>> 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''.