14 Backward induction commitment, spies, and first-mover advantages

Game Theory: Lecture 14 TranscriptOverview:We first apply our big idea--backward induction--to analyze quantity competition between firms when play is sequential, the Stackelberg model. We do this twice: first using intuition and then using calculus. We learn that this game has a first-mover advantage, and that it comes commitment and from information in the game rather than the timing per se. We notice that in some games having more information can hurt you if other players know you will have that information and hence alter their behavior. Finally, we show that, contrary to myth, many games do not have first-mover advantages.October 24, 2007 << backProfessor Ben Polak: All right, so today I want to do something a little bit more mundane than we did on Monday. I want to go back and talk about quantity competition. So in the first half of the course we talked about price competition. We talked about quantity competition. We talked about competition with differentiated products. I want to go back and revisit essentially the Cournot Model. So this was the Cournot Model: two firms are producing, are choosing their quantities simultaneously. Firm 1 is choosing Q1 and Firm 2 is choosing Q2. And all of this is just review so this is all stuff that's in your notes already. This is the demand curve. It tells us that prices depend on the total quantity being produced. So this is Q1 + Q2 and this is prices, then the demand curve is a straight line of slope b. That's what this tells us. Here's our slope –b. And we know that payoffs are just profits, which are price times quantity, revenues, minus cost times quantity, costs, we're assuming constant marginal costs.We did this model out in full in maybe the third week of class and we figured out what the best response diagram looked like. And if you remember correctly, this was the best response for Firm 1 taking Firm 2's output as given, and this is the best response for Firm 2 taking Firm 1's output as given, and there were a few other details in here. This was the monopoly quantity, this was the competitive quantity and so on, but this is enough for today. Actually, we had done a bit more than that, we'd actually worked out in class what the equations were for these best responses. Here they are. I'm not going to re-derive these today, but they're somewhere in your notes. We kind of crunched through some calculus and figured out what Firm 1's best response looks like algebraically, here it is. So this is theequation of this line, and similarly for Firm 2, so this is the equation of this line.Finally, we figured out what the Nash Equilibrium was, and there's no prizes here: the Nash Equilibrium in Cournot was where these best responses crossed, and this is the equation for the Nash Equilibrium. Have I made a mistake? The best response, oh thank you. The best response for Firm 1 is a function of Q2, exactly. Thanks Jake. So this is all stuff we did before, I want to go back to this model now to revisit it in the context of thinking about sequential dynamic games. So what we're going to do is--we're going to do is, we're going to imagine that rather than having these firms choose their quantities simultaneously, one firm gets to move first and the other firm moves after. Let's be clear, we're going to assume that Firm 1s moving first and the other firm--we'll assume Firm 1's going to move first--the other firm, Firm 2, is going to get observe what Firm 1 has chosen and then get to make her choice.So we're going to see what difference it makes when we go from this classic simultaneous move game into a sequential move game. This model is fairly famous and I'm almost certainly spelling this wrong, but it's due to a guy called Stackelberg. So what we're looking at now is the Stackelberg Model. So how do we want to think about this? A natural question to bear in mind is, assuming we're in this world of quantity competition, is it an advantage to get to move first, to set one's quantity first? Or is it an advantage to be able to wait, see what the other firm has done, and then respond? Is there an advantage in going first or is there an advantage in knowing a bit more about the other firm and being able to move second?That's going to be the question at the back of our minds for most of today. So how are we going to think of this? How are we going to figure this out? There shouldn't be a silence in the room. There should be an instance answer. How are we going to figure this out? We're going to use backward induction right. This is going to be an exercise in backward induction. We won't be able to draw a tree here because the game's too complicated because there's a continuum of actions, but nevertheless, we are going to use backward induction. So what does using backward induction mean here? Using backward induction means starting at the end and the end is what? The end here is Firm 2.Firm 1 is going to move first, Firm 2's going to observe that choice and then move. So the end of the game is Firm 2. So we're going to solve out Firm 2's problem first. We're actually going to do this entire analysis twice. We're first of all going to do this analysis a bit intuitively looking at pictures, and then I want to go back and crunch it out in the math. I want to get used to seeing that we can actually do it crunchily.This board is just review, so I'm going to get rid of it I think. I didn't manage to get rid of it, never mind. We're not going