Game Theory with Ben Polak: Five First Lessons (Transcript)

But there’s a catch. The catch is that that means you have to sign that legal release form. So you’ll see that you have in front of you a legal release form, you have to sign it, and what that says is that we can use you being shown in class. Think of this as a bad hair day release form.

You can’t sue Yale later if you had a bad hair day. For those of you who are on the run from the FBI, your visas run out, or you’re sitting next to your ex-girlfriend, now would be a good time to put a paper bag over your head.

Now, just to get you used to the idea, in every class we’re going to have I think the same two people. So Jude is the cameraman. Why don’t you all wave to Jude? This is Jude. And Wes is our audio guy. This is Wes. And I will try and remember not to include Jude and Wes in the classroom discussions, but you should be aware that they’re there. If this is making you nervous, if it’s any consolation, it’s making me very nervous.

So we’ll try and make this class work as smoothly as we can, allowing for this extra thing. Let me just say, no one’s making any money off this, at least I’m hoping these guys are being paid, but me and the TAs are not being paid. The aim of this is I think a good aim, it’s an educational project and I’m hoping you’ll help us with it. The one difference it is going to mean is that at times I might hold some of the discussions for the class coming down into this part of the room here to make it a little easier for Jude.

Admin: Grades and Problem Sets

BEN POLAK: How are we doing now on filling out those forms? Has everyone filled in their strategy for the first game? Not yet. Okay, so let’s go on doing a bit more admin. The thing you mostly care about, I’m guessing, is the grades. So how is the grade going to work for this class?

Thirty percent of the class will be on problem sets, thirty percent of the grade, thirty percent on the midterm, and forty percent on the final, so thirty, thirty, forty. The midterm will be held in class on October 17th, that is also on your syllabus. Please don’t anybody tell me any time after today you didn’t know when the midterm was and there before you it clashes with 17 different things. The midterm is on October 17th, which is a Wednesday, in class.

The problem sets, there will be roughly ten problem sets and I’ll talk about them more later on when I hand them out. The first one will go out on Monday, but it will be due ten days later. Roughly speaking, they’ll be every week.

The grade distribution, so this is the rough grade distribution. Roughly speaking, a sixth of the class are going to end up with As, a sixth is going to end up with A-, a sixth is going to end up with B+, a sixth is going to end up with B, a sixth is going to end up with B-, and the remaining sixth, if I’ve added that up right, are going to end up with what I guess we’re now calling the Presidential Great. Is that right?

That’s not literally true. I’m going to squeeze it a bit, I’m going to curve it a bit. So actually slightly fewer than a sixth will get straight As, and fewer than a sixth will get a season below, we’ll squeeze the middle to make them be more Bs. One thing I can guarantee from past experience in this class is that the median grade will be a B+. The median will fall somewhere under B+.

Now just as forewarning for people who have forgotten what a median is, that means half of you, not approximately half, I mean exactly half of you, will be getting something like B+, and below, and half will be getting something like B+, and above.

Admin: Textbooks

BEN POLAK: Now how are you doing in filling in the forms? Everyone filled in yet? Surely must be pretty close to getting everyone filled in. So last things to talk about before I actually collect them in, textbooks.

There are textbooks for this class. The main textbook is this one, Dutta’s book Strategy and Games. If you want a slightly tougher book, more rigorous book, try Joel Watson’s book Strategies. Both of those books are available at the bookstore. But I want to warn everybody ahead of time, I will not be following the textbook.

I regard these books as safety nets. If you don’t understand something that happened in class, you want to reinforce an idea that came up in class, then you should read the relevant chapters in the book and the syllabus will tell you which chapters to read for each class, or for each week of the class. But I will not be following these books religiously at all. In fact, they’re just there as backup.

In addition, I strongly recommend people read Thinking Strategically. This is a good bedtime reading. Do any of you suffer from insomnia? It’s very good bedtime reading if you suffer from insomnia. It’s a good book and what’s more, there’s going to be a new edition of this book this year and Norton have allowed us to get advanced copies of it. So if you don’t buy this book this week, I may be able to make the advanced copy of the new edition available for some of you next week. I’m not taking a cut of that either, there’s no money changing hands there.

Sections on the syllabus, sorry on the website, sign up as usual. Put yourself down on the wait list. If you don’t get in the section you want, you probably will get into the section you want once we’re done.

Game 1: A Simple Grade Scheme

BEN POLAK: All right, now we must be done with the forms. We done with the forms? All right, so why don’t we send the TAs with or without mics up and down the aisles and collect in your game number one. Not game number two, just game number one.

Just while we’re doing that, I think the reputation of this class, I think, if you look at the course evaluations online or whatever, is that this class is reasonably hard but reasonably fun. So I’m hoping that’s what the reputation of the class is. So you shouldn’t, if you think this class is going to be easy, I think it isn’t actually an easy class. It’s actually about a hard class, but I think I can guarantee it’s going to be a fun class.

Now one reason it’s a fun class is the nice thing about teaching game theory, quieten down folks, the nice thing about teaching game theory is you get to play games. That’s exactly what we’ve just been doing now. This is our first game and we’re going to play games throughout the course, sometimes several times a week, sometimes just once a week.

We’ve got all these things in, everyone handed them in? So I need to get those counted. Has anyone taken the Yale accounting class? No one has aspirations to be at one person has? I’ll have a TA do it. It’s all right, I’ll have a TA do it. So Kai, can you count those for me? Is that right?

Let me read out the game you’ve just played.

Game 1: A simple grade scheme for the class.

Read the following carefully without showing your neighbor what you are doing, put in the box below either the letter α or the letter β. Think of this as a grade bid. I will randomly pair your form with another form and neither you nor your pair will ever know with whom you were paired. Here’s how the grades may be assigned for the class. Well they won’t be but we can pretend.

If you put α and your pair puts β, then you will get an A and your pair a C. If you and your pair both put α you’ll both get B-, if you put β in your pair which is α, you’ll get a C in your pair and an A, and if you and your pair both put β, then you’ll both get B+.

