ECON 159: Game Theory
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ECON 159 - Lecture 7 - Nash Equilibrium: Shopping, Standing and Voting on a Line
Chapter 1. Bertrand Duopoly: Standard Model [00:00:00]
Professor Ben Polak: All right, so last time we started to study imperfect competition. We looked at the Cournot model, but the bigger theme is the study of how firms compete outside of the sort of easy cases that you study in 115: so outside of the example of monopoly, where there is only one firm and so there’s not much competition; and outside the case of perfect competition where there are so many firms, it’s as if each firm is a price taker. We’re going to continue with that today, and in fact you’re going to continue with it on into the homework.
So last time, we looked at Cournot. It was the model we started with. We looked at two firms competing in quantities. And just to review a little bit what we did last time, the kind of exercise there–so I think, is this right, I think probably went a little fast last time. I was kicking myself afterwards. People can nod or shake their heads. I think I went a bit fast, so I’m sorry for that. If there was a time to go too fast it was probably last time, because it turns out this is something that’s covered well in the textbook, so it’s covered in chapter six of the textbook. But the kind of exercise we did last time is a useful exercise for those of you who are Econ majors.
The way we solved that Cournot model was we did three kinds of things. We did something very nerdy, namely, we just played with calculus and algebra. That was sort of mathy. We did something a little bit less nerdy, that is, we drew some pictures that represented what we’d found. And we did a third thing, which we tried to match this up to the economic intuition about monopoly, and perfect competition, and demand curves, and so on. And this exercise of being able to work in these three different modes, economic intuition, graphs, and kind of nerdy high school math is a lot of what we want to get you used to as Economic Majors, just to be able to translate easily between those.
Again, I apologize for going too fast but it’s still a useful exercise I think, so have a look at it in the book. Now back to the lessons and away from the nerdiness a second, what did we learn? Well at the end of the day we learned that in the Cournot Equilibrium, things were, as we perhaps might have anticipated–things sat naturally between the extreme cases. So the amount of output produced by the industry was somewhere between the case that would be under monopoly and under perfect competition. It was more than under monopoly, less than perfect competition. Prices in the industry, if we’d gone back and checked, would sit between. They would be lower than the monopoly prices, but higher than the perfect competitive prices.
Industry profits would be in between. Industry profits would be less than monopoly. They must be less than monopoly, since that’s the highest they could ever be. And they’re higher than perfect competition, which of course, has zero profits in this case. Consumer surplus, the benefits flowing to consumers lie in between. So in fact the model, in addition to the sort of nerdiness of the model, it ended up with a result we kind of believe in. If you have imperfect competition, it’s somewhere between perfect competition and no competition. But, what’s the “but”? The “but” is that there are other ways we could model imperfect competition, and, as we’re going to see today, they yield different answers.
So the first thing we’re going to do today is look at a different form of competition which is called Bertrand competition. The Cournot was competing in quantities and Bertrand is competing in prices. Just a quick check and this is not for the camera, how many of you have seen Bertrand competition before? So, a good many of you. So this is going to be review still again, and for those of you for whom it isn’t review don’t worry. We’ll do this relatively quickly but I hope I won’t rush it like I did last time. If it’s any consolation, when I feel I’ve rushed a lecture, I have a sleepless night. So if you had a sleepless night because you’re worried about it, I’m having a sleepless night too about that.
So there’s two firms, just as there was before, and just as there was before–so the players are those two firms, and just as there was last time, they’re producing an identical product. So the products we mentioned last time were Coke and Pepsi, but you could think of other products that are pretty much identical. Just as last time, we’re going to assume that costs are constant marginal costs. We’ll assume that constant marginal cost just as it was last time is equal to C. Just to remind those people who are not Economics Majors what that means, it means if I produce 1 unit then it costs me C to produce, if I produce two units it costs me 2C, 100 units a 100C and so on. Now what’s different here from last time? I’ve already told you but I’ll repeat it.
This time, however, instead of setting quantities, instead of just deciding how much Coke and Pepsi to produce and spewing it out in the market and letting prices take care of themselves, this time the firms are going to set prices and let quantities take care of themselves. So the strategy sets this time are prices. Again, so as we don’t get confused, normally we use S for strategy, but let’s use P since they’re prices. So they’re going to P1, will the price of Firm 1, and P2 will be the price of Firm 2. This strategy set–formally, let’s just simplify it here–let’s assume that for each Firm i they can set their price anything bigger than 0 and anything less than 1, just to keep life simple.
So this is a little bit more realistic perhaps. Perhaps more realistic to think of firms actually thinking about what prices to set and then adjusting the quantities they actually ship based on demand. Depends on–I don’t want to be too rigorous about that because you could imagine the other way around, but if you talk to managers this sounds more realistic, at least for retail goods like Coke and Pepsi. So we know how prices are set, they’re set by the firms, so the next question is where do the quantities come from? So how do–where do the quantities from? And they’re going to come from demand. And let me use a big Q(P) to be the total quantity produced–sorry–the total quantity demanded in the market. So this will be total quantity of goods produced by Firm 1 and goods produced by Firm 2 that are consumed. So this is the total quantity of Coke plus Pepsi.
Now we’re going to–notice there’s no subscript on this P and you’ll see why in a second. So we’ll assume just to make life simple, that the total quantity produced is 1 –P. And I’ll say this and then write it more carefully, where P is the lower of the two prices. So total quantity demand is 1 -P where P is the lower of the two prices. Now to make that lower-of-the-two-prices comment a little bit more rigorous, let’s figure out what the demand actually is for these firms. So what’s going on here? So let’s look at the demand for Firm 1, which is going to end up being the quantity that they sell. So that’s going to be Q1 and what’s that going to be? Well it’s going to 1 -P1 if they’re the low price firm. So if P1 is less than P2–so if they’re the low price firm–if Coke is the low price firm, only Coke sells in the market. There’s no Pepsi in the market and 1 -P1 quantity of Coke is sold.