to be using this board. This is just what we did do in the simultaneous move game, so we'll get rid of it.So in the sequential move game we're going to start by analyzing the move of Firm 2. So imagine yourself as the manager of Firm 2, you're coming along to make your output decision. The output of Firm 1 is already set. So analyze Firm 2 first, Firm 2 sees Q1 and now I must choose Q2. So what is Firm 2 going to do? So I claim we already know this. We've already solved this problem. When did we solve this problem? Anyone know when we solved this problem? The problem of what Firm 2's going to do. Well we already solved it about a month ago when we looked at the simultaneous move game, because what we worked out then was what is Firm 2's best response for any particular choice that Firm 1 makes? We already solved out that problem. It took us a while to solve out, but basically it was to maximize Firm 2's payoff, taking as given Firm 1.We already know what the equation looks like and let's just remind ourselves what the picture looked like, just a repeat of the picture we had before. We said for any particular choice of Firm 1, Firm 2's best response can be drawn on a best response diagram, and looked like this. It's exactly the picture we have up there. So this is the best response for Firm 2 taking as given the choice of Firm 1. We even know the equation of it--I won't bother rewriting that. We already know the equation. So in some sense Firm 2's problem is a problem we've already seen. We already worked out months ago what Firm 2 should do, taking if Firm 1's output is given and that's exactly the problem Firm 2 finds herself in.She wakes up one morning, Q1 has been set already, and now she must choose Q2 to maximize her profits, so she's going to choose her best response. Just to remind you how we read this picture, for any particular choice of Q1 we go up to the line and look across, this tells us what Q2 will respond. So if Q1 chooses this amount then Q2 will choose this amount. If Q1 chooses this amount then Q2 will choose this amount and so on. So there's no mystery here. We already know what Firm 2's going to do. So by definition, the best response of 2 to Q1 tells us the profit maximizing output of Firm 2 taking Q1 as given. All right, so we've done the second step for this already, we already know what Firm 2's going to do. Of course, the additional step here now is that Firm 1 knows that Firms 2's going to do it.Firm 1's going to move first and Firm 1 knows that after she sets her quantity Q1, Firm 2 will respond by choosing her corresponding quantity, which is the best response to it. So if Firm 1 knows that if Firm 1 wereto choose this quantity, then Firm 2 will respond by choosing this quantity, and Firm 1 knows that if she chose this smaller quantity, then Firm 2 will respond by choosing this larger quantity. Is that right? So Firm 1 can anticipate how Firm 2 is going respond to each of these choices. So let's just make that clear. So in particular, if Firm 1 was to choose Q^1, I'm not suggesting it should, but if Firm 1 was to choose this Q^1, then Firm 1 knows that Firm 2 will produce this quantity, which is the best response to Q^1, and if Firm 1 were to choose Q^^1, then Firm 2 will respond by choosing the best response to Q^^1.So this is pretty straightforward so far, but what we're able to see now, is the problem facing Firm 1, which is the interesting problem. The problem facing Firm 1 is, what quantity should Firm 1 choose knowing that this is how Firm 2 is going to respond? Before we solve this out mathematically, I just want us to think it through a little bit. So the first way I want to think this through is, is to make the following observation. From Firm 1's point of view, Firm 1 knows that any Q1 she chooses leads to a response on this line by Firm 2. That's what Firm 1 knows. So Firm 1 is effectively choosing points on this line. Let me say it again, so what's actually happening is Firm 1 is choosing Q1 and Firm 2 is responding by choosing a Q2 that puts them on this line. But in effect that means Firm 1 is choosing points on this line.So you could think of Firm 1's problem as, choose the joint output level on this line that maximizes Firm 1's profits. Think of Firm 1's problem as choose the combination of outputs on this line by choosing Q1 and then Q2 responds, choose the combination on this line that maximizes profits for Firm 1. So I'm belaboring this a little bit because it's a more general mathematical idea here. How many of you are in Econ 150 right now? So for those of you in Econ 150, this should be a very familiar kind of thing. This is a constrained optimization problem and you've been having constrained optimization problems rammed into you for the last month or so, so this is an example of a constrained optimization problem.You have to choose a point but you can't choose a point freely: you have to choose a point on the line. Okay, so let's talk about it a bit more before we do the math. Let's actually redraw it again since I made a mess of this picture. So one thing you might want to ask is, in making this choice for Firm 1, should Firm 1 choose more or less or the same as it used to choose when the problem was simultaneous? So let's put in again what it used to choose when the problem was simultaneous. I'll put it in just faintly. So here's our old Cournot picture--looked like this--and this was the quantity that Firm 1 chose in the Cournot game, so let me call that Q1C. So certainly