ECON 159: Game Theory
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ECON 159 - Lecture 4 - Best Responses in Soccer and Business Partnerships
Chapter 1. Best Response: Penalty Kicks in Soccer [00:00:00]
Professor Ben Polak: So last time we introduced a new idea and the new idea was that of best response. And what was the idea? The idea was to think of a strategy that is the best you can do, given your belief about what the other people are doing: what your opponents are doing, what other players are doing. And you could think of this ─ you could think of this belief as the belief that rationalizes that choice. So if you have a boss you might want to ─ and he or she is going to ask you why you chose the action you did. If you took a best response to some believe, you can say I took this action because I believed other people are going to do this. And since that was the best you could do under that belief, you’ll hopefully keep your job.
I promised that today we would look at the most important game in the world. And, as announced last time, the most important game is the penalty kick game. So this is a game that occurs in soccer and just to give an idea of how important it is for those people who are unfortunate enough not to be soccer fans here, the last World Cup was decided on penalty kicks. In England’s case, England goes out of every single World Cup and every single European competition because it loses on penalty kicks, usually to Germany, it has to be said. And more immediately, this weekend, as all of you are thinking the most event in the world was whatever was happening in Congress to do with Iraq, actually, the most important event in the world was taking place in England where my favorite team, Portsmouth, were playing Kaj, the head TA’s favorite team, Liverpool. And, about a third of a way through that game there was a penalty, and … I’ll let you know what happened later. So keep at the back of your mind that the real world example that matters here is Portsmouth versus Liverpool this weekend. (Kaj, the head TA is Scandinavian so I’ve got no idea why he’s supporting Liverpool anyway, but I think maybe he spells it like this or something like this).
So what we’re going to do is we’re going to look at some numbers that are approximately the probabilities of scoring when you kick the penalty kick in different directions. But just make sure everyone ─ do I need to explain what’s going on here? Is everyone familiar with this situation? There’s one guy who’s going to run up and kick the ball. The goal keeper is standing at the goal. And their aim is to kick it into the goal. That’s probably enough. You’ve all seen this right? If you haven’t seen this, go see it. I mean come on! So things you should do in life: read Shakespeare and see a soccer game.
So the rough numbers for this are as follows ─ and actually later on in the class I’ll give you some more accurate numbers, but these will do for now. There are three ways, the goal─ the attacker could kick the ball. He could kick the ball to the left, the middle, or the right. And I shouldn’t just say he here of course, I mean this is he or she but if I get that wrong going on, please forgive me for it. The goalie can dive to the left or the right. In principle the goalie could stay in the middle. We’ll come back and talk about that later. So this is the guy who is shooting, he’s called the shooter and this is the goalie.
These are roughly ─ well, let me put up the payoffs for this game and then I’ll explain them. So you’ll notice that I’m just going to put in numbers here and then the negative of the number and the numbers are roughly like this: (4,-4). So the numbers are (4, -4), (9, -9), (6, -6), (6, -6), (9, -9) and (4, -4). And the idea here is that the number 4 represents 40% chance of scoring if you shoot the ball to the left of the goal and the goal keeper dives to the left. So the payoff here is something like u1(left) if the goal keeper dives to the left is equal to 4, by which I mean there’s a 40% chance of scoring.
So the number for the–The payoff for the shooter is his probability of scoring and the payoff for the goal keeper is just the negative of that. Let’s keep things simple. As I said before, for now we’ll ignore the possibility that the goal keeper could stay put. So how should we start analyzing this important game? Well we start with the ideas we learned already several weeks ago now, or more than a week ago, which is the idea of dominant strategies. Does either player here have a dominated strategy? Does either player have a dominated strategy? No, it’s kind of clear that they don’t.
Let’s just look at the shooter, for example. So you might think that maybe middle dominates left, but notice that middle has a higher payoff against left than shooting to the left. It has as lower payoff if the goalie dives to the right. So, not surprisingly in this game, it turns out, that if the goalie dived to the left you’re best off shooting to the right, second best off shooting to the middle, and worst off shooting to the left. That’s if the goalie dives to the left. And if the goalie dives to the right, you’re best off shooting to the left, second best off shooting to the middle, and worst off by shooting to the right; and that’s kind of common sense.
Okay, so if we had stopped the class after the first week where all we learned to do was to delete dominated strategies, we’d be stuck. We’d have nothing to say about this game and as I said before, this is the most important game, so that would be bad news for Game Theory. But luckily, we can do a little bit better than that. Before I do that, let’s just take a poll of the class. How many of you, if you were playing for, I guess it’s going to be America, which is a sad thing to start with, never mind. You guys are playing for America and you’re taking this penalty kick and it’s the last kick in the World Cup, how many of you, show of hands, how many of you would shoot to the left? How many of you would shoot to the middle? How many of you would shoot to the right?
We’ve got kind of an even split there, pretty much an even split. We’re going to assume these are the correct numbers and we’re going to see if that even split is really a good idea or not. So how should we go about thinking about this? What I suggest we do is we do what we did last time and we start to draw a picture to figure out what my expected payoff is, depending on what I believe the goalie is going to do. So this is the same kind of picture we drew last time.
So on the horizontal axis is my belief, and my belief is essentially the probability that the goalie dives to the right. Now as I did last time, let me put in two axes to make the picture a little easier to draw. So this is 0 and this is 1. And you probably have lines in your notes but I don’t, so let me just help myself a bit. So this is 2, 4, 6, 8, 10, so this is going to be 2, 4, 6, 8, and 10 and over here 2, 4, 6, 8, and 10, 2, 4, 6, 8, and 10. This would be the basis of my picture.
