Hi, I'm William Spaniel. Let's learn some game theory.
Today's topic is the prisoners dilemma and the strict dominance solution concept.
I cover this in lesson 1.1 of my textbook can learn more about that
by clicking on the video description
in any case
here's the situation
two suspects are arrested
the police think that they're trying to rob a store but the cops only have the
evidence to prove that the suspects were trespassing
thus, to charge that the contacts with the greater crime the police are going to need
one of the criminals rat out the other
and the way the police are going out was that a confession is by throwing them
into the prisoners dilemma
which looks like this
each of the convicts is going to be privately sequestered in interrogation
room and offered the following deal
if no one confesses to the robbery
the police are only to be able to charge the prisoners with trespassing, and the
punishment for that is one month in jail a piece
but if one confesses in the other one doesn't
the police will be lenient on the rat thanking him for his confession
and severely punish the quiet one that's gonna be twelve months in jail for the
one who kept quiet and the one who ratted out the other guy
is going to be able to walk free--he'll spend no time in jail whatsoever
finally if both confess then the police can actually punish them for robbery
and then that's going to result in a much greater jail sentence than
trespassing and this time, it's going to be eight months in jail apiece
instead of one had the only been able to charge them with trespassing
so the question looks like this
supposed to be used only one minimize the number of months they spend in jail. we're making an
assumption here
we're supposing the criminals only care about themselves for the moment. if
that's the case should they confess to the police?
now we saw a lot of information in the last two slides but we can condense it
into just this little payoff matrix right here and this is what we call a prisoner's
dilemma
so we have two players: player one, player two
each of them has to strategies: they can either confess or keep quiet
and based off those strategies you end up at different outcomes so if both
keep quiet then they both spend one month in jail, so they both eagles get negative one right
there
if one confesses and the other one keeps quiet the one one who confesses, in this case player
two, gets no time in jail and player one gets twelve months in jail so that's a
negative twelfth and it's the reverse situation over here
and finally both if confess then you end up over where they both spend eight months in jail
and of course if you get confused about the payoffs, then just notice that these are
color coordinated where player one is
blue and player two is red
now also in game theory we have the following convention: we make player one a man and
player two a woman
the reason we do is so we can use pronouns like he and she to differentiate
between the players without
creating any extra confusion
so that aside, let's actually solve this game
and the way we're gonna do that is by noting how player one and player two should
respond
based off what the other player is going to do so let's look at this for a second like this
this is just highlighting player one's payoffs here
supposed player one knew that player two is going to keep quiet he can just
predict this. he saw into the future he knew that players two was going to keep
quiet. What should he do?
well if we look at the payoffs on the screen here, we see that he's better off
confessing than he is keeping quiet because he spends no time in jail if he confesses
and one month in jail if he keeps quiet. So 0 is greater than -1, he should
confess if player two keeps quiet.
now suppose player one knew that player two was going to confess
think about what he should do here.
Well, if he keeps quiet he will twelve months in jail; if he confesses, he will spent eight
months in jail
so in response to player two confessing player one is also better off confessing
if you string these two pieces of information together
whether player two keeps quiet or confesses, player one is always
better off confessing
confessing always leads to a better outcome. We say that player one's
confess strategy strictly dominates
his
keep quiet strategy and so in general strategy x dominates strategy y
if generates greater payoff than y regardless what the other players do
so in this case this is confess strictly dominating
keep quiet
so keep quite as strictly dominated
now, rational players never play strictly dominant strategies like keeping quiet
the reason for that is why would you want to keep quiet when you could confess and
do better and so in general why would you want to play y would you play x
instead?
and so in the future as we go through these videos we are going to make sure
that
these players are never going to be playing strictly dominated strategies--it just doesn't
make sense for them to ever do that
and so to finish solving the players' prisoner's dilemma here we know that
player one is going to confess
so all that's left is to figure out what players two is going to do here.
So imagine that player two knew that player one is going to keep quiet
we can see that she's in the exact same boat as player one is. If player one keeps
quiet she's also better off confessing because 0 is greater than -1
but on the other hand if player one is going to confess and player two could
see that in advance again players two would also want to confess because
-8 is better than -12. And so you end up starting out with four
different outcomes possible here but we know that there's only one sensible
outcome
and that is for both the players to confess
now when you first show this to people who are new to game theory, they will ask you "okay, that
doesn't make sense. Why they did this outcome right here
when this outcome when they both keep quiet is mutually better for them?"
And it's true this is a better outcome for both the players collectively, right?
Because -1
and -1 is better individually and collectively than confessing
and then you get eight months in jail apeace
but the reason that this outcome isn't sensible is because each of them can
individually do better by switching off to confess
and so there isn't as stable sort of solution here because they can always do
better by flipping their strategies and going to confess
even if they agree to do is in the in the time before they actually went
to the interrogation rooms
those sorts of agreements don't actually withstand the test of time because as
soon as they get into the room and actually had to confess or keep quiet
it's in their individual interest
to confess and not keep quiet
so that's why we think that this is the only outcome that's sensible when both of
the players confess
so that is the prisoner's dilemma and strict dominance in a nutshell
and in the next video we will start talking about how we can
use the strict dominance idea
to get to even more complicated games' solutions. Join me then.