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Welcome back. One small aside from the last session,

Nash equilibrium is often also referred to as

a Cournot-Nash equilibrium because

the assumptions where we ended up in a Cournot outcome,

each firm was choosing the best output knowing what the other firm had chosen,

that's the same concept that we saw in the last session with the Nash equilibrium.

Now we can turned to the most famous game theoretic construct, the Prisoner's Dilemma.

It's also the classic example where individual players pursuing their self-interest

end up in an outcome where all are worse

off than they could be if they behaved cooperatively.

And it's the best counterexample to Adam Smith's invisible hand theory.

In Adam Smith's invisible hand theories,

each person pursuing their best interests would end up maximizing the welfare of society.

The Prisoner's Dilemma concept comes from a setting where two people,

let's call them Sid and Nancy,

have been arrested, and the police suspect

they're part of an international drug cartel and they've done some serious crimes.

But the evidence that they've picked up from the arrest is for more minor infractions,

not the more serious crimes that they'd like to pin a conviction on Sid and Nancy for.

So what a district attorney will do to try to get the two to confess to rat on each

other is to set up the payoffs that end up forming a prisoner's dilemma.

And the way the payoffs work is if Sid and Nancy don't confess,

and the two are typically separated and then interrogated individually,

if neither of them confess,

then the payoffs they'll get is both doing two years in

prison for the lesser crime that they've been picked up by the police for.

if Nancy confesses, the district attorney will say, "Look,

if you turn state's evidence and give us the confession that allow us to really

stick it to your partner Sid and Sid doesn't confess,

then it's Sid that does 15 years in prison,

and the district attorney offered

a lesser sentence in return for the confession to Nancy,

one year as opposed to the two years.

Analogously, if Nancy doesn't confess,

Sid is offered a similar deal where

if you turn state's evidence and give us the confession on the more major crime,

we'll stick Nancy in prison for 15 years,

and you'll do only one year.

And if they both confess on each other,

then there's enough evidence to send them both to jail for

a longer time period, 10 years.

Where will we end up? From the prisoner's perspective,

they would be best off each going to jail for two years.

But we'll see that the outcome in

this prisoner's dilemma setting is they'll both go to jail for 10 years,

the more extended time period. How can we see this?

And let's erase the screen.

In this type of setting,

both players have a dominant strategy.

Both players find it better to confess.

Let's take Sid first.

If Nancy, if you knew your partner for sure was going to confess,

what's your best option is sit in?

You then choosing between 15 years, and 10 years.

You'd rather do less time.

Your best option is to confess if you knew for sure your partner was going to confess.

Likewise, if Nancy doesn't confess,

what's your best strategy if Sid?

You're looking at a reduced time in prison of

one year versus the two years you're going to do if you also don't confess.

Sid has a dominant strategy of confess.

Likewise, with Nancy, and we'll let you figure that out on your own to test yourself.

So, we'll end up in a dominant strategy equilibrium,

also a Nash Equilibrium,

the broader subset, each doing 10 years time.

Each person pursuing their self-interest,

we've ended up with an outcome where they're both

worse off than they would have been if they wouldn't squealed on each other.

This Prisoner's Dilemma, set this Prisoner's Dilemma framework has wide applicability.

It shows up, for example,

when we look at representative democracies and why did they run deficits,

where a representative wants to get his or her district's fair share of spending.

But if every representative from their district wants to do likewise,

we end up with more government spending than otherwise would be desired.

The analogy would be if we went to a hundred of us

look the same numbers in the US Senate out to dinner,

and before we might have bought a cheeseburger and a Diet Coke for six dollars.

And yet, if we agree to share the bill with everybody,

that steak may look a lot better than the cheeseburger.

And we may opt for a glass of wine or two.

Stuff that cost $40,

if everybody has that same incentive across a

hundred people at dinner and we're sharing all the costs equally,

we end up with a $4,000 bill with a hundred diners for dinner.

Each of us responsible for our share,

$40, whereas before we individually would have spent only 6 dollars.

We've gotten our fair share and yet we end

up with indigestion and a pretty big bill individually at the end.

What if we try to restrain our spending and still buy

the cheeseburger and the Diet Coke and only spend $6?

You'll feel like a chump in that case,

not getting your fair share.

And yet, you still end up being stuck with close to a $40 price tag,

your expenditures won't have much of an impact on the overall bill.

You see this in the healthcare legislation too where any

time the payments come out of a third party,

where if they don't come directly out of your own pockets,

you end up facing a prisoner's dilemma.

If there's any risk of a particular illness and you need to

undertake tests but the costs of those tests don't come out of your own pocket,

your incentive will be to confess,

to be to spend that extra money.

All of us though,

we face a similar incentive with our third-party payers end up with

a very expensive healthcare system and also

one that tends to be bureaucratic where we have insurance companies

or the government trying to determine is this expenditure warranted or not.

Cartels face this exact same problem and we'll see this in the next example.

Let's take a simple example of Utopia and Artesia.

If both don't cheat and comply,

both end up earning profits of $20,

and yet each firm has an incentive to cheat.

Each firm has a dominant strategy.

Let's take the case of Utopia.

If it knew for sure Artesia was going to cheat,

it would be better off cheating too, $10 versus $5.

And if it knew Artesia was going to comply,

Utopia's payoffs are larger by cheating than not cheating, 25 vs. 20.

So Utopia's dominance strategy is to cheat,

so is Artesia's and you can test yourself on that score.

We end up with both firms making less than combined profit of 20

versus the 40 that prevailed if both had cooperated and behaved as a cartel together.

Note too that if we move from Utopia complying to Utopia cheating,

total profits in the lower right hand corner of 40,

Utopia makes more, $25,

but the overall profits decline to 30.

So, any cheating by one of the two parties in

this duopoly setting serves to reduce the overall profits,

but each firm has an incentive to cheat because

each firm makes more money than it did before by cooperating.

Stable 14.4 also should convince you why from a game theoretic perspective,

the more firms we have,

the greater the incentive to cheat.

Two fundamental reasons.

The more firms there are,

the more elastic the demand curve confronting any individual firm,

if it's a smaller share of the overall market.

So, as it expands output,

it makes more money than if demand is more elastic,

if it's a smaller share of the overall market.

It's also harder to detect when there are a larger number of firms,

so the other firms can't figure out as easily whether this firm is cheating or not.

And then, one other example that you might see of the prisoner's dilemma,

if you have courses at your university where you're graded on the curve,

and let's say in this case if two different students Scott and Kaitlin,

should they study four hours or one hour?

If the professor grades on a curve,

he or she will only allocate so many percentage A's in the course,

so, many percentage B's,

so many percentage C's.

The optimal outcome is

putting in less study time if you only care about the grade in the course.

And each of you earning a 60,

each of you earning a B,

because both of you put in that one hour.

If you both spent four hours,

you'd get a higher numeric grade but still end up with that same B,

more study time, same grade.

Why is this a prisoner's dilemma?

Let me erase it.

Each of the two students has a dominant strategy: to study more,

if you're graded on the curve.

If you knew for sure your fellow student was only studying an hour,

you'd want to study four hours,

get an 85, and get that A.

It's better than studying

only one hour like your fellow student and getting 60 like she does and with a B.

If you knew for sure Kaitlin,

your fellow student, was going to study four hours,

you'd similarly have an incentive to study four hours

like her and get the B as opposed to the C if the professor's grading on the curve.

Of course, this example doesn't apply in your economics courses where

the love of learning also motivates behavior,

but it may help you explain behavior of fellow students in some of your other courses.