So it starts with a possibility of shooting to the left. Let’s do this in red. So I shoot to the left and the goalie dives to the left, my payoff is what? It’s 4. If I shoot to the left and there’s no probability of the goalie diving to the right, which means that they dive to the left, then my payoff is 4, meaning I score 40% of the time. If I shoot to the left and the goalie dived to the right, then I score 90% of the time, so my payoff is .9. By the way why is it 90% of the time and not 100% of the time? I could miss; okay, I could miss. That happens rather often it turns out, well 10% of the time.
So we know this is going to be a straight line in between, so let’s put this line in. So what’s this? It’s the expected payoff to Player I of shooting to the left as it depends on the probability that the goal keeper dives to the right. And conversely, we can put in … well let’s do them in order.
So middle: so if I shoot to the middle and the goal keeper dives to the left, then my payoff is .6, is 6, or I score .6 of the time, and if I shoot to the middle and the goalie dives to the right I still score 60% of the time, so once again it’s a straight line in between. So this line represents the expected payoff of shooting to the middle as a function of the probability that the goal keeper dives to the right.
Finally ─ let’s do it in green ─ let’s look at the payoffs, expected payoffs, if I shoot to the right. So if I shoot to the right and the goalie dives to the left, then I score with probability .9, or my payoff is 9. Conversely, if I shoot to the right and he or she dives to the right, then I score 40% of the time, so here’s my payoff .4. And here’s my green line representing my expected payoff as the shooter, from shooting to the right, as a function of the probability that the goalie dives to the right.
Did everyone understand how I constructed this picture? Easier picture than the one we constructed last time. So what does everyone notice from this picture? What’s the first thing that jumps out at you from this picture? Assuming these numbers are true, what jumps out at you from this picture? Can we get some mikes up here? So Ale, can we get this guy? Stand up first, the guy in red. What’s your name? Don’t hold the mike; just shout.
Student: There’s no point at which the 6, at which it shooting in the middle gets a higher payoff.
Professor Ben Polak: Exactly, exactly. So the thing that I hope jumps out at you from this picture is (no great guesses about figuring out this is a ½), so if the probability that the goalie’s going to jump to the right is less than a ½, then the best you can do is represented by this green line, which is shoot to the right. So the goalie is going to shoot to the right with the probability less than a ½, sorry he’s going to dive to the right with the probability less than a ½, you should shoot to the right.
Conversely, if you think the goalie’s going to shoot to the right with probability more than a ½, then the best you can do is represented by the pink line, and that’s shooting to the left, or if you think the goalie’s going to dive to the right with the probability more than a ½, the best you can do, your best response is to shoot to the left. And there is no belief you could possibly hold given these numbers in this game that could ever rationalize shooting the ball to the middle. Is that right? So no: to say it another way, middle is not a best response to any belief I can hold about the goal keeper, to any belief.
So there’s a lesson here, and it’s pretty much (just to make the lesson resonate again): imagine there you are in the World Cup, you’re playing for England, you have to justify your actions not only to your teammates and your manager, and your boss, but to about 60 million rather angry fans. What’s the lesson here? I’m hoping it was going to be obvious, what’s the lesson here? The lesson is, do not shoot to the middle. Let me qualify that lesson slightly, unless you’re German. Germans can do whatever they like.
Now, it turns out that about a third of the game between my team Portsmouth and Kaj’s team Liverpool, this weekend there was a penalty. Portsmouth had a penalty and the guy who was going to take the penalty came up to kick the penalty and he kicked it to the middle and it was saved. So just confirming these actions, not only did that spoil my weekend but it also spoiled my opportunity to make fun of Kaj all week, so it was really a big deal. So this weekend a penalty was missed exactly by somebody ignoring this rule.
There’s a more general lesson here, and the more general lesson is, of course: do not choose a strategy that is never a best response to anything you could believe. The more general lesson, do not choose a strategy that is never a best response to any belief. Notice here, just to underline something which came up at the end last time, that doesn’t just mean beliefs of the form, the goalie’s going to dive left or the goalie’s going to dive right. It means all probabilities in between. So we’re allowing you to, for example, to hold the belief that it’s equally likely that the goalie dives left or dives right.
But if there’s no belief that could possibly justify it, don’t do it. And underlining what arises in this game, notice that in this game we’re able to eliminate one of the strategies, in this case the strategy of shooting to the middle, even though nothing was dominated. So when we looked at domination and deleted dominated strategies, we got nowhere here. Here, at least, we got somewhere, we got rid of the idea of shooting to the middle. Now if you can just persuade the English and Portsmouth soccer players of this lesson, I’d be very happy.
Chapter 2. Best Response: Issues with the Penalty Kick Model [00:15:14]
So before we leave it, I’ve been making a point in this class of coming back to reality from time to time, so this is a very simple model of the soccer game in reality. Let’s just try, any of you on the Yale Soccer Team? No? Have any of you played soccer for your college? One or two. Have you ever played soccer? How many of you have ever played soccer? Okay, good I was getting worried there for a second. So, one thing we said last time was when we put up a model and try and draw lessons from it, we should just take a step back and say, what’s missing here? So let’s try and get some kind of ─ I’ll come off the stage to make it easier for Jude. What’s missing here? What’s missing in this model of this piece of soccer, this game within a game? What’s missing here? Why is this not necessarily a hundred percent accurate model? I’ll need some mikes up here. Can you? You have to really shout because you’re miles from the mike there.
Student: You might be better kicking to the left or to the right depending on whether you’re right handed or left handed.
Professor Ben Polak: Good, so one thing that’s clearly missing here is I’m ignoring that in fact right footed players find it easier to shoot to their left, which is actually the goalie’s right. So right footed players find it easier to shoot to the left as facing, to shoot across the goal. Does everyone confirm that’s true? Yeah, anyone ever tried to this? It’s a little easier to hit the ball hard to the opposite side from the side which is your foot and that’s the same principle in baseball. It’s a little bit easier to pull the ball hard then it is to hit the ball to the opposite field. Yes?
