>> 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''.