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Path Dependent or Tipping Point
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Do curso por University of Michigan
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Path Dependence & Networks
In this set of lectures, we cover path dependence. We do so using some very simple urn models. The most famous of which is the Polya Process. These models are very simple but they enable us to unpack the logic of what makes a process path dependent. We also relate path dependence to increasing returns and to tipping points. The reading for this lecture is a paper that I wrote that is published in the Quarterly Journal of Political Science
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Scott E. PageProfessor of Complex Systems, Political Science, and Economics
Center for the Study of Complex Systems
Hi, we're now on our last lecture on path dependence and what I want to do is I'm
gonna relate path dependence to something else we've studied in this class which is
tipping points, and one of the many really fun things about models is once we've got
a bunch of models we can compare and contrast them. >> And if we think for a
moment about path dependent ticking points. They seem very closely related
concepts. So let's remind ourselves of what they are, what does path dependence
mean? Path dependence means that what happens along the way determines the
outcome. So formally we define this as outcome probably is going to depend on
what's happened in the past and we made a distinction between path open and outcome.
So does the outcome in this particular instance depend on what happened and then
path and equilibrium is what we have in the long run depend on what happens along
the way. So we think about elevating these to tipping points it's this one we're
going to want to. Focus on These path dependent equilibrium. It's the fact that
like, what happens in the long run depends on what happens along the path. Cuz let's
think about how we define tipping points. When you defined tipping points, we had
two types. We had these direct tips where the action itself moved things and then we
had contextual tips, like in the case of the forest fire model. So here I'm going
to relate it to the active tips, the direct tips, where somebody takes an
action that changes the probabilities of things happening. So remember in active
tip, we started out with, there's a 50 percent chance it could go to the left and
a 50 percent chance it could go to the right, and then if there's a little bit of
a tip this way, this then be. Becomes 100 percent and this then becomes zero%. So
this is saying the equilibrium of the system is now really likely to be over on
this side and it's not likely to be over here. So that's related our notion of
path-dependent equilibrium. Then the question is, what's the difference between
path-dependence? And tipping points, cuz in each case it seems like something that
happens along the way has an effect. Well let's think about how we measure tips. A
tipping point was a single instance in time where, where that long, long
equilibrium was gonna be suddenly changed drastically. So think about path depended.
Path dependent means what happens along the way. As you move along that path, how
does that effect where we're likely to end up. So each step may have a small effect,
but it's the accumulation of those steps that has the difference. With tipping
points, everything sort of moves along in expected ways but not getting a lot of
information. Then there's a singular event that suddenly tips the system abruptly
from case to the other. So you we measure tips, what we do is we have these measures
of uncertainty. We use the diversity index. Which just gave us the measure,
sort of what's the, sort of counter number, of different equilibrium we could
go to Or we used [inaudible] which was another measure we had that told us
uncertainty there was. How much information there was in the system. When
we measured tips we talked about there being an abrupt change in the likelihood
of outcomes. So let's see, just for fun. Let's go back and let's look at our
[inaudible] process and let's think about whether that really is [inaudible]
dependent or whether it has a tipping point. Whether there's initial decision.
Whether there's sort of events along the way that have a big effect on what's going
to happen. So remember in our. Process, right? We have in urn and we're picking
out. Red balls and I'm picking out blue balls, and if I pick out a red ball and I
add another red ball. So that's our player process. We wanna see is this thing path
dependent or does it have these abrupt changes that leads to tipping points. We
know we got this result that says, any probable distribution of red balls as an
equilibrium and it's equally likely. So if we want to think about doing a player
process and we think about how it works. So let's suppose I draw four balls from
the urn. And if I draw four balls from the urn, there's five things that could
happen. I could get zero red balls, I could get one red ball, I could get two
red balls, I could get three red balls, or I could get four red balls. Now the
probability of that. Since we know from that previous claim, they're all equally
likely. So the probability of each one of those is one-fifth. So my diversity index,
[inaudible] number was one over the square root of those probabilities squared. So
that's gonna be one over one-fifth squared, plus one-fifth squared, plus
one-fifth squared. Plus one fifth squared, plus one fifth squared. So that's equal to
one over five times one fifth squared. So, that's one over five, over five squared,
which is one over one over five, which is five. Well, we already knew that right? If
we got an equal distribution over five outcomes the diversity index just equals
five. So, diversity index equals five when we start this process. Okay, so let's,
let's suppose that the first ball I choose is red. Okay, let's work through the math
and what's gonna happen. Now I can say, remember I'm starting out with two red
balls. And one blue ball. Because I'm picking the first ball red. And I wanna
ask, what are the different outcomes I could get? Well one thing I could do I
could pick all red balls, in the next three periods and end up with four reds.