Student: Players don’t make their decision before, and then stick with it necessarily.
Professor Ben Polak: All right, so players are making decisions as they’re running up. I think that’s okay here, right? We can think of this as the decision happening at the instant at which you kicked. So you’re right that you could have made your decision back in the locker room, or you could have made the decision at half time, but ultimately what matters ─ let’s hope that goes away. We sure that it’s not off my mike? Just in case I’ll move my mike a bit lower. So I’m going to shout louder because my mike is now lower. It doesn’t really matter exactly when the decision is made. At the end of the day, the goalie doesn’t know the decision of the shooter and the shooter doesn’t know the decision of the goalie. So it’s as if that decision is made instantaneously as the shooter is running up. What else? Yeah, can we get this. Tae can you get this guy here? Stand up. Shout out.
Student: The goalie might stay in the middle.
Professor Ben Polak: The goalie might stay in the middle. That’s a good point, of course, I’ve abstracted from that here, and in fact, we’ll come back, I think, I’ll try and put that onto a problem set, but I think you’re right, it is an issue here. Anything else? Well let me put up some real numbers and we’ll see about how much the correspondence to what we’ve got here. So I gave you some numbers I made up actually a long time ago, but since I’ve been using this game in class, somebody went out and checked.
And it turns out that ignoring middle for a second, ignoring middle ─ so these are real numbers, and these numbers come from a paper in the AER by Chiappori and some co-authors and for everyone at Yale, I’ll make that paper available to you through JSTORE or through the Yale Library, so you can go look at it if you’d like. What they worked out was the following table. And again, we need to be a little bit careful here. So I’m going to put the left and right in inverted commas because actually what they did was they corrected for people’s natural direction and not natural direction. So the idea here is shooting to the left if you’re right footed is the natural direction, so left here means the natural direction. Of course, if you’re left footed it goes the other way, but they’ve corrected for that.
It turns out that the probabilities of scoring here are as follows, 63.6, 94.4, 89.3, and 43.7. So things are not–I haven’t given you the numbers for the middle but–So you can see that whoever it was who said, you’re slightly better off, you score with slightly higher probabilities when you kick to your natural side is exactly right. The thing is still not dominated and we could still have done exactly the same analysis, and actually you can see I’m not very far off in the numbers I made up, but things are not perfectly symmetric. I forget who it was who said that, but that does turn out to be true.
Certainly the goalie staying put is an issue, as I said we’ll deal with that in the problem set, but there’s another issue here. Let me just raise one more issue. One more issue is, you have another decision when you run up to hit this, hit the penalty other than just left and right. Someone whose played the game, what’s the other decision you really face? Can I get the woman here? What’s the other decision you face?
Student: You could kick up to the top corner.
Professor Ben Polak: Okay, you can kick up and down, that’s true. Okay, that’s true actually, that’s true. But I meant something else, that’s right, but I meant something else. What else is there? Try this guy here.
Professor Ben Polak: Well that’s getting subtle here. It’s a much more basic thing, what’s a more basic thing? What’s a more basic thing here? Take it, yeah right in front of you.
Professor Ben Polak: Speed, right. So another decision you face is do you just try and kick this ball as hard as you can or do you try and place it? That’s probably as important a decision as placing it, as deciding which direction to hit, and it turns out to matter. So, for example, if you’re the kind of person, (which is I have to say all I ever was), if you’re the kind of person who can kick the ball fairly hard but not very accurately, then it actually might change these numbers. If you can kick the ball very hard, but not very accurately, then if you try and shoot to the left or right, you’re slightly more likely to miss. On the other hand, as you shoot to the middle, since you’re kicking the ball hard, you’re slightly more likely to score.
Now, if this all seems like arcane and irrelevant detail, let’s just see why this matters in the picture, and then we’ll leave soccer, at least for today. So if you’re the kind of person who can kick the ball hard but not accurately, then it’s going to lower the probabilities of scoring as you kick towards the right because you might miss, and it’s going to lower the probability of scoring as you hit towards the left because you’re likely to miss, and it might actually raise the probability of your scoring as you hit towards the middle, because you hit the ball so hard it’s really pretty hard for the goal keeper to stop it. Here it goes in the middle, and if you look carefully there, I didn’t really make it clear enough, you can see, suddenly a strategy that looked crazy shooting to the middle, that suddenly started to seem okay.
It turns out, if you look at those dotted lines, there’s an area in the middle, the area between here and here, this little area here, you actually might be just fine shooting to the middle. So in reality, we need to take into account a little bit more, and in particular, we need to take into account the abilities of players to hit the ball accurately and/or hard. And if those people, if you’re interested in that ─ and I realize at this point I probably lost the interest of most Americans in the room, but for the non-Americans in the room, the people who are interested in the real world ─ as I said before, I’ll put that article online and that goes through all the gory detail of this.
I should just say that the data I just gave you is real data but it’s actually mixed ability data. This data comes half from the Italian league, which is pretty good and half from the French league, which sucks. So who knows how much we should trust it. Okay, so that was our example for the day and our first brush with reality for the day. Let’s clean the board and do some work. Do a bit more formal stuff here. So here we have an example but I want to go back to the generality and to a bit of formalism. By the way, I should tell you that the game ended nil-nil or 0-0. It’s a moral victory for me I think.
Chapter 3. Best Response: Formal Definition [00:24:06]
So I want to be formal about these things I’ve been mentioning informally. And in particular, I want to be formal about the definition of best response. I’m going to put down two different definitions of best response, one of which corresponds to best response to somebody else playing a particular strategy like left and right, and the other is just going to correspond to the more general idea of a best response to a belief. It’ll allow us to use our notation and just be a little bit more nerdy.
