# More on Nash equilibrium

## Video transcript

The idea of a Nash equilibrium
is important enough that I think it
deserves its own video. And you may or
may not know, it's named for John
Nash, who was played by Russell Crowe in the
movie "A Beautiful Mind." And it's a game
theoretical concept. And game theory
sounds very fancy, but it really is just
the theory of games. And this prisoner's dilemma
that we talked about in the previous video
really is a game. The different players
have different strategies, and based on their
interacting strategies, you end up in different states. You end up with
different outcomes. And here's a definition of Nash
equilibrium from Princeton. And that's a good place
to get the definition, because that's where John Nash
spent a good bit of his career. And it is defined, or
this definition says, it's a stable state
of a system that involves several
interacting participants-- in our prisoners' dilemma,
we had two participants-- in which no participant can
gain by a change of strategy as long as all the other
participants remain unchanged. So let's think about the
different states of this system right over here and think
of whether any of them meet this criteria. So let's say, let me
number this one-- let's say that is state 1. This is state 2. This is state 3, and this
is state 4 right over here. So if we are sitting, if
we are sitting in state 1, can any of the
participants change their strategy-- can gain
by changing their strategy, assuming the other
participant is constant? So if we are sitting here in
state one right over here, so Al is denying and
Bill is denying-- well, Al can improve his situation
by changing his strategy. He can go from denying,
which is the scenario here, to confessing. So Al can change his
strategy and gain. Or you could go the other. Assuming Bill was
constant in the denial-- or you could go the other way. If we're sitting here in state
4 and we assume Al is constant, Bill can improve his situation
by going from a denial to a confession. He can go from two
year to one year. So for both of those reasons--
if either of those were true, this would not be
a Nash equilibrium. But both of those are true. So this is definitely
not a Nash equilibrium. I gave two examples in
which a participant can gain by a change of strategy as
long as the other participant remains unchanged. This move was one example,
and this was a move by Al, with Bill's denial constant. This was a move by Bill,
with Al's denial constant. Not a Nash equilibrium. Now let's think about--
let's think about state 2. If we are sitting in state
2, assuming Bill is constant, can Al change to
improve his outcome? So can Al change to
improve his outcome? In state 2, Al was
only getting one year. If Al goes from a
confession to a denial he's going to get two years. So Al cannot change his
strategy and get a gain here. So far it's looking good. But let's think of it
from Bill's point of view. So if we are in state 2, if we
are in state 2 right over here, and we assume Al is
constant, can Bill do something that
changes things? Well, sure, Bill can go
from denying to confessing. If he goes from denying
to confessing he goes from 10 years in prison
to three years in prison. So I've given an example
of a participant who can gain by a change of
strategy as long as all of the other participants
remain unchanged. Both of them don't have
to be able to do this. You just need to
have one of them for it to not be a
Nash equilibrium. Because Bill can have a
gain by a change of strategy holding Al's strategy
constant, so holding Al's strategy in the
confession, then this is not a Nash equilibrium. So this is not Nash,
because you have this movement can occur to a
more favorable state for Bill holding Al constant. Now, let's go to state 3. Let's think about this. So if we're in state 3,
so this is Bill confessing and Al denying-- so let's first
think about Al's point of view. If we assume Bill is
constant in his confession, can Al improve his scenario? Well, sure. He can go from
denying, which is what would have to be in
state 3, to confessing. So he could move in this
direction right over here. And that by itself
is enough evidence that this is not a
Nash equilibrium. We don't even have
to think about Bill. And it's symmetric. There's actually nothing that
Bill could do in this scenario, holding Al constant, that
could improve things. Bill would not want to
go from here to here. But just by the fact that Al
could go from here to here, holding Bill constant,
tells you that this is not a Nash equilibrium. Now let's go to scenario 4. And you know where
this is going, because you watched
the last video. But it's a little--
I'm going through it in a little bit in more detail. In state 4, they
are both confessing. Now let's look at it
from Al's point of view. And we're going to
hold Bill constant. We're going to hold
Bill unchanged. So we're going to have
to stay in this column. We're going to say that,
assume that Bill's confessing. From Al's point of view,
if we are in state 1, can he change his strategy
to get a better outcome? Well, the only thing he could
do is go from a confession to a denial. But that's not going to do good. He's going to go from
three years to 10 years. So Al cannot gain by
a change of strategy, as long as all the
participants remain unchanged. Now let's think about it
from Bill's point of view. We're in this state
right over here. We're going to assume
that Al is constant, that Al is in the
confession mode. So Bill, right now, in
state 4, is confessing. His only option is to deny. But by doing that, he'll go
from three years in prison to 10 years in prison. So he's not going to gain. So he, too, cannot gain. So we've just found
a state in state 4 which no participant can
gain by a change of strategy as long as all other
participants remain unchanged. And this part is important. Because we're not saying that
both can change simultaneously. You are not, in
this payoff matrix, allowing a diagonal move. And so no participant can
gain, neither Al nor Bill, holding the other one constant. This is a Nash equilibrium. This one right here. And this is a stable state.