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# ECON 159: Game Theory

## Lecture 4

## - Best Responses in Soccer and Business Partnerships

### Overview

We continue the idea (from last time) of playing a best response to what we believe others will do. More particularly, we develop the idea that you should not play a strategy that is not a best response for any belief about others’ choices. We use this idea to analyze taking a penalty kick in soccer. Then we use it to analyze a profit-sharing partnership. Toward the end, we introduce a new notion: Nash Equilibrium.

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html## Game Theory## ECON 159 - Lecture 4 - Best Responses in Soccer and Business Partnerships## Chapter 1. Best Response: Penalty Kicks in Soccer [00:00:00]
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 u 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.
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.
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?
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, Ŝ So strategy Ŝ 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, Ŝ Once again, we can write it the other way, or Ŝ
## Chapter 4. Externalities and Inefficient Outcomes: The Partnership Game [00:29:59]
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 S When you’re working on your homework assignments, if your product, the thing you hand in was just S 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 S 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 S 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.
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 S 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 S
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 S 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 Ŝ 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 S 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 S
The way we read this graph, is you give me an S
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 S My algebra, which is often wrong, suggests that the solution is S 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.
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 S
## 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 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.
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. [end of transcript] Back to Top |
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