So Player i’s strategy, Ŝi (there’s going to be a hat to single it out) is a best response (always abbreviated BR), to the strategy S- i of the other players if ─ and here’s our real excuse to use our notation ─ if the payoff from Player i from choosing Ŝi against S- i is weakly bigger than her payoff from choosing some other strategy, Si’, against S- i and this better hold for all Si’ available to Player i. So in previous definitions, we’ve seen the qualifier, for all, be on the other player’s strategy. Here, the qualifier for all is on my strategy.
So strategy Ŝi is a best response to the strategy S- i of the other players if my payoff from choosing Ŝi against S- i, is weakly bigger than that from choosing Si’ against S- i, and this better hold for all possible other strategies i could choose. There’s another way of writing that, that’s kind of useful, or equivalently, Ŝi solves the following. It maximizes my payoff against S- i. So you’re all used to, I’m hoping everyone is used to seeing the term max. As I solve the maximization problem, how do I maximize my payoff given that other people are choosing S- i? Again, for the math phobics in the room, don’t panic, this is just writing down in words what we’ve already seen a couple of times already today, well today and last time.
Let’s generalize this definition a little bit, since we want it to allow for more general beliefs. So just rewriting, Player i’s strategy, same thing, Ŝi is a best response. But now let’s be careful, best response to the belief P about the other player’s choices, if ─ and it is going to look remarkably similar except now I’m going to have expectation ─ if the expected payoff to Player i from choosing Ŝi, given that she holds this belief P, is bigger than her expected payoff from choosing any other strategy, given she holds this belief P; and this better hold for all Si’ that she could choose. So very similar idea, but the only thing is, I’m slightly abusing notation here by saying that my payoff depends on my strategy and a belief, but what I really mean is my expected payoff. This is the expectation given this belief.
Once again, we can write it the other way, or Ŝi solves max when I choose Si, to maximize my expected payoff this time from choosing Si against S-i. What do I mean by expected payoff? Just in our example, so just to make clear what that expectation means, so for example, the expected payoff to Player i in the game above from choosing left given she holds the belief P is equal to the probability that the goal keeper dived to the left, times Player i’s payoff from choosing left against left, plus the probability that the goal keeper dived to the right, times Player i’s expected payoff from choosing left against right. Okay, so expectation with respect to P just means exactly what you expect it to mean. So this is a little bit of math, a little formality, but is everyone okay with that? I haven’t done anything here. All I’ve done is write down slightly boringly and nerdily, exactly what we already saw in a couple of occasions.
Chapter 4. Externalities and Inefficient Outcomes: The Partnership Game [00:29:59]
Professor Ben Polak: Thank you. So that right now is going to seem a little bit like a sudden blast of notation, so let’s just remind ourselves what we really care about is the idea, it’s not the notation, and let’s spend the next half hour applying these ideas to an application. So this application is not as important as soccer, but it’s a bit more Economicsy, so I can justify it under the Economics title of the class. So clearing off my soccer game. So imagine–What we’re going to look at is a game involving a partnership. So Partnership Game. And I believe this game is covered in some detail in the Watson textbook, or something very close to it is, if you’re having trouble.
The idea is this. There are two individuals who are going to supply an input to a joint project. So that could be a firm, it could be a law firm, for example, and they’re going to share equally in the profits . So one example would be a firm that they both earn, sorry, they both own, and another example would be two of you working as a study group on my homework assignment. So they’re going to share equally in the profits of this firm, or this joint project, but you’re going to supply efforts individually. So let’s just be a bit more formal.
So the players are going to be the two agents and they own this firm let’s call it. They own this firm jointly and they split the profits, so they share 50% of the profits each. So it’s a profit- sharing partnership. Each agent is going to choose her effort level to put into this firm. So, it could be that you’re deciding, as a lawyer, how many hours you’re going to spend on the job. So for most of you these decisions will be a question of whether you spend 20 hours a day at the firm or 21 hours a day at the firm, something like that. For most of you on your homework assignments, I’m hoping it’s a little less than that, but not much less than that.
So the strategy choices, we’re not going to do it in hours, let’s just normalize and regard these choices as living in 0 to 4, and you can choose any number of hours between 0 and 4. Just to mention as we go past it, a novelty here. Every game we’ve seen in the class so far has had a discrete number of strategies. Even the game, when you chose numbers, you chose numbers 1, 2, 3, 4, 5, up to a 100, there were 100 strategies. Here there’s a continuum of strategies. You could choose any real number in the interval [0, 4]. So you have a continuum of possible choices. That’s not going to bother us but let’s point out it’s there. So there’s a continuum of strategies. In principle, you could bill your clients for fractions of a second or fractions of a minute.
Let’s wonder what the firm profit is given by. So this partnership, this law firm, its profit is given by the following expression: 4 times the effort of Player I, plus the effort of Player II, plus a parameter I’ll call B, times the product of their efforts. This is their profit. And I won’t tell you what B is for now, but let’s just–I mean I won’t tell you exactly what it is–but I’ll explain it. We’ll assume that B lives between 0 and a 1/4 and it’s known, I just want to be able to vary it later. So what’s the idea here? The idea is Player I directly contributes profits to the firm by working, as does Player II. But they also contribute through this interaction term. How do we think of that interaction term? How do we think of that term B S1 S2?
When you’re working on your homework assignments, if your product, the thing you hand in was just S1 + S2, then you might think what? You might think there’s no point working in a study group at all. If the product is just the sum or multiple of a sum of the inputs, there wouldn’t be much of a point working in a team at all. It’s the fact you’re getting this extra benefit from working with someone else that makes it worth while working as a team to start with. Is that right? So we can think of this term has to do with complementarity, or synergy, a very unpopular word these days but still: synergy. So we’re going to assume that when you work together there are some synergies. Some of you are good at some parts of the homework, some of them are good at other parts of the homework. And so in this law firm, one of these guys is an expert on intellectual property and the other one on fraud or something.
