even relative to the Cournot quantity and produce all the way out here. Okay, so Patrick’s producing here but what about Mr. Coca-Cola guy? Now, Mr. Coca-Cola guy knows Patrick pretty well, they’ve been in the same industry for a while and Mr. Coca-Cola guy, whose name is?

Student: Steven.

Professor Ben Polak: Steven. Steven presumably knows that Patrick is actually going to produce this quantity–I’m sorry this quantity. So Steven, what should you produce?

Student: Increase my own production to match my best response line.

Professor Ben Polak: Good, so Steven, anticipating that Patrick is going to think that Steven’s a sucker but Patrick’s going to cheat on him and produce too much, Steven’s going to produce a best response to that and he’s going to produce this quantity here. So what is this? This is Player 2’s best response, so Player 1’s best response to Player 2 producing half monopoly output. But Patrick knows Steven pretty well; he knows those guys who run Coca-Cola are pretty smart guys, so Patrick knows that the guy who runs Coca-Cola is going to anticipate that the guy who runs Pepsi is going to cheat on the guy who runs Coca-Cola, and hence the guy who runs Coca-Cola is going to play a best response to Patrick’s cheat on quantity. So what quantity is Patrick going to produce?

Student: I’ll increase my production then; I’ll be pretty close to the Cournot equilibrium.

Professor Ben Polak: All right, so you’re going to go to here; so this point here is–I can’t even write this. This is the best response of Player 1 to the best response of Player 2, to the best response of Player 1 to Player 2 producing half monopoly output and there are lots of brackets here. Can anyone see where this process is going? It’s going back to the Nash Equilibrium. Each of them is trying to play a best response for what they anticipate the other person’s going to do. And in this game if they keep on doing that, it’s going to drag them back to Nash Equilibrium. But I want to be careful here. This won’t be true of all games. But in this game, playing best response to each other, figuring out that in fact the other guy is going to cheat on me, or the other guy is anticipating that I’m going to cheat on him, or the other guy is going to anticipate that I’m going to anticipate that he’s going to cheat on me, etc., etc., will eventually drag us back to the Cournot quantity.

So it’s going to be pretty difficult for us to sustain this joint monopoly output, this collusive agreement. We can’t let the courts enforce it. We’re Coca-Cola and Pepsi, so we really don’t want to have to bring the mafia in, although maybe we do, I don’t know. So basically, we’re stuck with this verbal agreement and we both have an incentive to cheat and produce more whatever it is, sugar water.

Now, in practice, this is not the only problem facing two firms who are trying to produce the monopoly output. In practice, when firms try and have these agreements, which are not contracts, to try and produce joint monopoly output, what else goes wrong? So you can imagine some firms trying to have an agreement. We know they can’t sign a legal agreement, but let’s take that off the table. So, for example, in around 1900 in America or a little bit earlier, let’s say 1880 in America, it really isn’t clear that it was illegal to write agreements to say we’ll restrict quantities. So firms did use to write that kind of agreements, but still something went wrong. What went wrong? So we’re getting a little bit further away from the game and more back into the real world. What kind of things went wrong? I want to get somebody who’s near a mike. Can I get the woman here with the Yale shirt?

Student: Without a contract there’s no means of enforcement.

Professor Ben Polak: All right, so it might be that–again, the problem with enforcement–although actually in 1880 America, it’s not clear that they couldn’t have enforced that contract. So one problem is unenforceability but that might–far off ago that might have been a problem. What else? The guy next to her.

Student: If they’re not at the competitive output they could undercut price and take the entire market.