one possibility is that Firm 1 could choose her Cournot quantity, she can certainly do that, and she knows that ifshe does that, Firm 2 will respond by choosing the best response of Firm 2 to Firm 1 choosing the Cournot quantity, but we know what that is. What's Firm 2's best response to Firm 1's Cournot quantity? It's Firm 2's Cournot quantity, right?So if Firm 2 does that, if Firm 1 chooses the Cournot quantity, then Firm 2 will also choose the Cournot quantity. So one thing that Firm 1 could do is effectively choose the old equilibrium. That's certainly something that's available to Firm 1. But Firm 1 could also do other things. Firm 1 could produce less than that or Firm 1 could do more than that. So who thinks Firm 1 should play--should choose the old equilibrium quantity? Who thinks Firm 1 should choose more than that? Who thinks Firm 1 should choose less than that? Let's just try it with the camera on you.So once again, how many people think that Firm 1 should choose the old equilibrium quantity? A dribbling of hands, and how about less than that?A few hands and then they went down again, and how about more than that? There's a majority for more--turns out more is correct, so that's good news. Why? Why do we think Firm 1 should produce more than it used to produce before? Any takers on this? Well let's think about it. As Firm 1 produces more, or if Firm 1 were to produce more, then Firm 2--my voice is going--then Firm 2 would produce what? Less. As Firm 1 produces more than her Cournot quantity, Firm 2's response is to produce less.Does anyone remember the jargon for this? What do we call games where the more I do of my strategy the less you do of yours? "Strategic substitutes," good. This is a game of strategic substitutes. What that means is that as Q1 goes up, Q2, the best response of Firm 2 to Q1 goes down. So what? We can look at that just by looking at the picture. Well the "so what" is, now we're in a sequential game. If Firm 1 produces more than her Cournot quantity she induces Firm 2 to produce less. That's what we just said and that's what? That's good for Firm 1. My producing more inducing you to produce less is good for me. It's going to keep prices higher in the market. Is that right?So let's just think it through again. In the Cournot equilibrium, the choice of Firm 1 was the best choice for Firm 1, taking the choice of Firm 2 as given. That was the old Cournot quantity. But now in the Stackelberg setting, the sequential setting, there's an additional feature. Firm 1 doesn't have to take Firm 2's output as given. There's an additional reason for producing at the margin, which is, at the margin if I produced some more units of output, that leads you to produce less which is good for me. So that suggests that I'm going to produce than I used to produce under the old assumption.So this suggests that Firm 1 should set Q1 bigger than Q1C to induce Q2 to be less than Q2C. So the first thing we've learned--we'll see this in the math later--is that Firm 1 will in fact produce more than they used to under Cournot, and that will result in Firm 2 producing less than Cournot. Now we've already got a lot on the board now, we can actually solve out intuitively the problem. Do we think that Firm 1's profits, by this procedure, are the same as they were under Cournot? Are they less than they were under Cournot, or are they more than they were under Cournot?So just let me say it again. Firm 1 is going first now. We've argued that Firm 1's going to produce more. Do we think that Firm 1's profits at the end of the day are going to be the same as they were under Cournot, higher than they were under Cournot, or lower than they were under Cournot? So let's have a poll again, let me get the camera on you guys. So who thinks their profits are going to be the same as they were under Cournot? Who thinks the profits have gone up? Who thinks the profits have gone down? We're in good shape here because indeed the profits have gone up. There's a very simple argument why the profits have to have gone up.How do we know the profits must have gone up? Let me actually--it's simple enough--let me grab a mike on this. How do we know the profits just must have gone up? There was a hand in the back, was there a hand in the back? Yes, way at the back. How do we know profits must have gone up here? Way at the back.Student: If Firm 1 was going to lower their profits they wouldn't have chosen to produce more.Professor Ben Polak: All right, good exactly. The fact that Firm 1 has changed their output, and in particular, are producing more, tells you they must be able to increase their profits by this maneuver. Let's just think that through. One option that was available to Firm 1 before was to set output at the Cournot level. If Firm 1 had set output at the Cournot level, that would have led Firm 2 to set output at the Cournot level, and in that case, profits would have been exactly the same as before. The fact that Firm 1 has moved away from that must mean there are higher profits available. Say it another way, Firm 1 could have had exactly what it had before, so it must be doing at least as well as it was doing before, and the fact it has changed means it must be doing better than it was doing before.So indeed, Firm 1's profits have gone up. We don't even need any math to prove that: it just must be the case logically. What must have happened to Firm 2's profits? What do you think has happened to Firm 2's profits?That's not so immediately obvious. It's