BEN POLAK: So that’s the thing you just filled in. Now before we talk about this, let’s just collect this information in a more useful way. So I’m going to remove this for now.

The Grade Game Matrix

BEN POLAK: We’ll discuss this in a second, but why don’t we actually record what the game is that we’re playing first. So this is our grade game and what I’m going to do, since it’s kind of hard to absorb all the information, just by reading a paragraph of text, I’m going to make a table to record the information.

So what I’m going to do is I’m going to put me here and my pair, the person I’m randomly paired with here, and α and β, which are the choices I’m going to make here, and on the columns α and β the choices my pair is making. In this table I’m going to put my grades. So my grades, if we both put α is B-, if we both put β was B+, if I put α and she put a β I got an A, and if I put β and she put an α, I got a C.

Is that correct? That’s more or less right? Yeah? Okay, well we’re here. Why don’t we do the same for my pair?

So this is my grades and the left hand table, but now let’s look at what my pair will do, what my pair will get. So I should warn people sitting at the back that my handwriting is pretty bad. That’s one reason for moving forward. The other thing I should apologize for at this stage of the class is my accent. I will try and improve the handwriting. There’s not much I can do about the accent at this stage.

So once again, if we both put α then my pair gets a B-, if we both put β then we both get a B+, and in particular my pair gets a B+. If I put α and my pair puts β then she gets a C, and if I put β and she puts α, then she gets an A. So I now have all the information that was on the sheet of paper that you just handed in.

Now there’s another way of organizing this that’s standard in game theory, so we might as well get used to it now on the first day. Rather than drawing two different tables like this, what I’m going to do is I’m going to take the second table and superimpose it on top of the first table. So let me do that and you’ll see what I mean.

What I’m going to do is draw a larger table, the same basic structure. I’m choosing α and β on the rows. My pair is choosing α and β on the columns, but now I’m going to put both grades in. So the easy ones are on the diagonal. We both get B- if we both choose α, we both get B+, if we both choose β. But if I choose α and my pair chooses β, I get an A and she gets a C. And if I choose β and she chooses α, then it’s me who gets the C and it’s her who gets the A.

So notice what I did here, the first grade corresponds to the row player, me in this case, and the second grade in each box corresponds to the column player, my pair in this case. So this is a nice succinct way of recording what was in the previous two tables. And this is an outcome matrix. This tells us everything that was in the game.

Class Vote: Alpha vs. Beta

BEN POLAK: So now seems a good time to start talking about what people did. So let’s just have a show of hands. How many of you chose α? Leave your hands up so that Jude can catch that, so people can see at home. How many of you chose beta? Wave your hands to betas.

There’s a beta here. It looks like a lot of, well we’re going to find out, we’re going to count, but a lot more alphas than betas. Let me try and find out some reasons why people chose. So let me have the alphas up again. So the woman who’s in red here, can we get a mic to the, yeah, okay we asked you, you’re not on the run from the FBI, we can ask you why.

So you chose alpha, right? So why did you choose alpha? So you wrote out these squares, you realized what your partner was going to do and responded to that. Any other reasons for choosing α around the room? Can we get the woman in here?

Try not to be intimidated by these microphones. They’re just mics. It’s okay. Yeah?

STUDENT: I think if I chose alpha, regardless of what my partner chose, I think there would be better outcomes than choosing alpha.

BEN POLAK: So let me see names a second. So your name was?

STUDENT: Courtney.

BEN POLAK: And your name was?

STUDENT: Clarice.

Reasons for Choosing Alpha

BEN POLAK: So slightly different reasons, same choice, alpha. Clarice’s reason, she said no matter what the other person does, she reckons she’d get a better grade if she chose alpha. So hold that thought a second. We’ll come back to, is it Clarice? Is that right? We’ll come back to Clarice in a second.

Let’s talk to the betas a second. Let me just emphasize at this stage there are no wrong answers. Later on in the class there’ll be some questions which have wrong answers. Right now there’s no wrong answers. There may be bad reasons but there’s no wrong answers.

Reasons for Choosing Beta

BEN POLAK: So let’s have the betas up again. Let’s see the betas. Oh come on. There was a beta right here. You were a beta right? You backed off the beta. Okay, so how can I get a mic into a beta? Let’s stick in this aisle a bit. Is that a beta right there? Are you a beta right there? Can I get the beta in here? Who has the beta in here? Can we get the mic in there? Is that possible? In here? Can you leave your hand up so that we can, there we go. Just point towards it, that’s fine. That’s fine, just speak into it, that’s fine.

STUDENT: I personally don’t like swings that much and it’s the B-B range, so I’d much rather prefer that to a swing from A to C. So that’s my reason.

BEN POLAK: So you’re saying it compresses the range. I’m not sure it does compress the range. I mean, if you chose alpha you’re swinging from A to B-, if you’re swinging from B you’re choosing from B-C. I mean, there’s similar kind of ranges but it’s certainly reasons. Other reasons for choosing? Yeah, the guy in blue here. Don’t hold the mic, just let it point at you.

STUDENT: Well I guess I thought we could be more collusive and kind of work together but I guess not. So I chose beta.

BEN POLAK: There’s a siren in the background so I missed the answer. Stand up and say it to me just here. Say it again.

STUDENT 1: Sure, my name’s Travis. I thought we could work together but I guess not. If you had chosen beta we would all have gotten B pluses but I guess not.

BEN POLAK: So Travis is giving us a different reason. He’s saying that maybe some of you in the room might actually care about each other’s grades. I mean you all know each other in class, you all go to the same college.

For example, if we played this game up in the business school, are there any MBA students here today? One or two? We played this game up in the business school. I think it’s quite likely we’re going to get a lot of alphas chosen, but if we played this game up in, let’s say, the Divinity School, and I’m guessing that Travis’s answer is reflecting what you guys are posing here. If you played in the Divinity School you might think that people in Divinity School might care about other people’s grades.

There might be ethical reasons, perfectly good, sensible ethical reasons for choosing beta in this game. There might be other reasons as well, but that’s perhaps the reason to focus on.

