Hi. In this lecture, I'm gonna talk about what is by far the most famous game in

game theory. It's called the prisoner's dilemma. The prisoner's dilemma has been

written about literally tens of thousands of times. Far more than any other game and

it's a really simple game. There's only two players and two actions. So what we're

gonna do in this lecture is we're gonna talk about why the prisoner's dilemma is

so interesting, and we're gonna talk about some of the applications of the prisoner's

dilemma in a variety of different disciplines. Okay. So let's get started.

How does the prisoner's dilemma work? There's two players, player one and player

two. Player one will be. Role player, player two will be the column player. Each

player has two actions. They can cooperate, or they can defect. So here's

what makes the game so interesting. If both players co operate, if they are nice

to one another, each one gets a payoff of four, so player one gets a payoff of four

and player two gets a payoff of four. However if player two is being nice, if

player two is co operating and player one defects, then player one will get a payoff

of six. So it's entire one interested effect and since this game is symmetric,

if player one is co operating and player two defects, player two will also get a

payoff of six. Each one has an incentive to defect. Now if they?re both defecting

their both gonna get payoffs of two [laugh]. So here's the funny thing, it's

in their collective interest to cooperate. It's in their individual interest to

defect. But if they both defect their both worse off. So that's what makes the game

so interesting. By the way, if you're wondering why this is called prisoner's

dilemma, the original story behind the game goes as follows: two people are

caught, and the police are pretty sure that they've committed a crime, so they

put these two quote unquote prisoners in jail, They put them in separate rooms, and

they say to each one of them, look, you can rat out your friend, you can defect.

Or, the two of you could coope rate with one another. And not rat each other out.

Now if the two prisoners cooperate with each other and don't rat each other out

then they're going to get fairly mild prison sentences, but if they defect. And

they basically say, no, you're right, he, we did it, we did it. Then, each one would

get off, but if they both did that, and they both rat on each other, then they're

going to be worse off. So that's the original story behind the game. But we're

gonna see that there's far broader applications than sort of little vignettes

about prisoners going to jail. This game has lots and lots of applications. But

before we get into it, let's talk about, a little bit about what is a prisoner's

dilemma and what's not a prisoner's dilemma. So here's a game that sometimes

you'll see written down as a prisoner's dilemma but it's not. Here again, if they

both cooperate, they get four. If they both defect, they get two. But now if

player 2's cooperating and player one defects, player one gets a payoff of nine.

So nine is a huge payoff. And so if you look at this game, you realize that 4-4,

although it's a good payoff, is no longer the best payoff if this game is played

many times. Because if we're playing the prisoner's dilemma several times, what

can. After years, the two players can alternate. They can go nine, zero and

zero, nine and their average payoff By alternating. You'd be four and a half

which is bigger than four. So this game, which people sometimes put down as the

prisoner's dilemma game, actually isn't. It's a game I like to call weak

alternation; where you've got an incentive to alternate as opposed to cooperate. So

in the prisoner's dilemma, it's gotta be the case that you're better off playing

cooperate, cooperate. So when people write down formal definitions of the prisoner's

dilemma, they don't use numbers like four, two and six. Instead they write down

things like T, F and R. So here's the idea. For there to be a prisoner's

dilemma, T is gonna be bigger than R because that means you're better off

cooperating. Been defecting. It's also the case that F Has to be bigger than T,

because that means that player one would rather defect if player two is

cooperating. And player two would rather defect if player one is cooperating. And

then the third thing we need is that two T. Has got to be bigger than F, because

we've got to make sure that you're not better off alternating than playing

cooperation. So this is formally how we write down a class of games that belong to

what we call the prisoner's dilemma. Of course we're assuming here that T and R,

of course, are bigger than zero. Okay, so that's it, that's all there is to the

prisoner's dilemma. Why is it so interesting? Why all this focus on the

game? Well again, let's go back to the two things I talked about before. What's the

efficient, outcome, what's the thing you'd really want to get? You'd like to get C-C,

you'd like to be the case that both people are getting four. But supposing they had a

weaker notion of efficiencies. Cause here, by picking 4-4, that's the one that sort

of both get the same payoff, and it's got the highest total payoff. You can think of

another notion of Efficiency, which is called pareto efficiency. Now path is

pareto efficient to a group of people if there's no way to make everybody better

off. So there's no way in which you can make every single person better off. Well

that's clearly true of the case four, four. There's no way to make everyone

better off. But it's also true of zero six and six zero. Let's see why. Suppose we're

sitting at zero six, is there any way to make everyone better off? No, Because if

we go here then player two is worse off. If we go down here player two is worse

off. If we go here player two is worse off. And if we go here player two is worse

off. So there's no way to do, make everyone better Than getting 0-6. Player

2's always gonna suffer. For the same reason 6-0 is also Pareto efficient.