So I’ve got the agents. I’ve got the strategy set. I know something about the profits of the firm. I need to tell you about their payoffs.
So the payoffs: the payoff for Player I is going to depend, of course, on her choice and on the choice of her partner, and it’s going to equal a ½–because they’re splitting the profits–so a ½ of the profits. So a ½ of 4 times S1 plus S2 plus B S1S2. She gets half of those profits but it also costs her S1 squared. So S1 squared is her effort costs, it’s her input costs. This is the effort cost. Similarly, Player II–everything’s symmetric here–Player II’s payoff is the same thing. This term is the same except we’re going to subtract off Player II’s efforts squared: S2 squared.
So you get the profits of the firm minus the disutility of having missed all that sleep. There’s a guy in about the fifth row there who’s missed too much sleep, so somebody just nudge him. That’s it, good. We won’t put him on camera just nudge him. That’s it, good. There you go. Next time we’ll use the camera for that. So now we have everything we want to analyze this firm and to analyze how things are going to work, either when you’re working on your homework assignments or in the law partnership. Again, just to make this relevant to you, I mean this is very stylized of course, but a huge number of businesses out there are partnerships and do have this kind of profit sharing rule and do have synergies. So this is a relevant issue in a lot of businesses.
Now we’re going to analyze this–no secret here–we’re going to analyze this using the idea of best response. That’s not a surprise to any of you since that’s where we started the day. So, in particular, I want to figure out what is Player I’s best response to each possible choice of Player II? What is Player I’s best response for each possible choice S2 of Player II? How should I go about doing that? How should I do that? So here what we did before was we drew these graphs with probabilities, with beliefs of Player I and the problem here is, previously we had a nice simple graph to draw because there were just two strategies for Player II.
Player II was a goalie, he could dive to the left or the right. Problem is here, that Player II has a continuum of strategies and trying to draw all possible probabilities over an infinite number of objects on the board is more than my drawing can do. Too hard. So we need some other technique. How are we going to find out Player I’s best response? Somebody? Wave your hands in the air, way back in the corner. Can I, can we, let’s get the mike. Stand up but wait for the mike to come to you. How are we going to do it? How are we going to figure out what Player I’s best response is? Shout loudly.
Professor Ben Polak: Good, okay. That’s certainly the first step. We’ve got that, actually we’ve got that. So here’s Player I’s payoff as a function of what Player II chooses and what Player I chooses, so we have that already. We have Player I’s payoff as a function of the two efforts and now I want to find out what is Player I’s best efforts given a particular choice of S2? Yeah.
Student: Take a derivative of S1.
Professor Ben Polak: Good, take a derivative and–
Student: Set it equal to zero.
Professor Ben Polak: Okay, good. So we’re going to use calculus. We’re going to use calculus of one variable. We’re just changing one variable, S1. How many of you–we won’t let the camera see you–how many of you have not–I’m not going to show hands at all. If you have not seen the calculus that I’m about to use on the board, or more likely, if you’ve forgotten it since high school, don’t panic.
There is a chapter in the back of the book, I think it’s chapter 25, that goes over this, it refreshes your memory of such calculus. And if you haven’t, if you’ve never seen it before, if you haven’t taken, for example, the equivalent of Math 112, come and see us. We’ll probably try and line up a quick calculus lesson, a special section for those people. So if what I’m going to do now is scary, come and see us and we’ll deal with it.
All right, what I’m going to do, we want to take a derivative of this thing. What we’re going to do is, we’re asking the question, what is the maximum, choosing S1, of this profit. Can I multiply the ½ by the 4 just to save myself some time? So the profit is 2 S1 plus S2 plus B S1 S2 minus S1 squared. We’re asking the question, taking S2 as given, what S1maximizes this expression and as the gentleman at the back said, I’m going to differentiate and then I’m going to set the thing equal to 0. So I’m almost bound to get this wrong on the board. So can you all watch me like a hawk a second?
So if we differentiate this object, I’m going to find a first order condition in a second. All right, so we differentiate. I’m going to have 2 still, and then this S1 is going to become a 1, and this S1 here is going to become a plus B S2, everyone happy with that? This S1 squared is going to become a minus 2S1. That was just differentiating. Everyone happy with the way I differentiated? Is this coming back from high school? The cogs are spinning now? To make this a first order condition, I’m going to say “at the best response,” put a hat over the 1. At the best response this is equal to 0. Yeah Tae, can you get the guy again.
Student: Wouldn’t that be 2, oh sorry, never mind.
Professor Ben Polak: Okay, you’re right to shout out because I’m very–I mean doing it on the board I’m very likely to make mistakes, but okay. So I differentiated this object, this is my first derivative and I set it equal to 0. Now in a second I’m going to work with that, but I want to make sure I’m going to find a maximum and not a minimum, so how do I make sure I’m finding a maximum and not a minimum? I take a look at the second derivative, which is the second order condition. So I’m going to differentiate this object again with respect to S1,. Pretend the hat isn’t there a second. And none of this has an S1 in it, so that all goes away. And I’m going to get minus 2, which came from here: minus 2 and that is in fact negative, which is what I wanted to know.
To find a maximum I want the second derivative to be negative. So here it is, I’ve got my first order condition. It tells me that the best response to S2 is the Ŝ1 that solves this equation, that solves this first order condition. We can just rewrite that, if I divide through by 2 and rearrange, it’s going to tell me that Ŝ1, or if you like, Ŝ1 is equal to 1 plus B S2. So this thing is equal to Player I’s best response given S2. Now I could go through again and do exactly the same thing for Player II, but I’m not going to do that because everything’s symmetric. So everyone happy with that?
