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.