Game Theory Part 1: The Prisoners' Dilemma

This video introduces game theory and goes through an example of the prisoners' dilemma. It discusses the concept of Nash equilibrium and introduces the idea of a repeated game.

For more information and a complete set of microeconomics videos, see
http://www.economistsdoitwithmodels.com/microeconomics-101

by Economists Do It With Models

>> Let's talk a little
bit about game theory.

Some times in economics
people want to be able

to describe situations

that involve what we call
strategic interaction.

Strategic interaction just means

that now not only does your pay
off, your profit, your utility,

how ever you want to think about
it depend on your own choices

but it also depends on the
choices of other people

in your market, in your
industry and so on and so forth.

Typical examples of strategic
interaction usually involves

decision among firms
regarding whether

to cooperate or to compete.

We're going to go
over and example

that has a slightly
different context known

as the prisoner's dilemma where
people are deciding whether

or not to confess to
a particular crime.

The set up of the prisoner's
dilemma is a tad bit contrived

but it goes as follows.

Imagine a situation in
which two people are brought

in for supposedly
committing a crime.

Now these two people are
held in separate cells

so they can't talk to
each other and even

if they could they couldn't
somehow contract on whether

or not they were going
to confess to the crime.

The people are then
brought in individually

and asked do you confess
or do you not confess?

We can represent the pay off's
to that sort of situation

in a table as follows.

You'll notice here that we
have player 1 and player 2.

I made things nicely color coded

such that we have player 1's
pay off's in terms of utility

in blue to match play 1 and
player 2's pay off's in terms

of utility in green here.

So you'll notice that if neither
player confesses they just sit

there and hold tight, they
each get a pay off of 10.

If the first guy keeps quiet
and the second guy rats him

out the second guy gets 15 while
the first player gets nothing.

The opposite happens here
if the first player rats

out the second one, now
the first player gets 15

and the second player
gets nothing.

And if they both try to rat
each other out, they both end

up with 5 meaning they're better
off than if they just sat here

and had the other guy
rat him out but not quite

as well off collectively
as if they both kept quiet.

The question then becomes given
this structure what's going

to happen.

In reality both players are
making the decision of whether

or not to confess at the same
time but let's just pretend

that they can guess or somehow
know what the other person is

going to do and we can ask a
number of hypothetical questions

as to what the best response
is for these players would be.

So let's take the
first case here,

say if player 1 confesses
what should player 2 do?

In other words what's
player 2's best response?

Well, we can go over here,

we say if player 1
confesses we're somewhere

in the bottom here and player 2
can either get zero by holding

out and being quiet or he
can get 5 by confessing also.

Five is strictly better than
zero so if player 1 confesses,

player 2 also wants to confess.

Now what about if
player 1 doesn't confess,

well if player 1 doesn't
confess we're up here

so player 2 again has two
options, he can get 10

by keeping quiet or he can get
15 by ratting out his buddy.

So 15 is better than 10 so
if player 1 doesn't confess,

player 2 still should confess.

Notice here that's interesting
that player 2 his best option is

to confess regardless
of what player one does

or alternatively put
player 2's best option is

to confess regardless of what he
thinks player 1 is going to do.

This type of situation is
called a dominant strategy

in that confess is
a dominant strategy

for player 2 meaning it's
always the best regardless

of what the other guy does.

Think about this the other way
around, say we make some guesses

as to what player 2 is going
to do and then when we say

in each case what's player 1's
best response in that situation.

So if player 2 confesses,

what's the best thing
for player 1 to do?

Say if player 2 confesses
we're over here

on the right somewhere we
say player 1 can either get 5

by confessing or 0 for being
quiet this problem is looking

strangely familiar, say
well 5 is better than 0

so player 1 is going
to want to confess.

Now if player 2 doesn't
confess what should player 1 do?

So if player 2 doesn't
confess, we're over here

on the left somewhere and
player 1 can either get 10

by being quiet or 15 by
ratting out his buddy,

well 15 is greater than 10 so
he's going to want to confess.

Notice here that
because we confessed

in both cases confessing
is also a dominant strategy

for player 1.

So here I've circled player
2's best responses in green

and I've circled player
1's best responses in blue

and you'll notice there's one
place here where they over lap

to say that in this situation
where both parties confess both

of them are responding
as best they can

to what they think the other
person is going to be doing.

We say that this situation
here is what's called a Nash

equilibrium; more formally put a
Nash equilibrium is a situation

where each player's
action is the best response

to the other player's actions.

In a situation where the players
are all moving simultaneously

this basically means that
each player is reacting best

to what they think the
other person is going to do

and they're actually
right in their guess

of what the other
person is going to do.

[ Pause ]

Notice here that the equilibrium
outcome actually...it doesn't

look as good as it could
because here we're saying

that any equilibrium when
people are acting according

to their own best interest each
of them ends up with a payout

of 5 where as if they only
cooperated they would each get a

payout of 10.

We can say here that there can
be a perato improvement going

from both parties confessing
to both parties staying quiet

in that both parties
would be made better off

and nobody would
be made worse off.

Unfortunately, due to
the competitive nature

of the this game that's
not what's going to result

because it's really hard when
there's no contracting involved

to guarantee regardless of
what the other party says then

when it comes down to it
they're actually going

to cooperate given that it's

in their interest
individually to not cooperate.

So one question that
economists like to think

about is then how can
cooperation be sustained

in the real world?

Well, one thing that's
important to remember here is

that in the real world this
game isn't played just once,

when you have firms interacting

with each other people making
these decisions often times they

have the chance to make the
decisions over and over and over

so when you have what's called
a repeated game you might have a

situation where people
start testing out the waters

to say well maybe if I
cooperate the other guy's going

to cooperate and then
we can keep this going

because to cooperate here and
hope for the best outweighs,

you know there's this
threat of well if you try

to screw me one time we're
reverting back here actually

gives in the long term an
incentive to cooperate.

So like I said it seems a little
bit artificial to be talking

about this context of
prisoners being interrogated

because really we're
talking about economics.

But it's very easy to see how
this situation could be relevant

in an economic context by just
replacing the intuition behind

some of the choices.

So what I did here is
set up the identical game

and have this model as
still player 1 and player 2

but now they have the
choice of whether or not

to cooperate or to compete.

And you can see here they'd both
do better off by cooperating

but they also all have the
private incentive to compete.

And you can notice here

that this situation is
actually pretty realistic

because at least in the United
States firms are not allowed

to contract on whether or not
they're going to cooperate,

that's called collusion,
it's illegal.

So they really are
simultaneously making

independent choices as
to how much to cooperate

with their ''competitors''.