It’s going to be 0 if P1 is bigger than P2. So if Pepsi is the low price drink in the market then no Coke is sold at all. It’s going to be (1 -P1)/2 in the case where Coke and Pepsi cost exactly the same. These should really be–realize that this is not exactly realistic. We’re making a lot of very strong assumptions here, but nevertheless, it’s going to be instructive to look at this model. So we’ve got the firms, they’re the players, I know what their strategies are, I know a little bit about the structure of the market, and I still need to tell you what payoffs are. And just as last time the firms are going to try and maximize profits. So what’s the profit for Firm 1 going to be? I won’t bother to write it separately for Firm 2, it’s going to be the quantity it sells times the price it gets for that quantity, minus that quantity it sells times the cost it incurs in producing that quantity.
So again, for Firm 1 it’s the quantity it sells times the price, this is its revenue, minus the quantity it sells times C, the cost, this is his cost. I can rewrite that a little bit more simply as Q1[P1 -C]. Where Q1, let’s keep it in square brackets, this Q1 is this object here. Does everyone understand what’s going on? I mean the algebra doesn’t really help the intuition here. The intuition is pretty straightforward. Coke and Pepsi are producing this stuff. They’re setting prices. If Pepsi is the low price then Coke sells nothing. If Coke is the low price it sells this amount and its profits are basically given by this equation. It’s basically just revenues minus costs. So what we want to do is we want to figure out what the Nash Equilibrium looks like in this market.
Just before we go any further, notice that really this is the same basic market we looked at before. Both times there’s a demand curve out there. In one model we thought of the firms setting quantities and the market determining prices, and here we have the firms setting prices and the market determining quantities, but the basic underlying economic structure of this is very, very similar. So to find the Nash Equilibrium we’re going to have to find best responses. So what we’re going to do together is to figure out what these best responses look like and let’s try and do that. So I want to figure out the best response of Firm 1 as a function of the price chosen by Firm 2. Best price for Firm 1 to choose given that Firm 2 is choosing P2.
I want to do this carefully and slowly, and before I even start, let me point out in passing that calculus is not going to help us here. So for those of you who are nerdy enough to understand the next sentence let me just say it. This is a discontinuous payoff function, so differentiating isn’t going to get you very far. For people whom that didn’t mean anything, it doesn’t matter. We’re not going to calculus today. Okay, so there were different cases to think about. So one case to think about is what if the other firm, you’re Coke, what if Pepsi is pricing below cost? You’re Coke, what should you price as if you’re trying to maximize your profits? Somebody try and wave their hands in the air if they know the answer. Pepsi’s pricing below cost, what’s your best response? The guy in there, do you want to shout out?
Student: Just get out of the market because it’s not worth it for you to sell anything.
Professor Ben Polak: Right, I’ve forgotten your name, I should know it by now. Your name is?
Professor Ben Polak: Sudipta says get out of the market. If the other guy is pricing below costs, you don’t want to sell anything, that’s the right intuition. How in this game do I “get out of the market?” You’re absolutely right, I don’t want to be involved in this market if the other guy is selling below cost. How do I operationalize that in this game? How do I actually manage? Ale is there somebody here, just wait for the mike a second and now shout it out.
Student: You could set your price above his price and then no one will buy.
Professor Ben Polak: The answer was set your price above his price. So the other guy is pricing below costs, the way which I avoid making losses is to set my price above his price. Let’s just make sure we understand this, if the other guy is selling below cost, the only way I can make any sales is to price below his price. I will then make sales but each of those sales I’m making a loss on. I don’t want to make losses, so I just “get out of the market,” and the way I “get out of the market,” is by pricing above the other guy. Everyone understand that? Okay, so now more interestingly, we leave a little gap here, what if the other guy is pricing above costs? What do I want to do if the other guy is pricing above costs? Can I get the guy with the beard here, shout it out.
Student: You set price below his but above or equal to cost.
Professor Ben Polak: All right, so if the other guy is pricing above costs, I want to set prices below his so that I steal the whole of the market and make profits on those sales. So where is–Suppose he’s pricing–We’ll say the prices are between 0 and 1, suppose he’s pricing at .8, what would be a good price for me to set? Same guy, yeah.
Student: As small an amount below his price as you can do.
Professor Ben Polak: As small an amount below his price as I can. So basically what I’m going to do here is I’m going to set my price to equal his price, minus a little bit, and I’ll use the letter ε to mean just a little bit. I’ll just undercut him a little bit and by just undercutting him a little bit, I’m going to get the whole of the market and I’ll make as much money as I can on those sales. Now that’s almost right. That’s kind of ninety percent of right, but it’s not quite right. I need to be a little bit careful. Now why do I need to be a little bit careful? This is almost right but it’s not quite right. Yeah, same guy.
Student: You want it, your price, to be at or above cost.
Professor Ben Polak: That’s certainly true, okay. So that’s true, fair enough. So I want to make sure this ε is not so big, so as it pushes me below cost, that’s certainly true. That’s correct. There’s another little issue here. So here’s the other issue, there’s a price in this market, we might want to think of as the kind of focal price as an interesting price, and that’s the price at which, the price I would choose were I the monopolist. Suppose the other guy didn’t exist, suppose Pepsi didn’t exist, so Coke has the whole market, then we would solve out this problem. We would solve out what the monopoly price is. That’s an exercise you did in 115. And notice that if Pepsi has priced above the monopoly price, suppose Pepsi has priced this good so high that it’s above the monopoly price, then, as the gentleman said, I can capture the whole market by pricing just below Pepsi.
