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Hi. Welcome back. In this lecture, I wanna flesh out a few more of the particulars about

Â the concept of path dependence. I want to relate it to an earlier notion of the Markov process,

Â which wasn't in any way path dependent. Remember we got a unique

Â equilibrium in that case. And I also want to relate it to chaos. And then finally

Â what I'd like to do is just flesh out a little bit more about why the distinction

Â between path dependence and phat dependence is so important, outside of the

Â context of the urn model. So, I wanna talk about it sort of in a real world setting

Â as opposed to within the context of that simple urn model. Okay, so let's get

Â started. Remember, when we talk about path dependence, what we're talking about is

Â the sequence of previous events influencing not only outcome in this

Â period, but possibly the long run equilibrium. So our definition of path

Â dependence is that the outcome probabilities depend on the sequences of

Â past outcomes. So in the case of a path dependent outcome where you see even the

Â outcome depends on it. In the case of path dependent equilibrium we're saying that

Â the long run equilibrium depends on the path past outcomes. Now remember when we

Â studied Markov processes in the previous lecture and in the Markov processes we

Â always got a unique equilibrium. And in the Markov process we made the following

Â assumptions. We said there's a finance head of states. We said there's fixed

Â transition properties within those states that you could get from any one state to

Â any other. And that it wasn't as. Simple cycle. So it didn't go A, B, C, A, B, C,

Â A, B, C. And then, given those assumptions, we get something called a

Â mark off conversions theorem that said that given A1 to A4, Markov process

Â converges to an equilibrium that's unique. Now remember that was a [inaudible]

Â equilibrium so it was moving between the states. It was still churning, but it was

Â a unique equilibrium. So why aren't Markov processes path dependent? Well, here's

Â why. This assumption right here, fixed transition probabilities. Remember in our

Â urn model, we've got this urn and as we, we've got red and blue balls in here. And

Â as we pick more red balls, we start adding more red balls. So the transition

Â probability. Change. So it's those fixed transition probabilities that are really

Â underpinning why the mark off process goes to a unique, even though it's a stochastic

Â equilibrium, goes to a stochastic equilibrium because we're not changing the

Â probabilities. The history of events doesn't change the probabilities. So, now

Â by comparing these two models, we see sort of, when does history matter? History

Â matters when it changes the transition probabilities. There's two ways of seeing

Â the effects of history on outcomes. One through the mark off process, by saying

Â it, history doesn't matter if the probabilities don't change. And the other

Â is through the urn models by showing history does matter if the outcome. Those

Â change. The probabilities change. What I want to do next, is relate this to chaos.

Â Now relating to chaos I got to begin by describing some recolor of recursive

Â functions, so recursive functions were sort of implicit in our mark off model and

Â in our urn model, but let me make it more formal. So in recursive function what

Â you've got is you've got an outcome at time T and there's an outcome function.

Â Math, and math acts into itself. So it's this process that's kind of moving on and

Â on and on. So you've basically got an outcome, you've got another outcome, and

Â another outcome, and another outcome. So one thing needs to be X plus two,

Â especially if you just go one, three, five, seven, nine, eleven. So in the urn

Â models that we had we had a precursor where we picked out, X could be either

Â blue or red, we picked out blue red, blue red, blue red. But what we got in each

Â period depended on what we picked previously. And in some cases on the whole

Â set. So in some cases, what you get in expect can only depend on a previous

Â variable, or it could depend on what happened in period one. What happened in

Â period two and what happened in period three. So you could have that X4 is a

Â function of period one, two, and three. That would be path dependent process. In

Â the simple recursive function, what happens in this period F of XT might only

Â depend on. We have X [inaudible]. Xt plus one might only depend on XT. So

Â [inaudible] only depends on the previous proof. We can use this regressive

Â functions, to describe processes that are chaotic. So when we talk about chaotic.

Â Chaos what we mean is, extreme sensitivity to initial conditions. So what that means

Â is if I start with two points. And are very, very close to one another. And then

Â I keep applying this recursive function, what I'm going to get is these paths are

Â going in very different ways. So two points that start near each other, end up

Â a long way away. So let's see an example of that, this is called a tent map. So let

Â X be in interval 01 and these round brackets mean that I don't include zero, I

Â don't include one. Now the function is defined as follows: F of X equals 2X, if X

Â is less than a half. So here's zero, here's one, here's one half. So it?s equal

Â to 2X if it's less than. And then it's 2-2x if X>1/2, so what that means is that

Â if X=1/2 I'm gonna get 2-, it actually equals one-half here so I'm gonna get one,

Â and an X=1/2 here I'm gonna get 2-1 which is also one so this looks like this and it

Â looks like. A tent. Hence the tent man. So the way it switches, if I start out at

Â point 2,1 and I apply it, I'm gonna get point 4,2. And then if I get to point 4,2

Â and I apply it, I'm gonna get point 8,4. Well then if I look at point 8,4 and I

Â apply it, then I'm gonna apply this. And I'm gonna get two minus two times. Point

Â 84 which is going to be two minus 1.68, which is going to be.32. So I'm going back

Â to.32 and I'm going to get.64. So that's how the tent map works. I just recursively

Â go through the function. Here's an example of the tent map, where I start with two

Â points that are very similar to each other. One is .4321, the other is .4322.