obvious, I think, that Firm 1's profits have gone up here because Firm 1 could have had the same old profits and has chosen something else. But it's not immediately obvious what happened to Firm 2's profits, is that right? Before we get to what's happened to Firm 2's profits let's go through an intermediate step. Let's try and ask what must have happened to total output in the market in this example--in this nice simple example.That's not immediately obvious either. Why? We've argued that Firm 1's output went up but Firm 2's output went down relative to Cournot. So it's not immediately obvious whether the sum of those two Q1 + Q2 went up or down. We'd like to know what happened to Q1 + Q2, total output, in the market. By the way, one particular reason we might care about this is of course consumers would like it to have gone up. Because if their total output has gone up, prices have gone down, and that's good for consumers. So if you're the regulator, if you're designing this industry--if you're working for the Justice Department, or if you're working for European Commission--you're going to want to know the answer when we switch from a simultaneous setting to an asymmetric setting where there's a leader firm and a follower firm, is that going to be good for consumers or bad for consumers?Well let's have a look. Well we know that Firm 1's output went up and we know that Firm 2's output went down, but can anyone tell me what happened to the total output and why? Let's have a poll again. Who thinks total output went down? Who thinks total output stayed the same? Who thinks total output went up? There's lots of abstentions. Let's try that again because too many abstentions. Who thinks total output went down? Who thinks total output stayed the same? Who thinks total output went up? That's pretty split. So I think total output--I know actually--that total output went up, and I claim I can see it on the picture. I claim if you stare at that picture you can actually see that total output must have gone up. Who's good at looking at a picture? Let me get the mike in here. The picture's there. Let me try this person. Your name is?Student: Andy.Professor Ben Polak: Andy go ahead.Student: Judging by the slope of the line you know that as it moves the amount that Q1 changes will be greater than the amount that Q2 goes down.Professor Ben Polak:Good, so what Andy said, what Andy's saying is look at the slope of the line along which we slid. When we went from Cournot to the Stackelberg equilibrium, we started here and we slid in thisdirection down this line: Q1 went up and Q2 went down. And what Andy's pointing out is we can see from the slope of this line that Q1 goes up more than one unit for every unit of reduction of Q2. Let me say it again, for every unit of increase of Q1 there's less than a proportion--sorry, say it again. For every unit of increase of Q1 there's less than a unit of decrease of Q2. Another way of saying it is the slope of this line is less than 1. Everyone see that?So we know by the slope of the line--we know that Q1 had to go up more than Q2 went down, which means total output went up, which means that prices went--what happened to prices as total output went up? That shouldn't be hard, everyone's taken 115 here, right? So when total output went up, prices went down, good. Demand curves sloping down is not as important as backward induction but it's still quite important. So prices went down. So therefore, we now are ready to say what happened to Firm 2's profits. Firm 2 is producing less than before. Firm 2's costs are the same, and prices have gone down, so what's happened to Firm 2's profits? They must have gone down as well.So Firm 2's profit has gone down, and we know that consumer surplus, CS, consumer surplus, has gone up. For those people who remember their 115, prices have gone down, quantities have gone up, so consumer surplus here has gone up. So we've analyzed everything qualitatively I can think of in this game without reference to any math at all. Is that right? We really haven't done any math there. We talked about slope of the line. I guess that's math in junior high or something, but we haven't really done any math here, right? Is that fair? We already have a pretty good intuition for what's going to go on in this market. We think Q1's going to go up. We think Q2's going to go down, we think that Firm 1's profits are going up. We think Firm 2's profits are going down. We think total quantity is going up.Now let's see if we're right. Let's go back and do the math. So I want to spend a bit of time grinding this out. So I don't claim that doing the math is fun, but I want to prove that we can do it, because otherwise everyone's going to either think that this was all kind of just blah, blah, blah, and/or people are going to be scared to do the math when it arrives on a homework assignment. So for now we're going to--more or less for the next few minutes--we're going to forget we're economists and we're going to turn into nerds. That isn't a huge transition, but we'll do it anyway.I've got the demand curve up there, so the demand the curve is still there it's P = A - B [Q1 + Q2] and I've got profits written up there, but let's put them somewhere more convenient anyway. So P = A - B [Q1 + Q2] and profit is equal to--profit for Firm i is equal to P Qi--C Qi. And what we're toldto do in backward induction is what? First of all, solve things out for Firm 2 taking Firm 1 as given, and then go back and solve for Firm 