However, then I’ll have the whole market, it’s as if I’m a monopolist but suddenly I’m pricing above the monopoly price, that can’t be right. I never want to price above the monopoly price, because I know the monopoly price, when I’m the monopolist, is the most profit I could ever make. Is that right? So we need to be a little bit careful. This answer that says I just undercut Pepsi, that’s true provided that Pepsi hasn’t gone above the monopoly price. If he’s gone above the monopoly price we need to be a little bit more careful. If P2 is actually above the monopoly price, then what’s my best response in that case? What’s my best response in that case? Yeah, can we get Katie who’s in green there, shout out.
Student: Pricing at the monopoly price.
Professor Ben Polak: Pricing at the monopoly price. So if Pepsi is dumb enough to price above monopoly, sure I will undercut Pepsi but I won’t undercut Pepsi by a penny, I’ll undercut Pepsi all the way down to the monopoly price and make monopoly profits. I’ll be very happy. I’ll be wondering what on earth is Pepsi up to, but that’s fine. So here my best response is to price at the monopoly price. There’s one other possibility here I’ve missed which is what if Pepsi prices at marginal cost itself? That’s the one missing case here. What if Pepsi prices at marginal cost? What’s my best response in this case? Pepsi’s pricing exactly at marginal cost, what’s my best response? Yeah, wait for the mike. This is Henry I think, shout out.
Student: To price at marginal cost as well.
Professor Ben Polak: I could price at marginal cost as well. How much profits will I make if I price at marginal cost as well? Zero profits, I’ll make zero profits, so that certainly is a best response, pricing at marginal cost as well. What else would be a best response? That’s correct, but what else would be a best response? Price above that. It doesn’t really matter, as long as I don’t price below it, I’m not going to make any money anyway. If I price below it I’m going to lose money. So this best response actually is a pretty complicated object, and we could, if we’re going to go like we did last time, we could take the time to draw this thing, but it’s a bit of a mess so I won’t worry about doing that right now.
So the best response is actually pretty complicated. Even though the best response is pretty complicated–and by the way, obviously the things are symmetric for Player II. Even though these best responses are pretty complicated it turns out that there’s only one Nash Equilibrium in this game. What’s the Nash Equilibrium? Somebody, what’s the Nash Equilibrium? Raise your hands. All you guys, there’s a huge number of hands who said they had seen it before. Who saw this model before? Fewer hands go up when they know there’s a question coming. There’s a different incentive. Who remembers what the Nash Equilibrium is? Yeah, there’s a guy, Tae there’s a guy right by you there.
Student: At C.
Professor Ben Polak: Shout out, even I can’t hear you.
Student: At C.
Professor Ben Polak: At C okay, so the Nash Equilibrium here, the Nash Equilibrium is for both firms to set their prices equal to marginal cost. You can check that both firms are then playing a best response, so that’s all right. If Firm II is charging C, then a best response for Firm 1 is to set price equal to marginal costs, and if Firm 1 is pricing at marginal cost, then conversely a best response for Firm 2 is to price at marginal cost. Slightly harder exercise is to check that nothing else is a Nash Equilibrium. Well let’s think about that for a second. Suppose Firm 1 is pricing at marginal cost and Firm 2 is pricing at something higher, at C + 3ε, a little bit higher. Suppose Firm 1 is pricing at marginal cost and Firm 2 is pricing at C + 3ε, is this a Nash Equilibrium?
Let’s think about Firm 2, first of all, is Firm 2 playing a best response to Firm 1? So I claim it is, let’s just check carefully. Firm 1 is pricing at marginal cost, the best response if the other guy is pricing at marginal cost is to price at marginal cost or above. Now this is marginal cost or above so this is the best response. So if Firm 2 is playing a best response, this is a best response for Firm 2, given that P1 is at C. What’s the problem here? Nevertheless I claim this is not a Nash Equilibrium. Why is it not a Nash Equilibrium? Who has an incentive to deviate? Can we get the guy way at the back by the door. Go ahead; shout out.
Student: Firm 1’s going to want to produce at C + 2ε.
Professor Ben Polak: Exactly, exactly. This is not a best response. This is not a best response. The reason is that the best response for Firm 1, if Firm 2 is charging C + 3ε is to price at C + 2ε, thank you. So it turns out it’s pretty to check that this is the only Nash Equilibrium in the game. Now what’s the lesson we’re going to draw from this? Well, as an exercise in Game Theory, that really wasn’t very hard. As an exercise in finding Nash Equilibrium, by this stage in the course most of you are looking at that saying, that wasn’t hard. So what’s the point here? The point is that in this game in which firms competed in prices, even though there were only two firms in the market, only one firm more than monopoly, we get a dramatically different result than we had last time.
In particular, we find that prices in the market are equal to marginal cost. We find that profit in equilibrium is equal to what? Zero. And we find there’s lots of consumer surplus because the prices are really as low as they ever could be. In fact, the outcome here, this equilibrium here is for all intents and purposes, the same equilibrium we would have had had there been thousands of firms in the market and had this been a perfectly competitive market. So even though there’s only two firms here, with price competition, identical products, we end up with an outcome that looks exactly like perfect competition, except for the fact there’s only two firms. The outcome is like perfect competition even though there’s only two firms.
That’s a pretty surprising result. It would suggest, think of this as a policy thing. So if you believe this model, if you think is really an accurate model of society, and you’re a regulator working in the Department of Justice, or you’re a judge trying to judge some monopoly case, or you’re a commissioner on the European Court or whatever trying to judge some competition case, all you’d worry about is getting one competitor in each market, two firms in each market and you’d be done. You wouldn’t worry about entry beyond two. Now my guess is we don’t believe that. We’ll come back to that in a second, but let me make a different remark before we get there. Is it me doing that. Let me move this down. Still doing it okay. It’s probably the wire. Let me just remove it, I’ll shout.
