In the last lesson,

we've learned that
every game has players,

actions or strategies,
and payoffs.

In this lesson, we're going to see
how to represent such a game

in a normal form.

And we're gonna do that
with an example

known as the Prisoner's Dilemma.

So the scenario of this follows:
there are two people who

committed
a crime together.

Maybe they've robbed a bank.

Unfortunately for them, they were caught.
And what the cops do

when they catch them
is they put them in the separate rooms

to interrogate them.

Each person then has one of two options.

They can either
confess to their crimes,

or they can keep their mouths shut
and not say anything.

The interesting part comes in,
is that the cops

don't quite have enough information

to put them away
for the full bank robbery.

So if they both keep quiet, then they'll
get away with the reduced sentence.

Hovewer, if they confess,
then the cops will

put them away
for a longer time.

And the key here is
if one player confesses,

the player that confesses is rewarded
for blowing his partner in.

So we're gonna
formalize this in the second,

but this is the basic
notion of what happens.

So, what are the players in this game?

Well, we can represent them
as criminal-1 and criminal-2,

but I'm just gonna use
player-1 and player-2.

And actions they can take

are either confess or don't confess.

I use 'D' to represent "Don't confess"

Now I want to put a little
note here on terminology.

Sometimes you'll see 'C' and 'D'
represent "Collaborate" and "Defect".

I was taught "confess"
and "don't confess."

When you use the terminology
"Collaborate" and "Defect",

the payoffs are flipped.

But if you're ever confused,
just look at the payoffs.

But in our cases, we always use
"Confess" and "Don't confess."

Okay, so now we wanna write
exactly what the payoffs are.

So, what are the payoffs to player-1?

So player-1's payoff

when they both confess,
when player-1 confesses

and player-2 confesses,

is equal to 2.

And we will say that this number
represents the years off

of the sentence that they
receive, 'cause remember,

player's won higher reward,

so we'll use the payoffs to mean
the years reduced off their sentence.

Now, player-1's payoff

when he confesses,
but player-2 doesn't confess,

is four.

Remember, player-1 is being rewarded
for blowing his partner in

when his partner's
being quiet.

So the cops are able to put
player-2 away for a long time,

player-1 gets a reduced sentence
because he confesses to the crime.

How about player-1's payoff...

when he doesn't confess, he stays quiet,

but player-2 confesses to the crime

and blows
his partner in.

In that case, player-1 only gets
1 year reduced off of his sentence.

Finally, if they both keep
their mouths shut

player-1 doesn't confess,
player-2 doesn't confess,

then they get 3 years
reduced off of their sentence.

Maybe they don't get
a full bank robbery charge

but they get something kind of

like an aggravated assault, something
that is a lesser charge than bank robbery.

Now I really think
what would that might be.

Okay, so now, this is the game,

we have the players,
the actions, and the payoffs.

I wanna mention also that player-2's
payoffs are going to be similar,

so what I mean is that
this is a symmetric game,

so when they both confess
player-2 also gets a payoff of 2,

when they both don't confess
player-2 also gets a payoff of 3,

and then these
are flipped for player-2.

So, for example, if player-1 doesn't
confess and player-2 does confess,

and player-2 gets a reward of 4
for blowing his partner in, and so on.

Okay, now we wanna represent
this game in its normal form.

So what I've drawn here is the players,
player-1 and player-2,

and each of their actions:
confess, don't confess.

The only thing left for a full
normal form specification

is to fill in
the payoffs.

So let's do player-2 first,
player-2 is red.

So, if they both confess,
player-2 confesses and player-1 confesses,

we've said that player-2 gets a reward,
in terms of reduced sentence, of 2.

If player-2 confesses
but player-1 keeps quiet,

so player-2 tell on player-1,

player-2 is rewarded
with the reduced sentence of 4.

If player-2 doesn't confess
but player-1 does confess,

player-1 blows in player-2,

then player-2 only gets
a reduced sentence of 1 year.

And finally, if they both keep their
mouths shut, nobody says anything

they both don't confess —
player-2 gets a reward of 3.

Now, we do the same
for player-1.

We've said that if they both confess,
player-1 also gets a reward of 2.

If player-2 confesses,
and player-2 doesn't confess,

then player-1 gets a reward of 1,
because player-2 told on him.

Similarly, if player-2 confesses —

I'm sorry, if player-1 confesses
but player-2 doesn't say anything,

player-1 gets rewarded
for telling on player-2

so he gets 4 years in terms of
his reduced sentence.

Finally, if they both
don't say anything,

they both get 3 years
reduced off of their sentence.

So, this is a normal form representation
of the Prisoner's Dilemma.

We have the players:
player-1 in blue,

player-2 in red;

the actions: confess
and don't confess;

and the payoffs.

Again, for example, we see
that if player-1 confesses

and player-2 confesses,
they both receive the payoff of 2.

And we can to this for any
possible strategy combination

to get the payoffs to both players
for any joint profile.

In a later lesson,
we're actually gonna find out

what the Nash Equilibrium
of this game is and solve it.