Hi. In this set of lectures, we're talking
about Lyapunov functions. And Lyapunov
functions give us a way to explain or
understand why some processes go to
equilibrium. What I wanna do in this very
short lecture is just clean up two
details, two questions that might be out
there lingering in the minds of people.
The first question is this: Can we say how
long it's gonna take the process to go to
an equilibrium? So we know it's going to
an equilibrium. Can we say exactly how
fast? We'll talk about that. And then
second: Does the process always stop at
the max or the min? So, remember when we
talked about Lyapunov processes, they can
either always be going up or always be
going down, until they reach an
equilibrium. The question is, if they're
going down, do they automatically hit the
floor? Do they necessarily hit the floor?
Could they stop above it? And if they're
going up, do they necessarily hit the top
or could they stop below it? That's the
second question. Both pretty straightforward. 
Both fairly easy to answer, but
it's worth cleaning up those details.
Let's look again at the formal definition
of the Lyapunov function, just so we can
answer these questions precisely. So some
function F, it has a maximum value.
If the process stops, then it's in
equilibrium. If it doesn't stop, then its
value according to [inaudible] increases by some amount
that has a value, by some amount at least
k. So it goes up at least k. So that means
that either the process is stopped or it's
going up k. And since there's a maximum, that
means that at some point the process has
to stop. So now I also get this question:
How long until the process reaches
equilibrium? That's really a fairly
easy question to answer. Suppose that we
start out with F of x1 equals 100, and
k equals 2. So that means that the
original value of the function is 100, k
equals 2, then I suppose the maximum
equals 200. So that means starting out at
100, the highest you can go is 200, and
it's got to go up at least 2 each
period. Well, what we can say is, is that
the number of periods has to be less than
50, because it's gotta go up at least two,
and the most it can go up is 100, and so
100 divided by 2 equals 50. So what we
get is, the maximum number of periods is 50. So
when we write down Lyapunov function, if
we could make k as big as we can possibly
make it and make the maximum as small as
we can possibly make it, then we can put a
bound on the number of periods. We can't
say for sure. The process could stop in
one period. It could stop in two periods.
It could stop in 47 periods. But we can
say for sure, is that the number of periods
is going to be less than 50. Now could
be, if I think really hard about the
model, I realize that, you know what, k
is not 2, but actually k is 4. That I
can show, it's got to go by at least 4 each
period. Well, if that's the case, then
instead of the number of periods being
less than 50, you could show it's less than
25. So if you want to put as tight a bound
as you possibly can on the time it's
going to take the Lyapunov process to converge,
what you want to do is: make k as big as
you can make it, and make that maximum value as
low as you can make it, and that will help
you make a tighter bound on how many
periods it's going to take to get to equilibrium.
But putting that bound on once you know
k and the maximum value and [inaudible]
minimum value as well. The
starting value is really straightforward,
it's just some really simple algebra.
The other question is a little bit harder. That is,
does the process necessarily reach a max
or minimum. And the answer here is going
to be no. Now that first reading where
people were choosing routes, in terms of
where to go in the city. That one it did
go to a minimum, it did go to
an efficient case always. We didn't prove
it but you can show that that's true. But
generally that's not going to be the case,
generally it can be the case that a
process can get stuck someplace less than
the max. So I'm going to explain this in
two ways. Let's go back and talk about a
rugged landscape model. Remember, in a rugged
landscape model there were peaks, so
here's a peak, here's a peak, here's a
peak. You can think of a Lyapunov function
as saying, I'm going to step up at least
some distance each period. It doesn't
necessarily mean that you're gonna get to
the highest peak. You could just go up a
particular hill, and it could be a
suboptimal peak. It doesn't seem necessary
that these processes would take you to the
maximal point, take you to the optimal
point. Again, there's a difference between
metaphor, and actually having a
mathematical example. So let's see if we
can come up with an example to show where
we get stuck at less than the optimal
point. To do that we're gonna go back to
our preference model. So you might be
noticing at this point, "uh-oh, I better have
paid attention to the earlier lectures",
cause you've done Langton's model, now we're
gonna do the preference model. So remember
in the preference model, individuals had
preference over different things. So
person one here. Here's person one. Person
one. They like apples more than bananas,
more than coconuts. And here's person two.
And they like bananas, coconuts, and then
apples. And then, here's person three.
They like coconuts, apples, and then
bananas. Let's suppose the following is
true: Person one has a banana. Person two
has a coconut. And person three has an
apple. And now we're going to have an
exchange market. We're going to ask, do
they want to trade? Can they trade to make
themselves happier? Well, it's clear they
could trade and make themselves happier,
but let's see if they can do it. So person
one is saying, "but I would like to have
the apple". Person one goes over to person
three and says, "hey, how about if I give
you this banana for your apple". And person
three says, "a banana? No way, 'cause I like
apples more than bananas." So they reject
the trade. So person one can't make any
trade that makes him better off. Person
two has the coconut, but they'd rather have
the banana. So they go to person one whose
got the banana and says, "hey person one,
would you like to have my coconut? How
about my coconut for your banana?" And
person one says, "the coconut? No way. I
like my banana more than the coconut. So
forget it." So no one's, person one is
not going to trade with person two. Now
person three has got the apple, and they'd
like person two's coconut. So they go to
person two and say, "hey person two, how
about if I give you my apple for your
coconut?" And person two looks and says,
"the apple? Forget it, I like my coconut
more than the apple, I'm not gonna trade
with you." So what we've got here is a
situation where person one has the banana,
person two has the coconut, person three
has the apple, none of them can make a
pairwise trade and be better off. One way
to understand that metaphorically, right, is
to think, here's the landscape where
they've got these certain things. They got,
they're stuck at this point. There's some
place they could get, they could be
better. It could be that if person one had
the apple, person two had the banana,
person three had the coconut, they'd be
better off. But they can't get there by
pairwise trades. They could do it if they
had a more sophisticated trade, where they
put all three things on the table and each
person grabbed the thing they wanted.
But through pairwise trades, they don't
get there. So what have we learned? What
we learn is that it's at least possible 
to put a Lyapunov function on a process and have
it stop at somewhere less than the optimal
point. Doesn't have to stop at the optimal point, it could stop below. That's
what we're seeing here. So we've answered
two important questions. The first
one is: Okay, we know it goes to equilibrium,
can we say how fast? And the answer is
yes. And the better bound we get on k, and
the better bound we get on the max, the
more accurately we can put a restriction
on how fast, how long it's going take. So,
we can put a tighter bound on how long
it's going to take, if we can estimate k
accurately, and if we can estimate the
maximum value accurately. We also learned
that it can stop a lot faster than that,
because of the fact that the process may
not get to that optimum value. Some
processes get stuck in suboptimal points,
and at least metaphorically, you can
understand it's being stuck in a landscape,
at a suboptimal peak, instead of climbing
the mountain, you're getting stuck
somewhere below the optimal point. Okay,
where we're going to go next, is we're
going to talk about another lingering
question. And that is, are there processes
where we don't know whether they go to
equilibrium or not? And the answer to
that, surprisingly, is going to be, yes.
There's some where it's just sort of hard
to figure out. [laugh] All right? Thanks.