Okay, I’m going to shout now, can people still hear me? Yep, okay thank you. So we’ll have to turn down the other mike a bit. So another thing to remark here is, even though we looked at essentially the same market as we looked at last time, and we didn’t make any really significant change–we just said instead of thinking of them pricing and setting quantities, think of them setting prices (happens to be just thought experiments)–we ended up with a radically different outcome. So another lesson here is: the same setting as last time, as Cournot, but with a different strategy set, a different way of thinking about what they’re doing led to a very different outcome. That’s worrying. It’s worrying because you can’t go away feeling comfortable about this.
You can’t think that it really could make so much difference in the real world how prices and quantities, and welfare, and profit is going to work out depending on some thought experiment about how I think about my strategy set. So there’s a little worry going in about not just this example but about the course, about the whole theme of learning Game Theory. So what’s going to save us here? What’s going to save us is if we inject a little bit more reality back in the model, we’re going to get back a more sensible result. So what we’re going to do now is we’re going to relax some of the assumptions of this Bertrand model and it’s going to do two things for us.
First, it’s going to give us an outcome that we believe. The outcome we believe I think, is that imperfect competition should look something between monopoly and perfect competition, it shouldn’t look like perfect competition. I think most of us don’t believe that two firms is enough to make it a perfect competition, that the regulator shouldn’t worry about the third firm. The second thing it’s going to do, from a more theoretical point of view, is it’s going to suggest to us that actually things aren’t quite as bad we thought. In fact, if we model this more carefully, we’ll get roughly the same prediction either way. So what’s the assumption we’re going to change?
Well before I say that, I should say I’m not going to change it, you are. So rather than go through this again a whole third time, I’ve gone through it once in quantities and once in prices, I’m going to get you guys to do it this time by having you do it on a homework assignment. Well why? Partly because I think it’s a good exercise, but also I don’t want this class to be a class where you sit there, with your cup of Willoughby’s coffee if needed to keep awake, and you watch me solve models because that’s not how you learn. What I want this class to be is a class where at least sometimes, like on this homework assignment, I set up a model for you, set up the story for you, and then you have to actually figure out how do I set this up properly and how do I solve it out? Because at the end of the day, if Game Theory is going to be useful for you in your later lives, whether it’s in dating or in running a company or whatever it happens to be, you need to able to go from the story to the model.
Chapter 2. Bertrand Duopoly: Product Differentiation [00:28:18]
Having said that, I’ll tell you what the homework assignment’s going to be about. The assumption we’re going to change is the idea that products are identical. So we’re going to look at a case, I’ll use a new board, where products are differentiated. I’m going to claim that assuming that products are not identical, is a pretty safe assumption for most of the world, for most things that you’re going to see in the world. So we’re going to look at what are called differentiated products. In particular, we’re going to look at a particular version of this that we’re going to call the linear city model. So what do I mean by products being differentiated?
So I started off with an example that’s pretty bad for this story, namely, Coke and Pepsi. Now be honest, how many of you in a blind taste test can taste the difference between ordinary Coke and ordinary Pepsi? Quite a few. That’s good for my case, so a number of you think those are different products. So a number of you, even if the price was just a tiny bit different might have a preference for one rather than the other, so without getting in trouble by putting the camera on you, how many of you would have a preference, if the prices were essentially the same how many of you would have a strict preference for Pepsi? How many would have a strict preference for Coke? How many would have a strict preference for Diet Coke? That’s amazing, that stuff’s–oh never mind.
Okay but you’re making my point. The point I’m trying to make is, products are not identical. For the most part there’s a little bit of difference between products and that’s actually going to–that is, if we inject that little bit of realism into the world–it’s actually going to help us. So what we would like, what would we like? We would like a model in which firms set prices because for the most part we think firms do set prices not quantities: not always but for the most part. But we’d like a model that yields an outcome that looks–that when you only have two firms looks somewhere between monopoly and perfect competition.
So we’d like an outcome that looks a little bit like Cournot, but we’d like the strategy set to be prices, and this is going to do the trick. This is going to turn out to do the trick as you’ll find out in your homework assignment. So how are we going to model this on the homework? The way we’re going to model differentiated products is to imagine, just to take a simple example, imagine a city and this city has one long straight road through it. So this city is not a city in New England, it’s a city in the Midwest, where everything’s flat and the roads just go completely straight. You can think of this as being a mile long. It doesn’t really matter, let’s just think of it as being a mile long.
We’re going to assume that consumers are evenly spread along this city. So there’s basically consumers everywhere, they’re evenly distributed and we’re going to assume that one of these firms, let’s call it Firm 1 sits at point 0 and the other firm, Firm 2, sits at point 1. Now, this is what you’re going to do in the homework assignment, but I’m just going to make the argument that you could also imagine firms sitting somewhere between 0 and 1. We could do a more general job if we wanted to, but for now, let’s assume that one of these firms has its shop at one end of the town, and the other one has its shop at the other end of the town.
So let’s think about a particular consumer, so suppose this consumer is here, at point Y say. So notice that this consumer is a distance Y away from Firm 1. So if she consumes from Firm 1 she has to walk a distance of Y. She’s a distance of 1 -Y from Firm 2, is that right? So if she consumes from Firm 2 she has to walk 1 -Y. That’s going to turn out to be key in our model as we’ll see in a second. So firms, as before, are going to set prices. I won’t write everything down because it’s all written down on the homework assignment which is already on the web, but in fact firms are going to maximize profits, aim to maximize profits, firms are going to have constant marginal costs.
We’ll make one other assumption to keep life simple. We’ll assume that each consumer buys one and only one product. Each consumer is going to buy one product, either from Firm 1 or from Firm 2. So the issue is going to be which firm does each consumer go and buy their product from. Which firm does each consumer choose? We’ll assume that each consumer chooses the product whose, and I’ll be careful here, whose total cost to her is smaller. What do we mean by total cost? Well let’s look at the consumer at point Y. For example, for the consumer at point Y, if they buy from Firm 1 then they pay the price P1, which is set by Firm 1, but they also have to pay a transport cost, the cost to them of having to walk all the way there and walk all the way back.
