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Hi, we're now on our last lecture on path dependence and what I want to do is I'm

Â gonna relate path dependence to something else we've studied in this class which is

Â tipping points, and one of the many really fun things about models is once we've got

Â a bunch of models we can compare and contrast them. >> And if we think for a

Â moment about path dependent ticking points. They seem very closely related

Â concepts. So let's remind ourselves of what they are, what does path dependence

Â mean? Path dependence means that what happens along the way determines the

Â outcome. So formally we define this as outcome probably is going to depend on

Â what's happened in the past and we made a distinction between path open and outcome.

Â So does the outcome in this particular instance depend on what happened and then

Â path and equilibrium is what we have in the long run depend on what happens along

Â the way. So we think about elevating these to tipping points it's this one we're

Â going to want to. Focus on These path dependent equilibrium. It's the fact that

Â like, what happens in the long run depends on what happens along the path. Cuz let's

Â think about how we define tipping points. When you defined tipping points, we had

Â two types. We had these direct tips where the action itself moved things and then we

Â had contextual tips, like in the case of the forest fire model. So here I'm going

Â to relate it to the active tips, the direct tips, where somebody takes an

Â action that changes the probabilities of things happening. So remember in active

Â tip, we started out with, there's a 50 percent chance it could go to the left and

Â a 50 percent chance it could go to the right, and then if there's a little bit of

Â a tip this way, this then be. Becomes 100 percent and this then becomes zero%. So

Â this is saying the equilibrium of the system is now really likely to be over on

Â this side and it's not likely to be over here. So that's related our notion of

Â path-dependent equilibrium. Then the question is, what's the difference between

Â path-dependence? And tipping points, cuz in each case it seems like something that

Â happens along the way has an effect. Well let's think about how we measure tips. A

Â tipping point was a single instance in time where, where that long, long

Â equilibrium was gonna be suddenly changed drastically. So think about path depended.

Â Path dependent means what happens along the way. As you move along that path, how

Â does that effect where we're likely to end up. So each step may have a small effect,

Â but it's the accumulation of those steps that has the difference. With tipping

Â points, everything sort of moves along in expected ways but not getting a lot of

Â information. Then there's a singular event that suddenly tips the system abruptly

Â from case to the other. So you we measure tips, what we do is we have these measures

Â of uncertainty. We use the diversity index. Which just gave us the measure,

Â sort of what's the, sort of counter number, of different equilibrium we could

Â go to Or we used [inaudible] which was another measure we had that told us

Â uncertainty there was. How much information there was in the system. When

Â we measured tips we talked about there being an abrupt change in the likelihood

Â of outcomes. So let's see, just for fun. Let's go back and let's look at our

Â [inaudible] process and let's think about whether that really is [inaudible]

Â dependent or whether it has a tipping point. Whether there's initial decision.

Â Whether there's sort of events along the way that have a big effect on what's going

Â to happen. So remember in our. Process, right? We have in urn and we're picking

Â out. Red balls and I'm picking out blue balls, and if I pick out a red ball and I

Â add another red ball. So that's our player process. We wanna see is this thing path

Â dependent or does it have these abrupt changes that leads to tipping points. We

Â know we got this result that says, any probable distribution of red balls as an

Â equilibrium and it's equally likely. So if we want to think about doing a player

Â process and we think about how it works. So let's suppose I draw four balls from

Â the urn. And if I draw four balls from the urn, there's five things that could

Â happen. I could get zero red balls, I could get one red ball, I could get two

Â red balls, I could get three red balls, or I could get four red balls. Now the

Â probability of that. Since we know from that previous claim, they're all equally

Â likely. So the probability of each one of those is one-fifth. So my diversity index,

Â [inaudible] number was one over the square root of those probabilities squared. So

Â that's gonna be one over one-fifth squared, plus one-fifth squared, plus

Â one-fifth squared. Plus one fifth squared, plus one fifth squared. So that's equal to

Â one over five times one fifth squared. So, that's one over five, over five squared,

Â which is one over one over five, which is five. Well, we already knew that right? If

Â we got an equal distribution over five outcomes the diversity index just equals

Â five. So, diversity index equals five when we start this process. Okay, so let's,

Â let's suppose that the first ball I choose is red. Okay, let's work through the math

Â and what's gonna happen. Now I can say, remember I'm starting out with two red

Â balls. And one blue ball. Because I'm picking the first ball red. And I wanna

Â ask, what are the different outcomes I could get? Well one thing I could do I

Â could pick all red balls, in the next three periods and end up with four reds.

