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.