We’ll give a name to that transport cost. We’ll say it’s TY². So Y is the distance they have to travel and TY² is their transport cost. So this object here we could think of as a transport cost. If the same consumer buys from Firm 2 she pays P2 + T times again the distance squared, so that’s going to be (1 –Y)² and once again this last term is a transport cost. Notice that these transport costs go up in the distance you have to travel and they go up pretty fast. They go up at rates squared. So what you’re going to do on the homework assignment is solve out this market.
You’re going to assume that firms set prices to maximize profits. You’re going to know what firm’s costs are. You’re going to work out what firm’s demands are going to look for each possible price they could set. And you’re going to solve out the whole Nash Equilibrium. And then we’re going to look at that Nash Equilibrium and you’re going to think, how does that compare to what I saw in the Cournot case we solved. So that’s on the homework assignment. But before I leave this, let me just point out that this is a little bit more general than it might appear. So here I’ve treated what makes products different as being where the shops are located. So here I’ve interpreted these terms here as transport costs, and I’ve interpreted what makes the products different is the fact that one of them is selling at one of the end of the town, and the other one is selling at the whitehead end of town.
But actually, we could consider this model more generally, and let’s just do so briefly here. So let me just redraw the town; here’s my town again. I don’t have to regard this product, this line as being distance along the high street of the town. It could be something else about the product. So, for example, in whatever it is you guys imagine makes Coke and Pepsi different, it could be that thing. Don’t quite know what that is, but whatever that is. Let me take an example that I understand better than I understand Coke and Pepsi. So imagine we’re talking about beer, think about beer, so this is the beer market. This distance here could be something like the alcohol content or the flavor of the beer.
You can imagine different products therefore, that are on the market, positioning themselves, or being positioned at different points on the line. So, for example, up here if you want beer flavor this might be Guinness, which I can’t even spell, but you know what I mean. If we think about the drinks industry more generally here, rather than just beer, this would be Guinness, this would be Poland Spring, this is water and everything else would be in between here. So if we just go a tiny distance in here, this is Bud Light and so on. You could put everything on this line, I think there may be, I’m not really allowed to do this on this model, the one thing I can’t do is, part of the truth is that Bud Light might be down here somewhere but I’m not allowed to do that.
So leaving aside the specific example of beer, you think about some product that has some dimension on which it varies, and we can use this model to see how competition is going to work in that market. But now notice that this transport cost is going to be a different interpretation. Now, instead of being the cost of traveling that distance to go and buy the product, what’s it going to be? It’s going to be that if my preferred beer flavor is here say, this is me, my preferred beer flavor is here, then if I end up having to consume Guinness, I have to pay the price of Guinness and I also incur some costs because Guinness isn’t the perfect beer for me, it’s a little bit too strong. If I consume Bud Light, then I have to pay for the Bud Light, I think it typically costs less, the price, the raw price is less than Guinness but I also pay an additional cost from the fact that Bud Light, when I drink it, causes me “disutility”: we’ll put it politely like that.
So basically, that transport cost is now the lack of pleasure caused by drinking a product that isn’t perfect for me. Does everyone understand the story here? So you’re going to figure out what happens in this story very generally, well not very generally, but in the particular case actually on the homework assignment, but I want you to understand that there’s this much more general story underlying this, and this is a pretty good model of a lot of markets. It’s the kind of model that you’d see again if you went onto graduate school.
Chapter 3. Perfect Competition Revisited: The Candidate Voter Model [00:40:13]
Now I want to spend the rest of today doing something quite different. But you’ll see it isn’t all that different. I’m going to go away from studying imperfect competition and go back and visit something we studied almost the first day or maybe the second day, and that’s elections. I want to go back to elections and why do I want to go back at this point? Well one reason I want to go back is you’ll notice I’ve been putting up these lines on the board and when we visited elections on the second day, we said that you could think of that line on the board, as being not just left wing, right wing politics but also some dimension of products. So I want to go back again, now we’re considering the line on the board as being flavor in beer, or location in a town, and I want to go back to politics now and go back to the interpretation we started with, so that left and right will end up being left-wing politics versus right-wing politics.
So I’m going to take the same basic idea back to the politics model. We’ll spend the rest of today in politics. So we’re going to be doing Political Science or Political Science as it meets Game Theory for the rest of the day. We’re going to study something called The Candidate-Voter Model. This model is going to look a lot like the models we looked at already a few weeks ago. In particular, as advertised, there will be a line and this line will go from 0 to 1 and the left hand side of this line will represent the left wing and the right hand side of the line will represent the right wing, so that’s the same as before. As before, as in the Downs or Hoteling model, we discussed already a few weeks ago, we’re going to assume that voters are evenly spread along the line. So we’re going to assume even distribution of voters on the line.
We’re also going to assume, just as we did last time, that voters are going to vote for the closest candidate. So all of these assumptions are the same assumptions we studied two weeks ago, or two and a half weeks ago, and which you actually studied in your first homework assignment. Is that right? But I want to go back there because what I want to do now is, I want to change some critical assumptions of that model and see that by making those changes, we’re going to get some very different outcomes. So, in particular, the new things here, two things, one the number of candidates is not fixed. So the number of candidates in this model is going to be endogenous. Previously, we looked at models where there were two candidates or on your homework assignment, three candidates. Now we’re going to allow the number of candidates to adjust itself.
