ECON 159.?Game Theory

Lecture?07. Nash Equilibrium: Shopping, Standing and Voting on a Line

https://oyc.yale.edu/economics/econ-159/lecture-7

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 learned that in the Cournot Equilibrium, 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?would be lower than the monopoly prices, but higher than the perfect competitive prices.?

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. There's two firms,?and?they're producing an identical product. We'll assume that constant marginal cost just as it was last time is equal to C, which?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.?

This time, however, 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.?

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?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.?

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. 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.?

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.?

If the other guy is pricing below costs, you don't want to sell anything, that's the right intuition.?I don't want to be involved in this market if the other guy is selling below cost.?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.?

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? 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.?

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. 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?I can capture the whole market by pricing just below Pepsi.?

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.?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? 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.?

What's the Nash Equilibrium? At C, 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. Nevertheless I claim this is not a Nash Equilibrium. Why is it not a Nash Equilibrium? Who has an incentive to deviate? Firm 1's going to want to produce at C + 2ε.?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.?

In particular, we find that prices in the market are equal to marginal cost. We find that profit in equilibrium is equal to 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.

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. 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.?

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.?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.?

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 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,?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.?

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.

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. Now I want to spend the rest of today doing something quite different.?

So I'm going to take the same basic idea back to the politics model. 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.?

Two?new things here, 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 the idea of the model is each voter is a potential candidate. So 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. 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.?

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. 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?

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?

Ss?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.?

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.?