The Missing Element: Payoffs

BEN POLAK: And perhaps the lesson I want to draw out of this is that right now this is not a game. Right now we have actions, strategies for people to take and we know what the outcomes are, but we’re missing something that will make this a game. What are we missing here? We’re missing objectives, we’re missing payoffs, we’re missing what people care about. So we can’t really start analyzing a game until we know what people care about, until we know what the payoffs are.

Let’s just say something now which I’ll probably forget to say at any other moment of the class, but today it’s relevant. Game theory, me, professors at Yale cannot tell you what your payoffs should be. I can’t tell you in a useful way what it is that your goals in life should be or whatever. That’s not what game theory is about. However, once we know what your payoffs are, once we know what your goals are, perhaps we can, game theory can help you get there.

So we’ve had two different kind of payoffs mentioned here. We’ve had the kind of payoff where we care about our own grade, and Travis has mentioned the kind of payoff where you might care about other people’s grades. And what we’re going to do today is analyze this game under both those possible payoffs. So to start that off, let’s put up some possible payoffs for the game. And I promise we’ll come back and look at some other payoffs later. We’ll revisit the Divinity School later.

Introducing the Payoff Matrix

BEN POLAK: So here once again is our same matrix with me and my pair, choosing actions α and β, but this time I’m going to put numbers in here. Some of you will perhaps recognize these numbers but that’s not really relevant for now.

So what’s the idea here? Well the first idea is that these numbers represent utiles or utilizes. They represent what these people are trying to maximize, what they’re trying to achieve, their goals. The idea, just to compare this to the outcome matrix, is for the person who’s me here, AC yields a payoff of, so AC is this box, so AC yields a payoff of 3 whereas B- yields a payoff of 0 and so on.

So what’s the interpretation? This is the first interpretation, the natural interpretation that a lot of you jump to straight away. These are people, people with these payoffs, are people who only care about their own grades. They prefer an A to a B+, they prefer a B+ to a B-, and they prefer a B- to a C. I’m hoping I’ve got the grades in order, otherwise it’s going to ruin my curve at the end of the year.

So these people only care about their own grades. What do we call people who only care about their own grades? What’s a good technical term for them? In England, I think we’d refer to these guys, whether it’s technical or not, as evil gits. These are not perhaps the most moral people in the universe.

Dominant Strategy: What Should You Do?

BEN POLAK: So now we can ask a different question. Suppose, whether these are actual payoffs or not, pretend they are for now. Suppose these are all payoffs, now we can ask not what did you do, but what should you do? Now we have payoffs. I can really switch the question to a normative question, what should you do?

Let’s come back to Clarice. Where was Clarice before? Let’s get the mic on you again. So just explain what you did and why again.

STUDENT: Why I chose alpha?

BEN POLAK: Yes, stand up a second. You chose alpha. I’m assuming these were roughly your payoffs more or less, right? You were caring about your grades. Why did you choose alpha? Just repeat what you said before.

STUDENT: Because I thought the payoffs, the two different payoffs that I could have gotten were highest if I chose α.

BEN POLAK: So what Clarice is saying, and it’s an important idea, is this, and tell me if I’m paraphrasing you incorrectly, but I think this is more or less what you’re saying, is no matter what the other person does, no matter what the pair does, she obtains a higher payoff by choosing α.

Let’s just see that. If the pair chooses α and she chooses α then she gets 0. If the pair chooses α and she chose β she gets minus 1, 0 is bigger than minus 1. If the pair chooses beta, then if she chooses alpha she gets 3, beta she gets 1, and 3 is bigger than 1. So in both cases, no matter what the other person does, she receives a higher payoff from choosing alpha, so she should choose alpha.

Does everyone follow that line of reasoning? That’s a stronger line of reasoning than the reasoning we had earlier. So the one I’ve now immediately forgotten the name of in the red shirt, whose name was?

STUDENT: Courtney.

BEN POLAK: So Courtney also gave a reason for choosing α and it was a perfectly good reason for choosing α, nothing wrong with it, but notice that this reason is a stronger reason, it kind of implies your reason.

Lesson One: Do Not Play a Strictly Dominated Strategy

BEN POLAK: So let’s get some definitions down here. I think I can fit it in here, let’s try and fit it in here.

Definition: We say that my strategy alpha strictly dominates my strategy β if my payoff from alpha is strictly greater than that from β, regardless of what others do.

Now it’s by no means my main aim in this class to teach you jargon, but a few bits of jargon are going to be helpful in allowing the conversation to move forward, and this is certainly one. Evil gits is maybe one too, but this is certainly one.

Let’s draw out some lessons from this, so you can still read that and we bring down and clean this board. So the first lesson of the class, and there’s going to be lots of lessons, is a lesson that emerges immediately from the definition of a dominated strategy and it’s this.

Lesson one of the course is: Do not play a strictly dominated strategy.

Why? Someone want to tell me why? Yeah, Alan, you want to get this guy? Yeah, stand up. Yep, this is right. Yeah?

STUDENT: —

BEN POLAK: I’m not sure that’s the possible answer.

The Prisoner’s Dilemma

BEN POLAK: I’m looking for something more direct here. So we look at the definition of a strictly dominated strategy. I’m saying never play one. What’s a possible reason for that? Can we get the woman there?

AUDIENCE QUESTION: You always lose?

BEN POLAK: Well, I don’t know. It isn’t about winning and losing. What else could we have? Can we get this guy in the pink down here?

AUDIENCE QUESTION: Well the payoffs are lower.

BEN POLAK: The payoffs are lower. Okay, so here’s an abbreviated version of that, and that’s perhaps a little bit longer. The reason I don’t want to play a strictly dominated strategy is if instead I play the strategy that dominates it, I do better in every case. The reason I never want to play a strictly dominated strategy is if instead I play the strategy that dominates it, whatever anyone else does, I’m doing better than I would have done.

Now that’s a pretty convincing argument. That sounds like a convincing argument. It sounds like too obvious even to be worth stating in class. So let me now try and shake your faith a little bit in this answer. Somebody was wanted by the FBI.