1. So exactly the discussion we've just had informally, we're now going to do more formally. So backward induction tells us, solve for Firm 2 first, taking Q1 as given.What is that problem? It's this sort of boring math problem, it says maximize by choosing Q2 the profits of Firm 1, so that's going to be A - B Q1--B Q2. That's the price times the quantity Q2, - C Q2. So this bit here is the price. These two terms together are revenues, and this term is costs. Now I could do that, I could grind that bit out, but we already ground out that bit out three or four weeks ago right, so I'm not going to grind it out again, we know how to do that. But by the way, let's just remind ourselves what we did. We differentiated with respect to Q2. We set the thing we found, the derivative, equal to 0. That was our first order condition. And then we solved for Q2. Is that right? So when we did that, we went through the first order condition and then we solved it, we know what we actually got.So what we actually got was that Q2*, if you like--or Q2 let's just call it--is equal to [A - C] / 2B - Q/2. In fact, I've already given it to you up there, it's up on that top board. It's the best response for Firm 2. So again, I could do this again, but since we did it a few weeks ago I don't want to redo it. Now the more interesting part, not thrilling, buta little bit more interesting. Now let's solve for Firm 1. So what is Firm1 doing? Firm 1 is also trying to maximize profits. Firm 1 is choosing Q1 and Firm 1--at least initially it looks like the same problem. It's A -B Q1--B Q2 Q1--C Q1, this is the same line we had before but now whereas Firm 2 was taking Q1 as given, Firm 1 knows that Q2 is given by this formula here. So what we're going to do is we're going to plug this Q2 into there.Now again, for those of you in 115, this isn't the only way to do it--I'm sorry 150--this isn't the only way that we could do it, we could also set up a Lagrangian equation, but for those people who don't know what that is, don't worry we're going to plug in today. So we're going to plug this in, and when we plug it in, we get a right old mess but let's do it anyway. So we're going to get max Q1, [A - B Q1--B [[A - C]/2B - Q1/2] - C] Q1. Everyone okay with that? I'm doing algebra on the board which is not fun but it's useful to do occasionally. So eventually what are we going to do? Eventually we're going to differentiate this thing, set it equal to 0, look at our first order condition and so on, just as we normally would.So eventually we're going to use basically 112 level calculus to solve this thing. Everyone remember how to solve a maximization problem? Yeah? But before we do that let's tidy up the algebra a bit. So this thingis--let's just tidy it up. So this is equal to max with respect to Q1--and notice I've got an A - C here and once I take this B inside the brackets I've got a--[[A -C]/2] here. So I've got an [A - C]--[[A - C]/2] so that's going to give me an [A - C]/2--and I'm really going to pray that the T.A.'s are watching me carefully and are going to catch my errors here. Okay, so I think I'm okay so far but please catch me.What else have I got? I've got a -B Q1 here and from in this bracket I've got a - - that's a + B Q1/2. So I have a –B Q1 + B Q1/2 so that's a –B Q1/2. So far so good and that whole thing is multiplied by Q1. Okay so far? Let's multiply out the bracket because otherwise I'll make a mistake. So this is the same as saying [A - C]/2 Q1--B Q1²/2. So far so good? Now we're at the level where even my very rusty memory of calculus will get us through, so let's try and do it. So what we're going to do is we're going to differentiate this thing with respect to Q1. So differentiating with respect to Q1--let's do it up here --, we get [A -C]/2 from this term and from the –B Q1²/2 we're going to get--the two's are going to cancel rather pleasantly--so we're going to get B Q1 from that term. Everyone with me so far?I'm going through this in sort of slow steps, I agree it's not exciting but I want to make sure I don't make a mistake. So to turn this into a first order condition what must be true about this derivative? At the maximum, what must be true about this derivative? Should be equal to 0, good, and we should just check the second order condition. How do I check the second order condition? I differentiate again and check it's negative, but if I differentiate again I'm just going to get -B. So -B is certainly negative all right, so second order condition is okay. So let's solve it out. Solving this for Q1, I get Q1 = [A - C]/2B. So we're leading in sheer boredom, it has to be done occasionally, Q1 = [A - C]/2B. We're not done yet. Now we want to go back and solve out algebraically for Q2.I know what Q1 is now, Q1 is [A - C]/2B. How do I find Q2? Somebody? Shout it out, how do I find Q2? I've got to plug it in. I'm going to go back and plug this Q1 back into this expression here, so I plug it back in, I'll get Q2 = [A - C]/2B - 1/2 [A - C]/2B for a total of [A - C]/4B. Is that right? That's what I have in my notes. This looks good. So I'm now done. I've now found the equilibrium. I've found that in thisleader-follower game, this Stackelberg version of quantity competition, Q1 is given by [A - C]/2B and Q2 is given by [A - C]/4B.Let's see how it matches up with the intuition we developed before without using any boring math. So first of all we're comparing Q1 and Q2 with what they used to produce, and what they used to produce is on the top board.。