So I could at the same–I could do the same kind of analysis but we know I’ll get the same answer. So similarly, I would find that Ŝ2 equals 1 plus B S1 and this is the best response of Player II, as it depends on Player I’s choice of effort S1. Okay, now I found out what Player I’s best response is to Player II, and what Player II’s best response is to Player I for each possible choice of Player II up here, and for each possible choice of Player I down there. Now, let’s see if we can get a bit further. And to get a bit further, let’s draw a picture.
What I’m going to do is I want to draw the two functions we just found and see what they look like. This is all in your notes already, so I can get rid of it. What I can do with is some more chalk. Excuse me. So what I’m going to do is, let’s draw a picture that has S1 on the horizontal axis and S2 on the vertical axis. And there are different choices here 1, 2, 3, and 4 for Player I, and here’s the 45º line. If I’m careful I should get this right 1, 2, 3, and 4 are the possible choices for Player I. Now before I draw it I better decide what B is going to be.
Okay, so I’m going to draw for the case–I’m going to draw the best response of Player I and I’m going to draw the best response for Player II in a minute, for the case B equals 1/4. So we said B was somewhere between 0 and 1/4, let’s draw the case for B equals 1/4. So the expression I want to draw first of all is the best response of Player I as a function of S2and we agreed that that was given by 1 plus 1/4 now, 1 plus 1/4 S2. So for each possible choice of S2, I’m going to draw Player I’s best response and we’ll do it in red. So if Player II chooses 0, what is Player I’s best response? Somebody shout out.
Professor Ben Polak: 1, okay. So 1 plus 1/4 of 0 is 1, so if Player II chooses 0, Player I’s best response is to choose 1. What if Player II chooses 4? If Player II chooses 4, what would be Player I’s best response? So it’ll be 1 plus 1/4 times 4, 1/4 times 4 is 1, so 1 plus 1 is 2, so Player II’s best response in that case will be 2. So if Player I chooses 4, Player II should choose, I’m sorry, Player II chooses 4, Player I should choose 2, and this is a straight line in between. So the line I’ve just drawn is the best response for Player I as it depends on Player II’s choice. Everyone happy with how I drew that? I’m assuming you’re taking it on faith that it is a straight line in between, but it is.
The way we read this graph, is you give me an S2, I read across to the pink line and drop down, and that tells me the best response for Player I. Now we can do the same for Player II, we can draw Player II’s best response as it depends on the choices of Player I, but rather than go through any math, I already know what that line’s going to look like. What does that line look like? Somebody raise your hand. Somebody? What will Player II’s best response look like as a function of Player I’s choice in the same picture? Someone we haven’t had before, we’ve had all these guys before, someone else. Yeah, there’s a guy in the middle, can we get in to him? Yeah, maybe it’s easier from that side. Shout out loud so the mike can hear you.
Student: It should be a reflection across the 45° line.
Professor Ben Polak: Right, exactly. So if I drew the equivalent line for Player II, which is Player II’s best response for each choice of Player I, we’re simply flipping the identities of the players, which means we’ll be reflecting everything in that 45º line. So it’ll go from 1 here to 2 here, and it’ll look like this. So this the best response for Player II for every possible choice of Player I, and just to make sure we understand it, what this blue line tells me is you give me an S1, an effort level of Player I, I read up to the blue line and go across and that tells me Player II’s best response. Okay, now we’re making some progress. What do we notice?
Remember we said that one of the lessons of today’s class, the second lesson. The first lesson was don’t shoot towards the middle of the goal and the second more general lesson was what? It was don’t ever play a strategy that is not a best response to everything. I admit I’m cheating a little bit here because I’m ignoring beliefs, but trust me that’s okay in this game. So are there strategies here that are never a best response to anything? Put another way, what strategies of Player I’s are ever a best response? Anybody? Well, let’s have a look.
If Player II chooses 0 then Player I’s best response is 1, and that’s as low as he ever goes. So these strategies down here less than 1 are never a best response for Player I. If Player II chooses 4, then the synergy leads Player I to raise his best response all the way up to 2, but these strategies up here above 2 are never a best response for Player I. Is that right? So the strategies below 1 and above 2 are never a best response for Player I. Similarly, for Player II, the lowest Player I could ever do, is choose 0, in which case Player II would want to choose 1, so the strategies below 1 are never a best response for Player II. And the strategies above 2 were never a best response for Player II.
So let’s actually–you might want to be a little bit gentle in your own notebook–but on my board let’s get rid of all these strategies that are never a best response. So all of these strategies for Player I are gone, and all of these strategies for Player I are gone. You might want to not scribble quite so much on your own notebook, but still. And all these strategies for Player II are gone, and all these strategies for Player II are gone, and what’s left? A lot of scribble is left. What’s left? So I claim if you look carefully there’s a little box in here that’s still alive. I’ve deleted all the strategies that were best–that are never best responses for Player I and all the strategies that are never best responses for Player II, and what I’ve got left is that little box.
But I can’t see that little box, so what I’m going to do is I’m going to redraw that little box. So let’s redraw it. So it goes from 1 to 2 this time. I’m just going to blow up that box. So this now is 1, 1 and up here is 2, 2 and let’s put in numbers of quarters, so this will be, what will it be? It’ll be 5/4, 6/4 and 7/4, and over here it’ll be 5/4, and 6/4, and 7/4. And let’s just draw how those, that pink and blue line look in that box. This is just a picture of that little box, so it’s going to turn out it goes from–the pink line goes from here to here and the blue line goes from here to here. We can work it out at home and check it carefully, but this isn’t that incorrect.