So how about the following argument? Look at the payoff matrix again and suppose I reason as follows. Suppose I reason and say, if we, me and my pair, both reason this way and choose α, then we’ll both get 0. But if we both reasoned a different way and chose β, then we’ll both get 1. So I should choose β, 1 is bigger than 0, I should choose beta.

What’s wrong with that argument? That argument must be wrong because it goes against the lesson of the class and the lessons of the class are gospel, right? They’re not wrong ever. So what’s wrong with that argument? What’s wrong with that argument?

STUDENT 2: Well, because you have to be able to agree, you have to be able to speak to them, but we aren’t allowed to show our partners what we wrote.

BEN POLAK: So it involves some notion of agreeing. Yes, good. Good.

So certainly part of the problem here, with the reasoning I just gave you, the reasoning that said I should choose β because if we both reason the same way, we both do better that way, involves some kind of magical reasoning. It’s as if I’m arguing that if I reason this way and reason myself to choose β, somehow I’m going to make the rest of you reason the same way too. It’s like I’ve got ESP or I’m some character out of, what do you call it, the X-Men. Is that what it’s called, the X-Men?

Now in fact, this may come as a surprise to you, I don’t have ESP and I’m not a character out of the X-Men and so you can’t actually see brain waves emanating from my head and my reasoning doesn’t affect your reasoning.

So if I did reason that way and chose β, I’m not going to affect your choice one way or the other. That’s the first thing that’s wrong with that reasoning. What else is wrong with that reasoning? Yeah, that guy down here. Yeah?

STUDENT: I’m not going to.

BEN POLAK: Well how about this? Even if I could, even if I was that guy in the X-Men or the Matrix or whatever it was, who could reason his way into making people do things, even if I could make everyone in the room choose β by the force of my brainwaves, what should I then do? I should choose α. If these are my payoffs I should go ahead and choose α because that way I end up getting 3.

So there’s two things wrong with the argument. One, there’s this magical reasoning aspect, my reasoning is controlling your actions, that doesn’t happen in the real world, and two, even if that was the case, I’d do better to myself choose alpha.

So nevertheless, there’s an element of truth in what I just said. It’s the fact there’s an element of truth in it that makes it very, it seems like a good argument. The element of truth is this, it is true that by both choosing α, we both ended up with B-s, we both ended up with payoffs of 0 rather than payoffs of 1.

Lesson 2: Rational Choice Can Lead to Bad Outcomes

It is true that by both choosing, by both following this lesson and not choosing the dominated strategy beta, we ended up with payoffs 0, 0 that were bad. That’s probably the second lesson of the class. So lesson 2, and this lesson probably wouldn’t be worth stating if it wasn’t for sort of a century of thought and economics that said the opposite.

So rational choice, in this case people not choosing a dominated strategy, people choosing a dominant strategy, rational choice can lead to outcomes that are, what do Americans call this, that suck. If you want a more technical term for that, and you remember this from Economics 115, it can lead to outcomes that are inefficient, that are Pareto inefficient, but suck will do for today.

Rational choice by rational players can lead to bad outcomes. So this is a famous example for this reason. It’s a good illustration of this point. It’s a famous example. What’s the name of this example somebody?

AUDIENCE QUESTION: This is called Prisoner’s Dilemma.

BEN POLAK: How many of you have heard of the Prisoner’s Dilemma before? Most of you saw it in 115. Why is it called the Prisoner’s Dilemma? Yes, so the guy here in orange. It’s okay, it can just point at you, it’s fine.

STUDENT: I think it’s whether or not the prisoners cooperate in the sentence they have, and if they kind of write out the other person then they can have less, but if they both write out then they end up losing large scale.

BEN POLAK: So in the standard story you’ve got these two crooks, or two accused crooks, and they’re in separate cells and they’re being interviewed separately, kept apart, and they’re both told that if neither of them rats the other guy out, they’ll go to jail for say a year. If they both rat each other out, they’ll end up in jail for two years. But if you rat the other guy out and he doesn’t rat you out, then you’ll go home free and he’ll go to jail for five years.

And you put that all down, you pretty quickly see that regardless of whether the other guy rats you out or not, you’re better off ratting him out. Now if you haven’t ever seen that Prisoner’s Dilemma, you can see it pretty much every night on a show called Law and Order. How many of you have seen Law and Order? If you haven’t seen Law and Order, the way to see Law and Order is to go to a random TV set at a random time and turn on a random channel.

This happens in every single episode, so much so that if any of you actually, I mean this might not be true at Yale, if any of you, or the TV guys, if any of you know the guy who writes the plots for this, have him come to the class, or I guess see the video now, and we can get some better plot lines in there.

Real-World Examples of the Prisoner’s Dilemma

But of course, that’s not the only example. The grade game and this is not the only example. There are lots of examples of Prisoner’s Dilemmas out there. Let’s try and find some other ones. So how many of you have roommates in your college?

How many of you have roommates? Most of you have roommates. So I’m guessing now, but I won’t make you show your hands because it’s probably embarrassing, but what is the state of your dorm rooms, your shared dorm rooms at the end of the semester or the end of the school year.

So I’m just guessing, having been in a few of these things over the years, that by the end of the semester, or certainly by the end of the school year, the state of the average Yale dorm room is quite disgusting. Why is it disgusting?

It’s disgusting because people don’t tidy up. They don’t clean up those bits of pizza and bits of chewed bread and cheese, but why don’t they tidy up? Well let’s just work it out. What would you like to happen if you’re sharing a dorm room? You’d like to have the other guy tidy up.

The best thing for you is to have the other guy tidy up, and the worst thing for you is to tidy up for the other guy. But now work it out, it’s a prisoner’s dilemma. If the other guy doesn’t tidy up, you’re best off not tidying up either because the last thing you want is to be tidying up for the other guy. And if the other guy does tidy up, hey the room’s clean, who cares? So either way, you’re not going to tidy up and you end up with a typical Yale dorm room.