The second assumption we’re going to make which is new is that we’re going to assume that candidates cannot choose their position. So the idea is, any candidate who stands in this election, you know who that candidate is, you know whether they’re right wing or left wing, so they can’t tell you they’re something else. So candidates cannot choose their position. So this is a subtle thing because you think about the current election, there’s a debate about whether Hilary Clinton, for example, can choose right now to be at the center of the democratic party given her past history of votes, for example, on the Iraq War. Or on the other side, there’s a debate about whether Mitt Romney can choose to be, I guess he’s trying to choose himself to be on the right hand wing of the republic party, given he has a record of governance, as being Governor of Massachusetts when he provided state healthcare for example.
So it’s quite difficult in the real world for candidates to position themselves. Voters tend to know that those candidates have track records. That’s something we’ve mentioned already three weeks ago, two weeks ago, we’re going to take that fix now, we’re going to assume you know who these candidates are. So what are we actually going to assume? We’re going to assume that each voter in the model is a potential candidate. So this is, I think, a nice idealized version of American politics. Each of you, who is above 18 and is an American citizen, and was born in America (maybe you have to be more than above 18, above whatever it is, whatever the rule is, whatever the Constitution says), each of you can potentially stand up now and say you’re going to run for President. I’m guessing all of you are ruled out by age actually. There’s some rule in the Constitution, but never mind, let’s pretend that’s not there.
So the idea of the model is each voter is a potential candidate. That’s a very appealing assumption I think in a democracy. So what I want to do is, I want to lay this out a little bit more formally as a game, and then, since we haven’t done anything like this for a week or so, we’re going to play the game. So you’re about to play this game, so pay attention please. So who are the voters, who are the players? The players are the voters. The voters, or candidates, whatever you want to call them, depending on where they stand, in this game they’re going to be you.
The key strategy here, essentially the strategy is going to be very simple. The strategy is: do you run or not? The reason that’s going to be the strategy is that voting’s not going to be difficult. You’re always going to end up voting for the candidate who’s closest. So the only really relevant strategy is to run or not to run, to stand or not to stand. So just to make that clear, voters vote for the closest running candidate first. And second, what did it mean to win? Well assume that we’re in a plurality election here, so the winner is the person who gets a plurality. You win if you get the most votes, in other words. We’ll assume that if there’s a tie that we flip a fair coin or a Supreme Court judge, whatever you want to take, whichever. Flip if tie.
The payoffs in this game are as follows. We’ll assume that there’s a prize for winning. So if you win the election you get a prize equal to B. We’ll also assume that there’s a cost of running, so if you enter this election, win or not, you incur a cost of C, and we’ll assume that B is greater than or equal to 2C, and actually for today, let’s just keep things simple, and assume it’s actually equal to 2C. But that’s not the only part of the payoffs. There’s also a part of the payoffs that’s analogous to forcing me to drink Bud Light or forcing the Pepsi drinkers to drink Coca-Cola. What’s that? That’s if, regardless of whether I run or not, if some other candidate is elected other than me, then they’re not going to have my ideal policies. So it’s going to cause me unhappiness, it’s going to cause me disutility having a candidate win, who’s far away from me.
So there’s going to be an extra thing and, so if you’re position is X, if you are at X on that line and the winner of the election is at Y, than you pay a cost of –|X –Y|, the absolute distance between you and the winning candidate. So again, if you’re at X and the winner is at Y, it hurts you minus the distance between X and Y, in terms of your unhappiness, about having a winner who’s far away from you, winning. So let’s do an example, just want to make sure we understand the payoffs.
So Example 1, if you’re at X, let’s just call you Mr. X, just to make it clear, so Mr. X is the person at position X. If he enters the election and he wins the election then his payoff is B because he entered -C,–sorry B because he won, -C because he had to pay his election expense. But the winning candidate is him so he gets no disutility from the winning candidate being someone else. Second possibility, if Mr. X enters but Mr. Y wins, then X’s payoff is what? He still incurred -C because he ran and he also has a cost of |X-Y| because he doesn’t like Mr. Y winning.
Third, if Mr. X stays out but Mr. Y wins then Mr. X’s payoff, he doesn’t win so he doesn’t get B, he doesn’t lose C because he didn’t run, but he still has this disutility -|X –Y| because he doesn’t like Y winning. Does everyone understand this game? Everyone understand the game? Anyone not understand the game at this point? In principle, we could play this with the whole class, but let’s single out a particular row of the class so I’m going to come down here and I guess eventually, well I’ll grab it in a minute. So I’m going to use this row, I think, everyone in this row stand up a second, and this row stand up, this is the row of potential voters. They’re the voters. They’re also the potential candidates.
So we need to have a convention here, whether we’re viewing this from the perspective of people behind them or people in front of them, let’s view it from the perspective of the people in front of them. So I got an R, I’ll do it the other way around. People from behind that have the perspective. So this gentleman here who’s name is Andy. So Andy is our crazy right-wing guy and Sudipta is our crazy left-wing guy, everyone else is in between. There’s 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 people here. Let me add, let me make it 17 people so we have an odd number, so go and sit in that row. Okay, now we have 17 people it’s a bit easier.
Just want to avoid obvious ties. So how is this going to work? Everybody sit down again but don’t put your–keep your books handy because this is going to be pretty aerobic. So these are the potential voters and each of them is in the position that Fred Thompson was in, two weeks ago in the Republican Primary, when he’s deciding whether to run or not. But in the real world that’s a sequential game, people decide to run in some order, we’re going to assume this is all simultaneous. So what we’re going to do is we’re going to hold essentially this game. We’re going to ask everybody in this row whether they want to enter or not. Let’s make some real numbers.
Let’s assume that B is equal to $2, so the spoils of government are $2 and entering costs you $1, so C is $1. And we’ll assume that the amount of disutility you get from being far away from the winner is 1/17th of the distance, each of these, each place is worth 1/17th of a $1. Everyone understand? So whatever 1/17th of a $1, roughly $.05 for each position away. Everyone understand in this row what we’re doing? I’m not getting any nods, you guys over here you understand what we’re doing? So on the count of three, anybody who’s entering this election is going to stand up. Ready? One, two, okay what? That’s the opposite of what’s happening with the Florida Primary. Yeah question?