There's no other payoff that we can get that makes player one better off, 'cause

that's player 1's highest payoff. In f act, the only thing in this game that

isn't Pareto efficient is 2-2. And the reason 2-2 isn't Pareto efficient is

'cause we can make every better, everyone better off by going to 4-4. So what we get

in this game is there's only one outcome that's really bad if we use Pareto

efficiency as our criterion. And that outcome is 2-2. So the only thing that

isn't a good outcome in this game is 2-2. But, if we think of this in terms of game

theory, there's this notion of Nash equilibrium. And Nash equilibrium, it

seems that people optimize. Well let's look at the strategies that people should

follow. If I'm player one, and player two right here is cooperating, then I should

defect because six is better than four. But if player two is defecting, I should

also defect because two is bigger than zero. Now the same thing's two by two. If

player 1's cooperating Player two should defect, 'cause six is bigger than four.

And if player one is defecting, player two should defect, 'cause two is bigger than

zero. So the Nash Equilibrium here is 2-2. Well, this is why the game is so

interesting. There's three efficient outcomes, in some sense 4-4, 0-6, and 6-0.

Any one of these things you could argue, on moral grounds, is sort of an okay or

efficient outcome. Because there's no way to make everyone better off. But the only

equilibrium outcome here is two; two is the one inefficient thing. So what you

have is incentives don't line up with what we wanna have occur socially, so there's

this disconnect. And it's that disconnect that is so interesting to people when they

study the prisoner's dilemma. 'Cause what we wanna do is we'd like to have a way to

have the outcomes that we get in a situation align with our social

preferences. And what happens in the prisoner's dilemma is our individual

preferences point in one direction, which is toward defect. And our social

preferences head in another direction, our collective preferences, which is to

cooperate. It is that tension that creates so much interest. Now that's not gonna be

true of most games. Right? So here's another game called the self interest

game. Where again if I look at player one and player two, they've each got two

strategies, A and B. And what you see is that player one would rather play B if

player 2's playing A, And if player 2's playing B. We would also like to play

deep. For the same reason if player one is playing A you get to player two would

rather play B. Cuz six is bigger than four. And if player one is playing B.

Player two would also like to play B. So what we get is the equilibrium in this

game Is 7-7. So if people are strategic, what you're gonna get is 7-7. But that's

also the only pareto efficient outcome. So here what we get is the pareto efficiency

lines up with incentives. So the self-interest game, nobody writes many

papers about this, [laugh], because it's obvious what's gonna happen. You're gonna

get 7-7, and that's what we'd like to get. So therefore, there's no dilemma. There's

no problem. But the prisoner's dilemma has a problem because what we are going to get

is 2-2 and what we want is 4-4. So the question is how do we get it. Before I go

on to talking about how we get corporations [inaudible] let?s talk a

little bit about why it has been of so much interest. The reason is it sort of

applies to a lot of settings. Here's one, let's think about arms control. You can

think about corporations, spending money on education and you can think about

defections spending money on bombs. So now you got two countries, Country One and

Country Two. Both countries would prefer if they were both spending their money on

education. They can't help themselves and they spend money on bombs and everyone

else is worse off. Look at price competition between two firms. So you got

two firms competing. Maybe two ice cream stores right next to each other.

Cooperation would be, let's keep prices high, and the firms make money. Defection

would mean that they lower their prices to get more customers. Well, if they both

lower their prices, they're both worse off. So what they'd prefer to d o is have

high prices, and get higher payoffs. But it's in their individual interest to have

lower prices, and so they end up being worse off. Now in this case, the

prisoner's almost interesting because even though the firms are worse off, the

consumers are better off. So don't really care too much about this type of

prisoner's dilemma, at least not as much as we care about the situation of taking

money that could go to education and spending it on bombs. Similar logic holds

for technological adoption. So let's think of two banks. And the banks can decide

either not to buy ATM machines or to buy ATM machines. If they don't buy ATM

machines Both make profits of, let's say, four. However, if one of'em buys an A-,

buys ATM machines, and puts them up all over town, everyone will go to that bank.