Am I being unfair? Are your dorm rooms all perfect? This may be a gender thing but we’re not going to go there. So there are lots of prisoner’s dilemmas out there. Any other examples?

AUDIENCE QUESTION: Okay, and divorce struggles.

BEN POLAK: Okay, you’re too young to be worrying about such things but never mind. Yeah, okay, that’s a good example because hiring lawyers and bringing in big guns. What about an economics example? What about firms who are competing in prices? Both firms have an incentive to undercut the other firm, driving down profits for both. The last thing you want is to have the other firm undercut you and that’s going to push prices down. That’s good for us, the consumers, but bad for the firm, bad for industry profit.

Remedies for the Prisoner’s Dilemma

What remedies do we see? We’ll come back to this later on in the class but let’s have a preview. So what remedies do we see in society for prisoners’ dilemmas? What kind of remedies do we see? Let me try and get the guy here and right in front of me.

STUDENT: Collusion.

BEN POLAK: Collusion. So firms could collude. So what prevents them from colluding? So one thing they could do, presumably, is they could write a contract to these firms. They could say, I won’t lower my prices if you don’t lower your prices, and they could put this contract in with the pricey lawyer who’s taking his day off from the divorce court, and that would secure that they wouldn’t lower prices on each other. Is that right? So why wouldn’t that work? Why wouldn’t writing a contract here work? It’s against the law, it’s an illegal contract.

What about you and your roommates? How many of you have a written contract stuck with a magnet on the fridge telling you when you’re supposed to tidy up? Very few of you. Why do you manage to get some cooperation between you and your roommates even without a written contract?

STUDENT: It’s not legally enforceable.

BEN POLAK: Well it probably is legally enforceable actually. This guy says it’s not legally enforceable. He probably could have a written contract about tidying up. The woman in here.

AUDIENCE QUESTION: Repetition, you do it over and over?

BEN POLAK: Yeah, so maybe even among your roommates, maybe you don’t need a contract because you can manage to achieve the same end by the fact that you’re going to be interacting with the same person over and over again during a time at Yale. So we’ll come back and revisit the idea that repeating an interaction may allow you to attain cooperation, but we’re not going to come back to that until after the midterm. That’s way down the road, but we’ll get there.

Now one person earlier on had mentioned something about communication. I think it was someone in the front. So let’s just think about this for a second. Is communication the problem here? Is the reason people behave badly in this, or badly, people choose alpha in this game here? Is it the fact that they can’t communicate?

Suppose you’ve been able to talk beforehand. Suppose the woman here whose name was Mary had been able to talk to the person next to her whose name is Erica, and they’d said, suppose we know we’re going to be paired together, I’ll choose beta if you choose beta. Would that work? Why wouldn’t that work? Yeah?

STUDENT: There’s no enforcement.

BEN POLAK: There’s no enforcement. So it isn’t a failure of communication per se. A contract is more than communication. A contract is communication with teeth. It actually changes the payoffs.

So I could communicate with allies on agreements, but back home I’m going to go ahead and choose Alpha anyway, all the better if he’s choosing Beta. So we’ll come back and talk about more of these things as the course goes on, but let’s just come back to the two we forgot there.

So the collusion case and the case back in Law and Order with the prisoners in the cell. How do they enforce their contracts? I mean they don’t always rat each other out and some firms manage to collude. How do they manage to enforce those contracts? Those agreements, how are they enforced? It could be they trust each other, although if you trust a crook that’s not a good idea. What else could it be? The guy here again with the beard, yeah?

STUDENT 2: It might not be a zero sum game.

BEN POLAK: Well, but this is the game right? So here’s the game.

STUDENT 2: No, but the way they value each decision.

BEN POLAK: So the payoffs may be different. I have something simpler in mind. Suppose they have a written contract, or even an unwritten contract. What enforces the contract for colluding firms or crooks in jail?

STUDENT 5: If the crook gets off scot free in five years when the other guy gets out he might run into a situation.

BEN POLAK: So a short version of that is, it’s a different kind of contract.

Indignant Angels: A New Set of Payoffs

If you write someone out in jail, someone puts a contract out on you. Tony Soprano enforces those contracts. That’s the purpose of Tony Soprano, the purpose of the Mafia. The reason the Mafia thrives in countries where it’s hard to write legal contracts, let’s say some new parts of the former Soviet Union or some parts of Africa, the reason the Mafia thrives in those environments is that it substitutes for the law and enforces both legal and illegal contracts.

So I promised a while ago now that we were going to come back and look at this game under some other possible payoffs.

So let’s, no I wasn’t under a contract but let’s come back and fulfill that promise anyway. So we’re going to revisit, if not the Divinity School, at least some people who have more morality than my friends up in the business school. We’re going to ask for the same grade game we played at the beginning, what would happen if players’ payoffs looked different? So these are possible payoffs too. I’ll give these a name.

We called the other guys Evil Gets, so we’ll call these guys Indignant Angels. I can never spell indignant. Is that roughly right? Does that look right? I think it’s right, indignant isn’t it?

Indignant angels, and we’ll see why in a second. So here are their payoffs. Once again, the basic structure of the game hasn’t changed. It’s still I’m choosing α and β, my pair is choosing α and β, and the grades are the same as they were before. They’re hidden by that board but you saw them before.

But this time the payoff was follows. On the lead diagonal we still have 0, 0 and 1, 1, but now the grades here are minus 1 and minus 3, and here they’re minus 3 and minus 1. What’s the idea here? These aren’t the only other possible payoffs, it’s just an idea. Suppose I get an A and my pair gets a C, then sure I get that initial payoff of 3, but unfortunately I can’t sleep at night because I’m feeling so guilty.

I have some kind of moral conscience and after I’ve subtracted off my guilt feelings I end up at minus 1. So think of this as guilt, some notion of morality. Conversely, if I chose a beta and my pair chose an α, so I end up with a C and she ends up with an A, then I have a bad time explaining to my parents why I got a C in this class and I have to say about how I’m going to be president anyway. But then in addition, I feel indignation against this person. It isn’t just that I got a C, I got a C because she made me get a C.

