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

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

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

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

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

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

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

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

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

the sequence of previous events influencing not only outcome in this

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

dependence is that the outcome probabilities depend on the sequences of

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

physical recursive that produce this extreme sensitivity to initial conditions.

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

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

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

Okay. Thank you very much.