And so if this bank defects, bank one defects, they're gonna get a higher payoff

of six. But if bank two comes in and puts in ATM machines, what's happened? Well

now, both of them have probably the same customers they had to start out with, but

they've spent all this money on ATM machines. So they're actually worse off.

In fact, it could even be worse than that. Because [inaudible] maybe before, part of

the reason they made such profits is because they got geographic grants. People

who lived around the bank shopped at that bank. Now that there's ATM machines

anywhere, people can shop whatever bank they want. And that's created more price

competition, like our previous prisoner's dilemma game. And so the banks end up not

doing as well. So again, they prefer not to buy the ATM machines. They can't help

themselves. They're both worse off. But again, in this case, like the price

deter-, the price competition case, we end up with the consumers being better off.

Let's look at political campaigns now. Suppose you got two candidates running for

office and it's a pretty even race. They could both run purely positive ads. And

what happened is they get some vote total but they also had some really shining

reputations. But now one of th ''em thinks, I'm gonna go negative. I'm gonna

bring up the fact that this one person went on a whole bunch of junkets paid for

by private industry. And by doing that I'm gonna win the election. But now the other

person goes negative as well and your back to having a 50, 50 chance of winning the

election. And your tarnished, your reputations tarnished. So, your payoff is

lower. So you'd both be better off if you only ran positive ads but you can't help

yourself and so you run the negative ads. It's a prisoner's dilemma, Food sharing.

Let's think of the simple case where there's two of us, and we can decide

either, share food or not share food. So if we both share food, then we're gonna

get a payoff A four. But if I don't share with you and you share with me, I'm gonna

get a payoff of six, so I'm much better off. Now if neither one of us shares,

we're worse off because the thing is, by me not sharing with you I don't get to get

your food. Now why would your food be different than my food? Well, it's two

reasons. One is, it could be that you actually, you have apples and I have pears

so we can get both types of food and be better off. And the other thing it could

be that maybe some years I do well and you do poorly, and some years you do well and

I do poorly, and so by sharing with one another, we spread the risk. Either one of

those logics applies; we're better off collectively sharing. However, we're

individually better off if we don't share and the other person shares with us. So

again, it's a straight prisoner's dilemma. And then finally, Adonis treadmills, So

let's suppose, you're thinking about something like getting a fancy watch. Well

it could be that we both just wear nice digital watches and they keep time and

everything's fine. But then, I decide, you know what? I'm gonna go buy this fancy,

handmade watch for $5,000. I'm gonna take all of my savings and go buy a handmade

watch. No why, why would I benefit from that? I'm gonna benefit from that because

people are gonna look at me and say, wow. Look how succ essful Scott is. He's got

that fancy watch. And you're gonna feel badly, cuz you're gonna feel like, wow,

you know, I don't look as cool as Scott looks cuz I have this cheap Plastic

digital watch. So then what happens is, you go buy A fancy handmade watch. Well,

we're both wearing fancy handmade watches, and neither one of us looks any cooler

than the other, but we've spent all the money that we could have used to send our

kids to college, let's say on watches. So, we're both worse off Classic prisoner's

dilemma. Okay, so in this lecture, what have we seen? We've seen the prisoner's

dilemma's a really simple game. Two players, two actions, We've seen it

applies to a whole bunch of settings, from arms control, to buying watches, to

sharing food, to price competition, to technological adoption, tons of

applications. And we've seen the reason it's so interesting is because what people

are gonna tend to do Is give us an outcome that isn't efficient. What we'd like to do

is typically have situations in the real world where the outcomes we get are the

good outcomes but when we have a prisoner's dilemma, the outcome we get may

be the bad outcome. What we want to do next, is see how do we get cooperation,

how do we get that four, four. We're going to see there's a variety of ways in which

that can be achieved. Alright, thank you.

The Prisoners' Dilemma Game

Model Thinking

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