So that moral indignation takes us down to minus 3. So again, I’m not claiming these are the only other possible payoffs, but just another possibility to look at.

Classroom Exercise: What Would You Do?

So suppose these were the payoffs in the game. Again, suspend disbelief a second and imagine that these actually are your payoffs, and let me ask you what you would have done in this case. So think about it a second.

Now write it down, write down what you’re going to do on the corner of your notepad. Just write down at alpha or beta what you’re going to do here. You’re not all writing. The guy in the England shirt isn’t writing. You’ve got to be writing to him in an England shirt.

Show it to your neighbor. Let’s have a show of hands. Again, I want you to keep your hands up so that Jude can see it now. So how many of you chose α in this case? Raise your hands, come on, don’t be shy, raise your hands.

How many chose β in this case? How many people abstained? Not allowed to abstain, let’s try it again. Alpha, alpha in this case, no abstentions here. Beta in this case, so we’re roughly splitting the room.

Someone who chose alpha, can I raise the alphas again? Let me get this guy here, Ale, can we get this guy here? So why did you choose alpha?

AUDIENCE QUESTION: You would minimize your losses. You would get 0 or negative 1 instead of negative 3 or 1.

BEN POLAK: So this gentleman’s saying a good reason for choosing alpha in this game is it’s less risky. The worst case scenario is less bad. What about somebody who shows beta? A lot of you chose beta. Let’s have a show of hands on the betas again.

Let me see the betas again. Raise your hands. Can we get the woman here? Can we ask her why she chose beta?

AUDIENCE QUESTION: Because if you choose Alpha the best case scenario is you get 0. So I’m sure it takes more chances.

BEN POLAK: That’s a good counterargument. So the gentleman here was looking at the worst case scenario and the woman here was looking at the best case scenario, and the best case scenario here looks like getting a 1 here.

Now that’s also a different question. Is one of the strategies dominated in this game? No, neither strategy is dominated. Let’s just check. If my pair chooses α, then my choosing α yields 0, β-3, so α would be better. But if my pair chooses β, then α yields minus 1, β yields 1, in this case β would be better.

So α in this case is better against α and β is better against β, but neither dominates each other. So here’s a game where it’s, we just changed the payoff, we have the same basic structure, the same outcomes, but we imagine people cared about different things and we end up with a very different answer. In the first game it was kind of clear that we should choose α and here it’s not at all clear what we can do, what we should do.

Coordination Problems and the Lesson That Payoffs Matter

In fact, this kind of game has a name and we’ll revisit it later on in the semester. This kind of game is called a coordination problem. We’ll talk about coordination problems later on.

But the main lesson I want to get out of this for today is a simple lesson. It’s the lesson that payoffs matter. We change the payoffs, we change what people cared about and we get a very different game with a very different outcome. So the basic lesson is that payoffs matter, but let me say it a different way.

So without giving away my age too much, well I guess it will actually, when I was a kid growing up in England there was this guy, there was a pop star, a slightly post-punk pop star called Joe Jackson, who none of you will have heard of because you’re all about ten years old, not my fault, and Joe Jackson had this song in which he had the lyric, something like, “you can’t get what you want unless you know what you want.” You can’t get what you want unless you know what you want, and as a statement of logic that’s false. It could be that what you want just drops into your lap without you knowing about it. But at the same time, with a strategy, it’s a pretty good idea. It’s a good idea to try and figure out what your goals are, what you’re trying to achieve before you go ahead and analyze the game.

So payoffs matter. Let’s put it in his version. “You can’t get what you want till you know what you want.” Be honest, how many of you have heard of Joe Jackson? Oh, it makes me feel old. Oh man, okay. It goes down every year.

Mix and Match: Evil Git vs. Indignant Angel

So far, we’ve looked at this game as played by people who are evil gits, and we’ve looked at this game as played by people who are indignant angels. But we can do something more interesting. We can imagine playing this game on a sort of mix and match.

For example, imagine, this shouldn’t be hard for most of you, imagine that you are an evil git but you know that the person you’re playing against is an indignant angel. Say again, imagine that you know you’re an evil git but you know that the person you’re playing against is an indignant angel. So again, imagine that you know you’re an evil git, but you know that the person you’re playing against or with is an indignant angel. What should you do in that case? What should you do in that case?

What should you do in that case? Who thinks you should choose α in that case? Let’s have a show of hands in the room again. Keep your hands up so you can see. Who thinks you should choose β in that case? Who’s abstaining here? What am I to abstain in this class? It’s a complete no-no. So we’ll allow some abstention on the first day but not beyond today.

Let’s have a look. Let’s analyze this combined game. So what does this game look like? It’s an evil git versus an indignant angel, and we can put the payoff matrix together by combining the matrices we had before. So in this case, this is me as always, this is my pair, the column player.

My payoffs are going to be what? My payoffs are going to be what? My payoffs are going to be evil git payoffs, so they come from the matrix up there. So if someone’s going to help me with reading it off there, that’s a 0, a 3, a –1 and a 1. My opponent or my partner’s payoffs come from the indignant angel matrix, so they come from here.

There is 0, a −3, a −1 and a 1. Everyone see how I constructed that? So I’ll just remind you again, the first payoff is the row player’s payoff, in this case the evil git, and the second payoff is the column player’s payoff, in this case the indignant angel.

Well now we’ve set it up as a matrix, let’s try again that question I asked before. Suppose you’re the row player here, you’re the evil git, those are your payoffs, you’re playing against an indignant angel, what would you do?

So once again, no abstentions this time. Who would choose α? Let’s have a show of hands again. Keep your hands up a second. Who would choose β? Very few β’s, but mostly α’s.

α I think is the right answer here, but why? Why is α the right answer here? So yeah, can we get this guy here?

AUDIENCE QUESTION: It’s the dominant strategy.