So what are the odds that I get four reds? Well that's gonna be there's a two-thirds
chance, that this first ball's red. There's a three-fourths chance the second
ball's red. And there's a fourth-fifths chance the third ball's red. So if I
cancel all that stuff out I end up getting there's a two-fifths chance, of ending up
with four red balls. Now I can ask, what are the odds that I get Two reds and a
blue? Well again here two reds and a blue, it could go blue red, red, red blue red or
red, red blue. They're all equally likely. So let's just do one of them. The odds of
getting the red ball are gonna be two thirds, the odds of getting the next red
ball are three fourths and the odds of getting the blue ball are one fifth. So if
I cancel all this out I get one over ten. Remember there's three possibilities of
that, the blue ball can be here, here or here so I get that there's a three tenths
chance. Three over ten, then I get three red [sound]. And I could say, what are the
odds that I get one red and two blue? Well again the odds of picking a red ball are
two thirds, the odds of picking the blue ball are one fourth for the first one, two
fifths for the second, so we'll end up getting here is. This, these things cancel
out, so I get one over fifteen. But remember again, there's three
possibilities of where the red ball can be. So then I multiply that by three. So
that's one fifth. So that two over ten. So there's a two tenth chance that I get two
red. And finally I can ask what's the odds, what's the probability that I get
all three blue, and here the probabilities are. There's a one third chance of getting
the first blue, a two fourth chance of getting the second blue and a three fifth
chance of getting the third blue and these things cancel out, and I end up getting
one over ten. So what I end up with is the probability of getting four red. Is four
over ten, probability three red is three over ten, probability getting two red is
two over ten, and the probability of getting one red. Is one over ten. So this
is my new probability distribution. So let's go back on computer diversity index.
So remember initially, this is what we had initially. We had a diversity index equal
to five. Now we've got four-tenths, three-tenths, two-tenths, and one-tenth.
So how do we compute the diversity index? We just take one over four-tenths squared.
>> Plus three tenths squared [sound], plus two tenths squared [sound], plus one tenth
squared [sound], and that's going to equal one over sixteen over 100 [sound] plus
nine over 100 [sound], plus four over 100 [sound], plus one over 100. So that's
equal to one over 30 over 100 which is equal to 100 over 30, Which means that now
our diversity index is three and a third. Well, if we think about this, we started
off with an adversity index of five. Now we have an adversity index of three and a
third. So this movement from five. To three and a third suggests that something
happened along the path affecting where we're gonna go. There's path dependence
but it not an abrupt tipped. In abrupt tipped would be if we went from five to,
say, one point two or five to one where one single event get rid of a whole bunch
of uncertainty. So really, the difference between path dependence and different
points is one of degree. When you think of path dependence, what we mean is things
along the path change where we're likely to go. So we move from, you know, this set
of things to this other set of things but in a gradual way. Tipping points mean that
there's an abrupt change. [inaudible] something, there was a whole bunch of
things we could have done. Now we're likely to move to something entirely
different, or something that was unexpected, or that uncertainty got
resolved because of what's gonna take place. Okay, so we've covered a lot here
with tipping points. We've covered path dependent equilibrium, path dependent
outcomes, path dependence, fact dependence. Markov processes, chaos,
increasing returns and now tipping points. And we've done most of this using simple
earn models, which is nice, cuz there's a lot going on in this. In this area, right.
There's a whole bunch of different concepts related to [inaudible] but the
nice thing is most of this stuff we're able to understand through this very
simple model using urns. And this is one of the real advantages of using models,
right. We had this sort of amorphous idea of path dependence. We thought it was
related to something like increasing returns. It also seems somehow logically
close to notions related to chaos and to tipping points and it seemed not unlike
our markup process model. What we're able to do by constructing these simple urn
models is to flesh out all the differences between the concepts and really get a
deeper, more subtle understanding of exactly what path-dependence is and even
use some of our measurements for tipping points to see exactly how path dependence
unfolds. Okay, thanks a lot.
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