Â Well, again, first I double it, then I apply 2-2X. And then I double that, 'cause

Â it's less than a half. And I apply 2-2X and so on. And notice after I do that just

Â a few periods, these two points are now a long way away from each other. So the tent

Â map ends up being chaotic, because there's extreme sensitivity [inaudible] initial

Â condition, just by being a [inaudible]. Teeny bit different on the fourth decimal

Â point, you end up a long way away just after eleven iterations of the function.

Â Now you can see this graphically as follows. Originally you can't even see the

Â blue line, because it's hidden behind the red line. So, this is the same things I

Â just put in. And over time, these two paths end up being very different. This is

Â extreme sensitivity to initial condition. Notice this is not path dependence. Why is

Â this not path dependence? Well let's go back. This tent map is just a fixed

Â recursive function. Once I choose my initial point, once I choose my four,

Â three, two, one, or .4322, then I know exactly what's gonna happen. So this is

Â extremely sensitive to initial conditions but it's not path dependent because all

Â that matters is the initial point. Now to find the path as being the initial point

Â then yes it's path dependent, but nothing that happens along the way really has any

Â effect on what's going to happen in the long run because we know what's going to

Â happen. Once we choose the initial point we've just got a fixed function. So chaos,

Â in its standard form means extreme sensitivity to initial conditions. So the

Â initial point matters. And if I apply this function over and over, tiny differences

Â in the initial point will vary [inaudible], by a lot later on. Path

Â dependence means, what happens along the way influences the outcome. So it's

Â typically not a deterministic process, 'cause what happens along the way has an

Â impact on the outcome. [sound] So let's step way back for a second. We think of a

Â process as being independent, if outcomes don't in anyway, depend on, the past

Â history of outcomes. We can think of a process as depending on the initial

Â conditions if the outcome or state in a, in a later [inaudible] depends only on the

Â initial state. It's completely deterministic. So this, this independence,

Â is a probabilistic concept. That, you know, there's a 50 percent chance of

Â getting a red or blue ball each period. With chaos, extreme sensitivity to initial

Â conditions, we're saying, it's deterministic. We know what's gonna happen

Â once we get to the initial point. And all that matters is that initial point. Path

Â dependence means that the outcome probabilities, what happens in the long

Â run, depends on what happens along the way. And finally, we have, then, fact

Â dependence, means that outcome probabilities don't depend on the order in

Â which things happen. It only depends on the set of things. So in our [inaudible].

Â [inaudible] What happens in, if we've got 24 red balls. In six blue balls, sitting

Â in this urn. It doesn't matter what order they appeared in, all that matters is the

Â number that there are. So, that's the difference between. Path dependent and

Â fact dependent. Now when historians or, you know, institutional scholars think

Â about path dependence, they often think in terms of the sequence of events hap,

Â mattering. Not just the set of things mattering. They also think that things

Â aren't independent, and they think that although conditions matter, they're not

Â the only thing that matters. So they don't think it's the case that once we write the

Â Constitution, that, you know, what then plays out is completely deterministic. So

Â they tend to side with things being path dependent. Why? Why do they think. Path

Â dependence and nothing else. Well, independently there's no structured

Â history so that doesn't make any sense. Extreme sensitivity to initial conditions

Â in undeterministic process doesn't make any sense either. So that means that fate

Â is just completely predetermined by a few initial choices. So, it really comes down

Â to path versus fact. Path says the sequence matters, fact says the set

Â matters. Why do they think it's the path not the set? That's I think. Seemingly a

Â deep question. These nice urn models have held us, make us think about it. Well, one

Â reason that they think that this is too zippy, that early events. Have larger

Â importance. Let's think about some events. So let's suppose that this was what

Â American history looked like. In 1814 we gave women the right to vote and then in

Â 1823 we had a civil war to get rid of slavery. In 1880 we finished the

Â transcontinental railroad across the United States. In 1894 we find gold in

Â California. In 1923 we decide to buy the Midwest from France, so previously we put

Â this transcontinental railroad through, we had to negotiate with France to put it

Â through, what later became the Louisiana Purchase and then in nineteen. In 67 we

Â have a brutal war with England for independence. And it's hard to imagine,

Â that if this was the sequence of events, that I'd be sitting here giving you this

Â course right now. That it's probably the case with American society would look

Â extremely different then it does now. In particular, I might be speaking French,

Â because I would be in what was formerly the Louisiana Purchase, and we'd probably

Â look a lot more like Quebec. In here in the United in Michigan, than we look Then

Â I look now. Okay, so. When we think about these ideas. Path dependence. Fact

Â dependence. Independence. Sensitivity to initial conditions or chaos. What we see

Â is these simple models help us organize our thinking about the world might look

Â like. And we understand why a lot of historians focus so much on path

Â dependence. Because it seems the most reasonable. We also see why people who,

Â you know, study gambling in casinos consider independence. And we see why a

Â lot of physicists are interested in things like chaos, because there are actual

Â physical recursive that produce this extreme sensitivity to initial conditions.

Â So, these simple models help us make sense of a lot of concepts that are actually

Â fairly closely related logically, and the URN model, in particular, helps us make,

Â draw bright lines between path dependence, PHAT dependence and Independence.

Â Okay. Thank you very much.

Â