BEN POLAK: Good. Actually nothing has changed from the game we started with. The fact I changed the other guy’s payoffs didn’t matter here. Alpha was dominant before. It dominated beta before and it still dominates beta. Let’s just check. If my opponent chooses Alpha and I choose Alpha I get 0, Beta I get minus 1, so Alpha would be better. If my opponent chooses Beta and I choose Alpha I get 3, Beta I get 1, once again Alpha is better. So as before, Alpha does better than beta for me, regardless of what the other person does, alpha dominates beta.

What was the first lesson of the class? What was the first lesson of the class? Shout it out please. Right, so you should all have been choosing in this game. You all should have chosen alpha, so the one person who didn’t will let him off for today.

So alpha dominates beta here.

Indignant Angel vs. Evil Git: Iterated Dominance

Let’s flip things around. Suppose now, harder to imagine, but let’s try it. Suppose now that you are an indignant angel and you’re playing against, and you know this, you’re playing against an evil git. You’re an indignant angel, so you have the payoffs that are still there and you’re playing against an evil git, which is the payoff we covered up but we’ll reproduce them.

Let’s produce that matrix. By the way, this is beginning to sound like a wrestling match, I don’t mean it to. Let’s try here, α, β, α, β, pair me. So my payoffs this time are the Indignant Angel payoffs, so mine are 0, -1, -3, and 1, and my opponent’s payoffs are, what would have been my payoffs before? They come from the other matrix. Let’s just show you it.

They come from this matrix so they’re going to be 0, -1, 3, 1, 0, -1, 3, 1. I took the second payoff from that matrix and made it the second payoff in this matrix. Everyone see how I did that? Once again, the row player is the first payoff and the column player is the other payoff.

All right, what should you do in this case? You’re the indignant angel, you’re playing a game against this evil git. What should you do? Write down in your notepad what you should do. Show it to your neighbors, you can’t cheat, or you can cheat but you’ll be shamed in front of your neighbor. And raise your hand, let Jude see it, raise your hands and keep them up if you chose α now.

How about if you chose β now? So 1 or 2 β’s, mostly α’s. Well let’s see it, let’s reason this through a second. Does α, does my α dominate my β? No, in fact α doesn’t dominate β for me.

If my pair chooses α then α gets me 0, β-3, so α does better, but if my pair chooses β then α gets me minus 1, β gets me 1, in this case β is better. So once again, as we saw before, alpha is better against alpha, beta is better against beta, there’s no dominance going on here.

Nevertheless, at least 90% of you chose alpha here, and that’s the right answer. Why? Why should you choose alpha here? Somebody, can we get the guy with the beard here? Yeah, wait for the mic. There you go, great.

AUDIENCE QUESTION: We’ve acknowledged that alpha is a dominant strategy for my opponent so we must choose based upon knowing that my opponent’s going to choose alpha.

BEN POLAK: Good, good. Your name is?

AUDIENCE QUESTION: Henry.

BEN POLAK: So Henry’s saying, sure I don’t have a dominated strategy. My alpha doesn’t dominate my beta, but look at my opponent. My opponent’s alpha dominates her beta.

Dominant Strategy and Strategic Thinking

BEN POLAK: If I choose alpha and she chooses alpha to get 0, beta she gets minus 1, alpha is better. If I choose beta, if she chooses alpha she gets 3, beta 1, again alpha is better. For my opponent, alpha dominates beta. So by thinking about my opponent, by putting myself in my opponent’s shoes, I realize that she has a dominant strategy, alpha. She’s going to choose alpha and my best response against α is to choose α myself.

So here, this time my α does not dominate β, but my pairs’ choice of α, my pairs’ choice of alpha dominates her choice, or her possible choice of β. So she will choose α and once I know that she’s going to choose α, it’s clear that I should choose α and get 0 rather than β and get −3. So I should choose α also.

Okay, so now we’ve seen four different combinations. We’ve seen a case where an evil git was playing an evil git, where an indignant angel was playing an indignant angel, and we’ve seen both the flips of those, the evil git versus the indignant angel, and the indignant angel against the evil git.

The Essence of Strategic Thinking

BEN POLAK: Why are we doing this? Because there’s an important lesson here. What’s the lesson here? The lesson is, it comes from this game, that a great way to analyze game, a great way to get used to the idea of strategic thinking, perhaps even the essence of strategic thinking, is the ability to put yourself in someone else’s shoes, figure out what their payoffs are, and try to figure out what they will do. In a sense, this is the first difficult lesson of the class.

It’s easy to spot when a strategy is dominant more or less. It’s pretty easy to figure out you have to know about your own payoffs, but the hard thing in life is getting you to come out of your own selves a bit, realizing it’s “not all about you.” You’ve got to put yourself in other people’s shoes to figure out what they care about and what they’re going to try and do so you can respond well to that.

Now while we’re here, let’s just mention that things will get more complicated in a world where I don’t actually know the payoffs of my opponent. It’s much easier to figure out my own payoffs than to figure out my opponent’s payoffs.

I might not know whether I’m playing someone who’s an evil git or an indignant angel, so I’m going to have to figure out what the odds are of that in doing this exercise. And that gets, we’re going to come back to that idea too, way at the end of the class. That’s getting a bit ahead of ourselves but we’ll get there.

Real-World Experiments: The Prisoner’s Dilemma

BEN POLAK: Now it turns out that this game, this Prisoner’s Dilemma, with the alphas and betas, or essentially the same game, has been played many, many, many times in experiments. So out there in the real world, we can do this here, out there in the real world when they do these experiments, they find out that roughly 70% of people choose α and roughly 30% choose β. Roughly, almost a third choose beta.

Now what do we think is going on? Why is it that a third of the people who seem to be choosing a dominated strategy, or is it? What’s going on there? Why do you think 30% of people are choosing beta? Anybody?

AUDIENCE QUESTION: They might be motivated by the fact that every person who chooses beta raises the average score.

BEN POLAK: They could be moral people. So one possibility is this 30% of people in the real world who choose beta are just nice people. What else could it be?

AUDIENCE QUESTION: I know this might be changing the game a little bit, but if you ever expected to play the same game with the partner, you have more.