Student: I’m sorry, who stands up on the count of three?
Professor Ben Polak: You guys have to decide whether you, okay. So if you’re going to, if you’re running you stand up. If you’re standing you stand. Let’s try again one, two, three. Stay standing and I’ll come around a second. So here’s our election, and it’s between a, which way do I want to do it? This is our left wing candidate, this is our moderate left wing candidate whose name is?
Professor Ben Polak: Our kind of centrist candidate, maybe even this centrist candidate who’s name is?
Professor Ben Polak: Beatrice. So this is the election, who’s going to win this election? Beatrice is going to win this election. Let’s just check that. Beatrice is going to get all of these right wing votes. Beatrice is also going to get one and a half of the votes in the middle, so this vote and half the Pierson guy’s vote and, I’ve forgotten your name already, again? Alex is going to get all the left wing votes, but there aren’t enough left wing votes here, right? So Beatrice is going to win. So Beatrice won this election, is this array a Nash Equilibrium? Is this an equilibrium? How do we know it’s not an equilibrium? Katie, how do we know this is not an equilibrium?
Student: Because Alex’s best response to her running is to not run.
Professor Ben Polak: Absolutely. So Alex here, Alex ends up with a payoff of what? Alex ended up with a payoff of -C, actually more than that, -C minus the distance between him and Beatrice, so that’s another $1, $.05, $.10 or even $.20 whatever it is, and he would have done better to have not run. If he hadn’t run, Beatrice would still have won, but it would still have cost him the $.20 of unhappiness at Beatrice winning, but at least he would have saved himself the C of running. Everyone see that? Let’s try it again. Okay, same row, but don’t hesitate so much this time. Count of three, you’re just going to go or not. Everybody in this row close their eyes so you can’t look around you. Close your eyes, you can do all sorts of nasty things, close your eyes. Myrto you’re still in this row okay. So one, two, three. You can open your eyes again, so now what happened, we ended up with two centrist candidates. Let’s see what happened, so we have Beatrice and your name is Claire Elise.
Student: Claire Elise.
Professor Ben Polak: Claire Elise is going to get her own vote plus 1, 2, 3, 4, 5, 6, 7, 8, 8 votes and Beatrice is getting 1, 2, 3, 4, 5, 6, 7, how did I end up with eight, I thought I had an odd number. 1, 2, 3, 4, 5, 6, 7 there must have been an odd number before. So Claire Elise is going to win this election because she got one more vote. Now here we ended up with two centrist candidates, which is a result pretty close to what we saw in the Hoteling model, is that right? Is this an equilibrium? So check that I counted right, 1, 2, 3, 4, 5, 6, 7, 8 and 1, 2, 3, 4, 5, 6, 7. I guess I did count right, so Claire Elise turned out to win this. I counted that correctly. There’s another vote hidden behind there, I didn’t see you.
Let’s try again. So 1, 2, 3, 4, 5, 6, 7, 8. I apologize Beatrice won this election, I apologize. Beatrice won this election. Is this an equilibrium? How do we know this isn’t an equilibrium? Because Claire Elise would rather not have run, is that right? Let’s try once more, down again, close your eyes again, just to keep life interesting, let’s switch to this row. Can we switch to this row okay? So this is our row now, everyone understand this. Close your computer so you can get up without destroying thousands of dollars of technology. This is our row and at the count of three we’re going to see who’s going to run. Everyone understand the game? Close your eyes, 1,2,3.
It’s obvious he’s going to win this election, is that right? Just to ask the question, is this an equilibrium? Let’s count. 1, 2, 3, 4, 5, 6, 7, 8, 9 he’s going to get 10 votes, including himself there are 10 votes here. I can’t count correctly, 1, 2, 3, 4, 5, 6, is he actually the center guy here? I don’t think he is the center guy here. So I claim, if I’ve counted correctly, that this actually isn’t an equilibrium. Who can deviate and do better here? This guy, stand up a second, whose name is?
Professor Ben Polak: Had John entered, if I’ve counted correctly, let’s pretend I have, John would actually have won this election because John is actually further to the center. Okay, so let’s, thanks for a second guys, I’ll come back to this in a second. Let’s try and figure out how to analyze this game. We’ve played it three times, let’s try and figure out what the equilibria looked like in this game. This game that’s going on more or less right now in the primaries, well I guess the entry stage of it is gone now. So first of all, is there a Nash Equilibrium in which no candidates stand? Is there a Nash Equilibrium in which nobody stands? No. How do we know that’s not a Nash Equilibrium? Somebody? Yeah, how do we know that’s not a Nash Equilibrium? Shout it out.
Student: Because then somebody’s definitely better off by standing up.
Professor Ben Polak: Right, if nobody stands, each and every possible candidate would do better individually, so any particular voter would do better standing. They would win the election. Is that right? So clearly it’s not an equilibrium for nobody to stand. That’s good because we don’t see that very often. Is there a Nash Equilibrium in which one and only one candidate stands?
Student: If they’re directly in the center.
Professor Ben Polak: All right, so given that we chose an odd number of people in the row, if exactly one candidate stands and that candidate is the center candidate, then that’s an equilibrium. How do we know that’s an equilibrium? Somebody give me the argument why that’s an equilibrium? It’s a good, in fact, it’s a correct guess. But how do we know that’s an equilibrium? That’s the guess part. How do we check that’s an equilibrium?
Student: If anybody on either side of them chose to run, they would lose.
Professor Ben Polak: They would lose, exactly. So if anybody on either side of them runs, that person would simply lose. It wouldn’t make any difference to the outcome. If a person on the right stood, so let’s just try it again. So we figured out that, Beatrice stand up a second. So here’s our potential equilibrium with just Beatrice standing, and if somebody to the, which way do we do it, somebody to the right of Beatrice was to stand, so ma’am if you stand a second, and your name is?