So what I’ve done is I’ve redrawn the picture we just had and blown it up. And have any of you seen that picture before? Anyone here seen that picture before? That’s the picture we just had, except I’ve changed the numbers a bit. Once I deleted all the strategies that were never a best response and just focused on that little box of strategies that survived, the picture looked exactly the same as it did before, albeit it blown up and the numbers changed. So what have we done so far? We said players should never play a strategy that’s never a best response to anything, so we threw those away. Now what’s left? What should we do now?
So some of the strategies that we didn’t throw away were best responses to things, but the things they were best responses to have now been thrown away. Is that right? This should be something familiar from when we were deleting dominated strategies. The strategies I’m about to throw away now, they’re not–it isn’t that they’re not best responses, they are best responses to something. But the things they were best responses to, we know are not going to be played, because they themselves were not best responses to anything. So what strategies do I have in mind? What strategies am I about to throw away?
Well, for example, for Player I we know now that Player II is never going to choose any strategy below 1, and so the lowest Player II will ever choose is 1, and it turns out that the lowest Player II would ever do in response to anything is 1 and above, never leads Player I to choose a strategy less than 5/4. The highest Player II ever chooses is 2, and the highest response that Player I ever makes to any strategy 2 or less is 6/4, so all these things bigger than 6/4 can go. Let’s be careful here. These strategies I’m about to delete, it isn’t that they’re never best responses, they were best responses to things, but the things they were best responses to, are things that are never going to be played, so they’re irrelevant.
So we’re throwing away all of the strategies less than 5/4 for Player I and bigger than 6/4 for Player I, (which is 1½ for Player I) and similarly for Player II. And if I did this–and again, don’t scribble too much in your notes–but if we just make it clear what’s going on here, I’m actually going to delete these strategies since they’re never going to be played–I end up with a little box again.
So everyone see what I did? I started with a game. I found out what Player I’s best response was for every possible choice of Player II, and I found out what Player II’s best response was for every possible strategy of Player I. I threw away all strategies that were never a best response, then I looked at the strategies that were left. I said those strategies that were a best response to things that have now been thrown away, but not best response otherwise, I can throw those away too. And when I threw those away, I was left once again with a little box, and I could do it again, and again, and again.
If I go on doing this exercise again, and again, and again what am I going to end up with? Shout it out, what am I going to end up with? The intersection, right? If I keep on constructing these boxes within boxes, so the next box would be a little box in here. I’m not going to draw it, but it’s something like this. But if you keep on doing boxes within boxes, I’m going to converge in on that intersection. So if we know people are not going to play a best response–that’s never a best response, and we know if we people are not going to play something which is never a best response, and we know people are not going to play which is not a best response, which is not a best etc., etc., etc. We’re going to converge in, in this game, to just one strategy for each player, which is where they intersect.
So what we’re going to converge in on to, is the S1*, let’s call it in this case, is equal to 1 plus B S2* and that S2* is equal to 1 plus B S1*. Actually, we can do it a little better than that, since we know the game is symmetric, we know that S1* is actually equal to S2*. So taking advantage of the fact that we know S1 is equal to S2 (because we’re lying on the 45º line), I can simplify things by making S1* equal to S2*. So now I’ve got–actually, that looks like three equations, it’s really just two equations, because one of them implies the other. And I can solve them, and if I solve them out I’m going to get something like (let me just be careful) I’m going to get something like: 1 minus B S1* is equal to 1, or S1* equals S2* is equal to 1 over (1–B). And again, anytime I’m doing algebra on the board, someone should check me at home, so just have a quick look at that. Is that right? I think that’s right.
My algebra, which is often wrong, suggests that the solution is S1* equals S2* equals 1 over (1–B). But what I’m doing, it’s just math, there’s nothing interesting going on. I’m just trying to solve out for the equation of this point. So what did we learn here? We learned that in this game deleting strategies that are never best response, and then deleting strategies that are never best response to anything that is a best response and so on and so forth, yielded a single strategy for each player. Just one strategy for each player and that strategy was given by this equation.
So if we were management consultants working for Mckinsey or something, and we were brought in to advise you on your homework assignments, or this law partnership on their work practices, we would come down with a prediction that this is how much work you’re going to get. Question, is this amount of work a good amount of work or a bad amount of work? Here you are, you’re working for Mckinsey, you’ve been hired by Joe Smith and Ann Blogs to figure out their strategy, working on a problem in a team on working on my homework assignments. You figured out how much work they’re going to contribute. Is this a good amount of work? Are they contributing too much, too little?
Because the answer is, depending on, compared to what? So let me rephrase, are these people, are these pair of partners in the firm, or two students working on their homework assignments, are they working more or less than an efficient level? Let’s have a poll, who thinks more? Turn the camera out into the audience, let’s have a look. Who thinks more? Who thinks they’re working just right? Who thinks less? A lot of abstentions here. I think they’re working too little here compared to what’s efficient. I’ll get you to solve it out on a homework assignment, so you can actually prove that.
You can prove that, in fact, if you were writing a contract, if there was a social planner you’d work more. But let’s try and get to grips why. Why is it that when we see these law partners, or medical partners, or whatever it happens to be, or students together on a homework assignment, why is it we tend to get inefficiently little effort when we start figuring out the strategy and working through the game? I’m conceding the answer. I’m telling you they’re going to work too little. Why do they end up not working hard enough? Any takers? Can we get a mike in here, yeah.
Student: Because if they work any harder than that, then the other person is just going to slack off instead.
Professor Ben Polak: All right, so there’s something about that, there’s something. On the other hand, this isn’t really, I mean the intuition you’re giving me is kind of a Prisoner’s Dilemma intuition. Saying I’m going to let the other guy work and I’ll shirk. But there’s something, I think there’s something in that, but there’s a little bit more going on here, what more is going on? I think that’s a good first step. There is something of that. Yeah.