Professor Ben Polak: Let’s be careful. We’ll come back to that next time, but right now we’re competing in quantities. So we’re not fixing prices here, we’re not even naming price. We’re just producing Coca-Cola and Pepsi and for want of a better word, spewing it out into the market. What else goes wrong here? Well, one thing that goes wrong is that suppose Coke and Pepsi were able to–either to write a contract or whatever–to sustain joint monopoly output at this high price, this price which is actually–producing positive profits in the industry–it is going to turn out, therefore the price is going to be above costs. These firms’re making all the profits, what do you thinks going to happen in this industry? This industry that’s got a lot of profit around it, what’s going to happen in this industry? Yeah, the guy in the blue shirt here; Ale this guy here.

Student: Another competitor could join the market.

Professor Ben Polak: Another firm is going to come in and produce a similar product. So Dr. Pepper or something, is there one? Someone else is going to come in and produce cola in this industry and that’s exactly what happened in the U.S. at the turn of the last century. Companies did produce agreements to restrict, for example, the production of paper and the production of rubber, and the production of steel, and the production of iron, and the production of railway lines, actually, even quite complicated things. Sure enough what you see very quickly emerging is new entrants entering the market to say, hey these firms are restricting their quantities, we can get in there and make money. So new and a competitive fringe of firms is going to enter and drive prices down. Now, I’m talking about 1900 America where has that happened in the twentieth century more dramatically? We know the early example in sort of Golden Age America, but where else did we see a competitive fringe enter when there was a collusive agreement to keep prices–to keep quantities down?

Student: Airlines.

Professor Ben Polak: Good, so airlines is an example; we’ve seen that. Where else? Oil is probably the classic example. Airlines is a good example. Okay, so in oil we all know that in the late 60s, early 70s, OPEC was formed precisely to restrict quantity among the major oil producing countries, and very quickly we see companies take–other countries take advantage of this to start producing oil profitably. So who were the competitive fringe against OPEC? Anybody? The Brits were for a start, the Brits–England sort of–Scotland I should say. Britain anyway started producing oil in the North Sea; lots of countries in Latin America started looking and finding oil, and of course, Russian oil started being very, very profitable.

So there are various reasons why it’s hard to maintain this collusive agreement. One is there’s an incentive to cheat and another is that other firms are going to enter. Before we leave this, let’s just pose the question we started with. Suppose, in fact, we do end up back at the Cournot quantity. So here we are back at the Cournot quantity. How does this quantity compare and how do therefore the prices compare, and how do profits compare with monopoly prices on the one hand, and competitive prices on the other? So we know the quantity that’s going to be produced. We figured that out; here it is. We know the quantities going to be produced. Each firm is going to produce a - c over 3b and there are two such firms. So the total quantity produced in the market–the total quantity will be 2a - c over 3b.

How does that compare to the monopoly quantity? Well, the monopoly quantity was a - c over 2b and the competitive quantity was a - c over b, just to remind you of those, they’re on our picture. Here’s a - c over 2b and here’s a - c over b. So this is the total quantity produced on this equilibrium, in this Cournot Nash Equilibrium. How did it compare to the monopoly quantity and to the competitive quantity? Anybody? Just staring at the board, which is of these is bigger? So I claim that this total quantity being produced is less than the competitive quantity, but more in total than the monopoly quantity. I was about to get them the wrong way around. Total quantity being produced is less than would be produced under perfect competition, but more than would be produced under monopoly. Consequently, prices are going to go the other way and consequently prices are going to be highest under monopoly, lowest on the competition, and somewhere in between in this Cournot situation.

So from the point of view of the producers, this Cournot Equilibrium is worse than monopoly, but better than perfect competition. And from the point of view of the rest of us, the consumers, this Cournot quantity is worse than perfect competition but better than monopoly. The only rider on that being, if they’re producing Coca-Cola and you have any kind of concern for your teeth, maybe you shouldn’t produce so much anyway. So, so far what we’ve done is we’ve looked at this sort of classic, I admit not the most exciting game in the world, but classic application of Game Theory to imperfect competition. Next time I’m actually going to take this a bit further but I’m going to leave this game behind and look at other ways in which we could study imperfect competition using Game Theory.

[end of transcript]

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