Professor Ben Polak: So if Stacy stands, it doesn’t make any difference she just loses. Beatrice wins anyway; it just costs Beatrice some money. Conversely, if Beatrice doesn’t stand and the woman who’s one in from the side there, sit down a second Stacy and if this person stands, and your name is? If Sarah stands, it doesn’t help Sarah at all, she just loses. Is that right? So there is an equilibrium with exactly one candidate, the center candidate. Now that’s looking a little bit like the median voter model. It says there is an equilibrium with only one candidate standing and that candidate would be the median candidate, just like it was in the model we saw on your homework assignment and also in class.
However, we’re not done yet, there could be other equilibria. So let’s just try and think about whether there are other equilibria. So, for example suppose now, just to make life a little bit more interesting, suppose that the voters were actually two rows and just allow me the suspension of disbelief that these rows have the same number of people in them, I know they don’t really. So now what I want you to think of is that at every political position there are two voters, and hence two possible candidates. There’s two people at every position, these two people at this position, these two people at this position. Everyone understand that? Let’s just assume that the rows are the same even though they’re not and let’s examine the following thing.
Suppose Beatrice stands again, sorry Beatrice, and the gentleman in front of her whose name is? Robert stands as well, so stand a second. So now we have two candidates standing who are identical, politically, they’re identical. They’re right on top of each other. Now is that an equilibrium? After all that looks a lot like the Downs-Hoteling model, we’ve got two candidates exactly at the middle, is that an equilibrium? Let me try the guy up here?
Student: No, because if someone on either side of them were to stand up that person on the side would win.
Professor Ben Polak: Good, this can’t be an equilibrium because look what happens if Claire Elise stands a second. Claire Elise is going to win all of the right wing votes, ranging from crazy to moderate, and these guys are going to split the left wing votes. So the total sum of right wing votes is going to beat out half of the left wing votes. So it can’t be an equilibrium, the exact prediction of the Downs model, two guys right on top of each other is not an equilibrium. Sit down again guys. Let’s go back to one row again because it’s easy to work with.
Let’s try a slightly different pattern. So suppose, assuming I counted right, suppose that the following candidates enter. So Claire Elise enters and I’m sorry I don’t know your name, Jean enters so let’s have a look at this array. Assume I got it right and Beatrice is actually the center. Now is that an equilibrium? Who thinks that is an equilibrium? Who thinks it’s not an equilibrium? Who’s waiting to see how other people vote? Well let’s check. So there’s three possible types of deviation here that we need to check. We need to check another entrant from the outside, left wing or right wing, we need to check another entrant in the middle, there is only one possible one there, and there’s a third kind of deviation we should check, what’s the third type of deviation? One of these might choose not to run, one of you might choose not to run.
Let’s do them in turn. So suppose that we consider a deviation in which, and again I’ve forgotten your name in which Stacy stands as well. Stacy, just stand a second. So is this a profitable deviation for Stacy? No it’s not. Why isn’t it a profitable deviation for Stacy? So two reasons: one is that she loses, she’s not going to win this election by standing. But there’s a second reason why this is a really bad deviation. Why is it, what’s the second reason? Yes sir.
Student: The winner is going to be farther away from her.
Professor Ben Polak: Good, so by standing not only does she not win the election, but she actually causes the election to be won for sure by Jean who’s further away from her. So Stacy’s going to pick up the right wing votes, she’s going to split. The crazy right wing votes she’s going to pick up. The moderate right wing votes she’s going to split with Claire Elise, and Jean’s going to have all the left wing votes, extreme and moderate, and so from Stacy’s point of view this is a double bad. She doesn’t get to be President, sorry and you end up with a left wing President, which you didn’t like. Everyone see that? So clearly, that’s not, thank you, that’s not a profitable deviation and it’s also not a profitable deviation for people on the left.
Hang on a second. So I think it’s probably obvious why it’s not a good strategy for Beatrice to enter here. If Beatrice enters she loses. Everyone can see that? The person in the middle if they enter here loses, so they’re not going to want to enter. So the other possible deviation we have to consider, is a deviation of the form of one of these guys dropping out. So let’s consider it, so come back to Claire Elise. Is Claire Elise doing better in or out? Is Claire Elise doing better in this election or out? Let’s think it through a second, if she stays in what’s her payoff? She wins B with probability a half. She costs C for sure, that’s a wash because B, the way we’ve worked things out, B was $2, C was $1, so B/2-C is a wash.
In terms of the benefits of–the spoils of government versus the cost of running, it’s a wash for Claire Elise, whether she enters or not. But by entering she has a half chance of winning the election. So one way to think about it is, is the expected distance from her of the winning candidates, is: with probability of a half it’s herself so that’s nothing, no distance, and with probability of a half it’s two places away. So in expectation it’s one place away. If she drops out, Jean wins for sure, so the expected distance away from her of the winning candidate is two away. Just to say it again, so the cost and direct benefits of government are a wash for her. But by dropping out she insures that somebody further away from her wins for sure. That’s bad so she’s going to stay in.
The same argument goes the other way, so this is an equilibrium. Now we’re not done and of course we’re out of time, so what are we going to do when we come back next week? Don’t pack up yet, what are we going to do? We’re going to come back to this row and we’re going to figure out not just this equilibrium, but all equilibria. Before you go though, let’s just think about it a second, so you can think about it and talk about it at home, so what we just saw was an election in which we saw two candidates, Jean and Claire Elise, who both stood very close to each other. But a different question is–before you go: stand up my very left-wing guy, Sudipta, way at the end, and stand up my very right-wing guy. Here’s a candidate–here’s an outcome with two entrants in it, an extreme right-wing guy and an extreme left-wing guy. Is that an equilibrium? Think about it until we come back on Monday.
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