BEN POLAK: They could be thinking they’re going to play again.

AUDIENCE QUESTION: Long run payoffs are greater if you choose beta every time.

BEN POLAK: So it could be that they think this is actually, they haven’t understood the experiment and they think this is a multi-shot game, not a one-shot game.

What else could it be? What’s the simplest explanation? What’s the other obvious explanation? They could just be stupid, right? It could be, right? We’re allowed to say that in class.

Let’s be honest here, when we say experiments in the real world in game theory, or the ones you read about in the New York Times, the real world when it comes to experiments in economics really means undergraduates at the University of Arizona. I’m not making that up, it just does, they all are. And I don’t know anything about it. Any of you from Arizona? I don’t know. I don’t know whether the average undergraduate at the University of Arizona just has a sunny personality or whether they’ve spent too long in the sun. I just don’t know which it is. We can’t really distinguish from this.

Yale Students vs. the Average

BEN POLAK: How about in Yale? What’s our numbers here? How about in this class? Do you want to make your colleague here? Yeah. So 238 at Yale, this is Yale, versus 36. So even at my level of arithmetic that’s a lot less than 30%, that’s more like less than 15%. No, it’s about 15% I guess. So 238 chose α and 36 chose β.

Now there’s one more lesson in this class and this is going to be it. This isn’t the end of the class but one more lesson to take home. You guys are going to be playing games among each other today and until whatever it is, December 7th, whatever it is, the end of term. So look around each other, you better get to know each other a bit. What have we learned today about you guys?

The lesson here, lesson 5 is: Yale students are evil. Be aware of that when you’re playing games.

Game 2: Pick a Number

BEN POLAK: I want to play one more game today in the remaining minutes. It doesn’t matter if we finish a little bit earlier, but I want to try and get this game at least started. So do you all have game number 2 in front of you? Just while you’re reading that over, can I also make sure you’ve all got your legal forms and you’re going to sign? Don’t walk away with your legal forms. We need to get those collected in. So at the end of talking about this game, I’m going to collect in both the second game for the class and also the legal form. If you don’t have a legal form, if you’ve lost it or something, it’s online.

Let’s have a look at that second game. I’ll read it out for you. Game 2, pick a number. Everyone got this? Anyone not got this? Everyone got it?

Without showing your neighbor what you’re doing, put in the box below a whole number between 1 and 100, whole number between 1 and 100, integer. We will calculate the average number chosen in the class. The winner in this game is the person whose number is closest to two-thirds times the average in the class. Again, the winner is the person whose number is closest to two-thirds times the average number in the class. And the winner will win $5 minus the difference in pennies between her choice and that two-thirds of the average.

Now just to make sure you’ve understood this, let me do an example on the board. I’ve got one more board, that’s good. So imagine there were three people in the class and imagine that they chose 25, 5 and 60. So 25 plus 5 plus 60 is 90. People should feel free to correct my arithmetic because it’s often wrong. 90 right?

And 90, two-thirds of 90, so two-thirds of 90, two-thirds of, what do I need? Start again. I need to divide it by 3 to get the average, so the average is 30. Somebody is adding out. So the total is 90, the average is 30, am I right so far? So two-thirds of the average is 20. I’m looking desperately at the TA, is that right? Okay, so the average is 30 and two-thirds of the average is 20.

So who’s the winner here? Which number would have won here? 25 would have won. 25 would have been the closest and what would they have won? They’d have won 5 bucks minus 5 cents for a total of $4.95.

Playing for Real

BEN POLAK: Now to make this interesting let’s play this for real. So this of course relies on me having brought some money and being able to do this without dislodging the microphone. So I’m going to see if I have enough money to do this class for all. When we play this game in the old days, during the dot com boom with the MBA students you had to put 50 bucks on the table to get them interested. Graduate students, five cents will do it.

So this is a, there’s some bloke with a beard on this one. Yeah, this is Lincoln apparently, who knew? Lincoln, okay. So this is a $5 note and I’m going to put it in an envelope. I’m not cheating anybody, right? No magic tricks here. This is going to be the prize for this game and we’d better give this to someone we trust.

So prize for 159. Who do you guys trust? The camera guy. Okay, Jude. We know Jude’s going to be there next week. I’m giving it to Jude. You can’t see this on camera, people at home, but I’m giving it to Jude. I’m going to put it here. Jude has to therefore show up next week with the prize. I thought we should give it to the guy at the back who is the moral guy. Who is our moral guy at the back? Never mind, we’ll give it to you. We know you’re going to be here.

Has everyone put a number down? Any questions? Yes, just shout them out to me.

AUDIENCE QUESTION: So given that we only have one $5 bill, does that have to be one unique winner? And if so, how is that determined if we have multiple people who are positive?

BEN POLAK: That’s a good question. If there’s multiple winners we’ll divide it but we’ll make sure everyone has a positive winning. Good question. Given the number of people in the room there may be multiple winners. I accept that possibility.

The Five Lessons

BEN POLAK: Has everyone written down a number now? So hand your numbers to the end of the row, but don’t go yet. Hand your numbers to the end of the row. Before you go, I want five things from you. I want to know the five lessons from this class. Tell me what you learned. What were the five lessons? Without looking at your notes, what were the five lessons? Anybody? Shout out one of the lessons.

AUDIENCE QUESTION: Don’t play a strictly dominated strategy.

BEN POLAK: Anything else?

AUDIENCE QUESTION: Yale students are evil.

BEN POLAK: Two lessons down, three to go. The guy over here, yeah.

AUDIENCE QUESTION: Rational choice is going to lead to bad outcomes.

BEN POLAK: We put it more graphically before but that’s fine. Two more outcomes.

AUDIENCE QUESTION: Put yourself in other people’s shoes.

BEN POLAK: And I’m missing one. I can’t remember what I’m missing now.

AUDIENCE QUESTION: You can’t get what you want, you could, but it’s a good idea to figure out what you want before you try and get what you want.

BEN POLAK: Five things you learned today, hand in your numbers and the legal forms, and I’ll see you on Monday.

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