Student: If there are two people working together, there’s about half as much work for each person to do.
Professor Ben Polak: That’s true, but that would suggest it doesn’t matter if they slack off. What’s going on here, so go back to your Economics 115 or 150, if you took either of those courses. What’s the problem here? What’s underlying the problem? Let’s get this guy in the pink down here.
Student: They only capture fifty percent of their marginal benefit.
Professor Ben Polak: That’s the point. Good, well your name is?
Professor Ben Polak: So Patrick is giving, I think, it’s the correct answer here. The problem here isn’t really about the amount of work. It isn’t even, by the way, about the synergy. You might think it’s because of this synergy that they don’t take into account correctly. That isn’t the problem here. It turns out even without the synergy this problem would be there. The problem is what Patrick said. The problem is that at the margin, I, a worker in this firm, be it a law partnership or a homework solving group, I put in, I bear the cost, at the margin, I bear the full cost at the margin for any extra unit of effort I put in, but I only reap half the benefits. At the margin, I’m reaping, I’m bearing the cost for the extra unit of effort I contribute, but I’m only reaping half of the induced profits of the firm, because of profit sharing.
That leads all of us to put in too little effort. What’s the general term that captures all such situations in Economics? It’s an “externality.” It’s an externality. There’s an externality here. When I’m figuring out how much effort to contribute to this firm I don’t take into account that other half of profits that goes to you. So this isn’t to do with the synergy. It isn’t to do with something complicated. It’s something you knew back in 115. If you have profit sharing in a firm or profit sharing in homework assignment, or any joint projects, you have to worry about too little effort being contributed because there’s an externality. My efforts benefits you not just me.
While we’ve got this on the board, let’s just think a little bit more. What would happen if we changed the degree of the synergy? What would happen if we lowered B? So B is the degree to which the synergy across these workers, if we lowered B, what would happen to our picture? Let me redraw a picture unscribbled. We had a picture that looked something like this. This was S1 and this was S2. If we lowered the degree of synergy, what would happen to the effort level that we’d find by this method? What would happen? What would happen to the picture? Anybody, again this is a 115 kind of exercise, we’re going to be moving lines around. Yeah, Henry isn’t it? So let’s get a mike in to Henry.
Student: The lines will get shallower and eventually become horizontal and vertical respectively.
Professor Ben Polak: All right, good. So the pink line is actually going to get steeper, but I know what you mean. So the pink line is going to move towards the vertical, and the blue line is going to move towards the horizontal, and notice that the amount of effort that we generate, goes down dramatically, goes down in this direction. So if we lower the synergy here, not only do I contribute less effort, but you know I contribute less effort, and therefore you contribute less effort and so on. So we get this scissors effect of looking at it this way. We could draw other lessons from this, but let me try and move on a little bit.
Chapter 5. Nash Equilibrium: Preview [01:07:23]
We decided in this game to solve it by looking at best responses, deleting things that were never a best response, looking again, deleting things that were never a best response, and so on and so forth, and luckily, in this particular game, things converged and they converged to the points where the pink and the blue line crossed. What do we call that point? What do we call the point where the pink and the blue line cross? That’s an important idea for this class. That’s going to turn out to be what’s called a Nash Equilibrium.
So we know what it’s called. How many of you have heard the term Nash Equilibrium before? How many of you saw the movie about Nash? We’ll come back and talk about that a bit next time. So this is a Nash Equilibrium, but okay we know what it is in jargon, and we know, we kind of knew that was going to be an important point, because most of you have taken Economics courses before and you know that whenever lines cross in Economics it’s important, right? But what does it mean here? Why is it–what’s going on at that line? What does it tell us that the pink and the blue line cross? What makes that point special? What does it mean to say the pink and the blue line cross? Can I get the guy way back, like three rows behind you in purple? Shout out again.
Student: It means that neither player has an incentive to deviate from that point.
Professor Ben Polak: All right, well that’s correct. So let’s try and read that through. So I don’t know your name, your name is?
Professor Ben Polak: So Allen is saying if Player I is choosing this strategy and Player II is choosing her corresponding strategy here, neither player has an incentive to deviate. Another way of saying it is: neither player wants to move away. So if Player I chooses S1*, Player II will want to choose S2* since that’s her best response. If Player II is playing S2*, Player I will want to play S1* since that’s his best response. Neither has any incentive to move away. So more succinctly, Player I and Player II, at this point where the lines cross, Player I and Player II are playing a best response to each other. The players are playing a best response to each other. So clearly in this game, it’s where the lines cross.
Let’s go back to the game we played with the numbers. Everyone had to choose a number and the winner was going to the person who was closest to 2/3 of the average. (By the way, the winner’s never picked up their winnings for that, so they still can.) So in that game, what’s the Nash Equilibrium in that game? Everyone choosing 1. How do we know that’s the Nash Equilibrium of that game? How do we know that everyone choosing 1 is the Nash Equilibrium in the game where you all chose numbers? Well let’s just use the definition. If everybody chose a 1, the average in the class would be 1, 2/3 of that would be 2/3 and you can’t go down below 1, so everyone’s best response would be to choose 1.
Say it again: if everyone chose 1, then everyone’s best response would be to choose 1, so that would be a Nash Equilibrium. Did people play Nash Equilibrium when we played that game? No, they didn’t. Not initially at any rate, but notice as we played the game repeatedly, what happens? As we played the game repeatedly, we noticed that play seemed to converge down towards 1. Is that right? In this game, when we analyzed the game repeatedly, it seemed like our analysis converged towards the equilibrium. Now that’s not always going to happen but it’s kind of a nice feature about Nash Equilibrium. Sometimes play tends to converge there. Nash Equilibrium’s going to be a huge idea from now to the mid-term exam and we’re going to pick it up and see more examples on Wednesday.
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