Â So what are the odds that I get four reds? Well that's gonna be there's a two-thirds

Â chance, that this first ball's red. There's a three-fourths chance the second

Â ball's red. And there's a fourth-fifths chance the third ball's red. So if I

Â cancel all that stuff out I end up getting there's a two-fifths chance, of ending up

Â with four red balls. Now I can ask, what are the odds that I get Two reds and a

Â blue? Well again here two reds and a blue, it could go blue red, red, red blue red or

Â red, red blue. They're all equally likely. So let's just do one of them. The odds of

Â getting the red ball are gonna be two thirds, the odds of getting the next red

Â ball are three fourths and the odds of getting the blue ball are one fifth. So if

Â I cancel all this out I get one over ten. Remember there's three possibilities of

Â that, the blue ball can be here, here or here so I get that there's a three tenths

Â chance. Three over ten, then I get three red [sound]. And I could say, what are the

Â odds that I get one red and two blue? Well again the odds of picking a red ball are

Â two thirds, the odds of picking the blue ball are one fourth for the first one, two

Â fifths for the second, so we'll end up getting here is. This, these things cancel

Â out, so I get one over fifteen. But remember again, there's three

Â possibilities of where the red ball can be. So then I multiply that by three. So

Â that's one fifth. So that two over ten. So there's a two tenth chance that I get two

Â red. And finally I can ask what's the odds, what's the probability that I get

Â all three blue, and here the probabilities are. There's a one third chance of getting

Â the first blue, a two fourth chance of getting the second blue and a three fifth

Â chance of getting the third blue and these things cancel out, and I end up getting

Â one over ten. So what I end up with is the probability of getting four red. Is four

Â over ten, probability three red is three over ten, probability getting two red is

Â two over ten, and the probability of getting one red. Is one over ten. So this

Â is my new probability distribution. So let's go back on computer diversity index.

Â So remember initially, this is what we had initially. We had a diversity index equal

Â to five. Now we've got four-tenths, three-tenths, two-tenths, and one-tenth.

Â So how do we compute the diversity index? We just take one over four-tenths squared.

Â >> Plus three tenths squared [sound], plus two tenths squared [sound], plus one tenth

Â squared [sound], and that's going to equal one over sixteen over 100 [sound] plus

Â nine over 100 [sound], plus four over 100 [sound], plus one over 100. So that's

Â equal to one over 30 over 100 which is equal to 100 over 30, Which means that now

Â our diversity index is three and a third. Well, if we think about this, we started

Â off with an adversity index of five. Now we have an adversity index of three and a

Â third. So this movement from five. To three and a third suggests that something

Â happened along the path affecting where we're gonna go. There's path dependence

Â but it not an abrupt tipped. In abrupt tipped would be if we went from five to,

Â say, one point two or five to one where one single event get rid of a whole bunch

Â of uncertainty. So really, the difference between path dependence and different

Â points is one of degree. When you think of path dependence, what we mean is things

Â along the path change where we're likely to go. So we move from, you know, this set

Â of things to this other set of things but in a gradual way. Tipping points mean that

Â there's an abrupt change. [inaudible] something, there was a whole bunch of

Â things we could have done. Now we're likely to move to something entirely

Â different, or something that was unexpected, or that uncertainty got

Â resolved because of what's gonna take place. Okay, so we've covered a lot here

Â with tipping points. We've covered path dependent equilibrium, path dependent

Â outcomes, path dependence, fact dependence. Markov processes, chaos,

Â increasing returns and now tipping points. And we've done most of this using simple

Â earn models, which is nice, cuz there's a lot going on in this. In this area, right.

Â There's a whole bunch of different concepts related to [inaudible] but the

Â nice thing is most of this stuff we're able to understand through this very

Â simple model using urns. And this is one of the real advantages of using models,

Â right. We had this sort of amorphous idea of path dependence. We thought it was

Â related to something like increasing returns. It also seems somehow logically

Â close to notions related to chaos and to tipping points and it seemed not unlike

Â our markup process model. What we're able to do by constructing these simple urn

Â models is to flesh out all the differences between the concepts and really get a

Â deeper, more subtle understanding of exactly what path-dependence is and even

Â use some of our measurements for tipping points to see exactly how path dependence

Â unfolds. Okay, thanks a lot.

Â