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# ECON 159: Game Theory

## Lecture 6

## - Nash Equilibrium: Dating and Cournot

### Overview

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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html## Game Theory## ECON 159 - Lecture 6 - Nash Equilibrium: Dating and Cournot## Chapter 1. Coordination Games: Scope for Leadership and Strategic Complements [00:00:00]
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 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 Then how many of you saw 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 Same thing with the other person. Same thing exactly, except that he would rather meet at the
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 q 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 q 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 aq 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.
So what are we going to get? So this term here aq Good, so this is our key expression and I’m going to use this expression; I’m going to solve it for q 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 q 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 - q 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–
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.
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.
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 q
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 q 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, q ## 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.
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.
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?
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] Back to Top |
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