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long-run distributions over the set of outcomes, that are path-dependent. So the

Â outcome in a period might depend on what happened in the past, that's fine. But

Â that's different from saying, the long-run equilibrium, where the system is gonna go,

Â is gonna depend on the, on the process. So I'm gonna make a distinction between

Â path-dependent outcomes and path-dependent equilibrium. And I'm gonna do this using

Â this very simple class of urn models. So what is an urn model? An urn

Â model is always simple, it's got an urn that contains blue balls and it contains

Â red balls, and you pick balls out of the urn. So I might reach this urn, and I might

Â pick out a red ball. That's it. And then it's the probability or picking out balls of

Â different colors. So the simplest urn model is called the Bernoulli model, and that

Â works a follows: You just got a fix number of balls in the urn. So this urn has three

Â red balls and one blue ball. The probability of picking up a red ball is 3/4.

Â So if you think about this, this model can be used to explain a lot of

Â casino games like roulette and blackjack: what are the odds of getting a

Â face card? And you can think that there's, you know, 52 cards in the

Â deck, and 16 of those cards are face cards. So there's a 16 over 52 chance

Â that you're going to get a face card. So that's it, so the Bernoulli models are

Â really simple, probabilities stay fixed. And the probability in each draw in a

Â Bernoulli model is independent. So if I get a red ball this time, put that ball

Â back in the urn, that doesn't have any effect on the probability of getting a red

Â ball in the next period. So when you say something's independent, it means: what

Â happens now doesn't depend in any way on what happened in the past. What we want to

Â do is take this simple ideal of an urn model and use it to construct some

Â path-dependent processes. And the most famous path-dependent process is called the Polya

Â process. And here's how it works: Start with an urn with two balls, one red, one

Â blue. You pick a ball. So maybe I pick the red ball out. After you pick that ball,

Â you look at it, see its color, and then you put in another ball of that same color. Then

Â you do it again. So maybe the next time I pick another ball out, and again it's

Â red, so I add another red ball. Well, then maybe then next time, I pick a blue ball out,

Â and then add in a ball that's blue. Now if we think about this Polya process, what's

Â gonna happen is that over time, these probabilities are gonna change. So if we

Â think about the Polya process in action, what was, what's gonna happen is, we're

Â gonna get lots and lots of balls in here. Lots of red balls, lots of blue balls,

Â depending on what I pick out. And the probability of balls, of the different

Â colors in the balls can be changed; it's going to start out that the probability of

Â red balls is the half. Then it might go to two-thirds as I pick another one. Then it

Â might go to three-fourths if I pick another red ball, but then if I pick a

Â blue ball it's gonna go to three-fifths. So what we see is, the probability of

Â picking a red ball is going to change over time, and it's going to be path-dependent.

Â It's going to depend on the path of previous choices. What I'd like to do now

Â is state two results about the Polya process. I'm not going to prove them. In

Â the next lecture I'll prove why they're true. In between this lecture and the next

Â lecture what I'd like you to do is see if you can figure out why they're the case.

Â So do some examples, with, you know, just with picking three balls from the urn and

Â four balls from the urn, you know, four sequences along the path, and see if you

Â can see why these are true. So here's the first result, and it's sort of surprising.

Â The first result is, any probability of red balls and blue balls is an equilibrium on

Â the long run. So you can end up with 60 percent red balls, you can end up with

Â 4 percent red balls, you can end up with 99 percent red balls. Any one of

Â those things could happen, and they're all equally likely. So you're equally likely

Â to get 4 percent red balls or 85 percent red balls. That's amazing. And it

Â turns out, it's not very hard to prove. So I want you to see if you can figure out

Â why it's the case. Let me talk about why this is important though, in terms of the

Â Polya process, and also just sort of why the Polya process is important. So let's

Â think of the Polya process as a model of fashion, let's say. So suppose that

Â there's different two colored shirts you might buy. You might buy a red shirt or a

Â blue shirt. And so you look in the store window, and you see one red, one blue. And

Â you're not sure what to do, so maybe you choose red. Well, now your friend's going

Â to come buy a shirt. And she looks and she thinks well, I see one red and one blue in

Â the store window, but I see that my friend bought a red. So she sees two red and one

Â blue. And so maybe her probability of buying a red shirt is two-thirds because

Â she sees more red than blue. So then she buys a red. So now somebody else comes in and

Â they buy a shirt. And they're thinking, "okay, what's more popular, red or blue?" And they

Â see three people buying red and one person buying blue. And so they are three times as

Â likely to buy red or blue. What you're getting is a Polya-type process. Now,

Â what's interesting about this is that, if people make decisions in that way, what

Â this result tells us is: anything could happen, and anything is equally likely.

Â We're as likely to end up with 4 percent red shirts as 99 percent red

Â shirts. We could use this same model to think about people buying Macintosh

Â computers or IBM machines, right? Whether you use DOS or Mac? And the things that

Â you think about using, the probability that you buy a Mac machine is proportional

Â to the number of people who bought Mac machines in the past, you're going to get

Â a Polya-type process. So, what plays out, what happens in reality is gonna be very

Â contingent on the history, and anything could happen. So, that's result one.

Â See if you can figure out why it's true. Here's result two: Result two is that any

Â sequence of events that has R red balls and B blue balls is equally likely. So, if

Â I get red, red, red, blue, that has the same probability of happening as drawing

Â up blue, red, red, red. Why is that the case? Again, to use some examples, see if

Â you can figure it out. I'll prove it in the next lecture. So what this means, and

Â the reason this is interesting: It basically says, if you know the frequency

Â of red and blue balls. That, that doesn't tell you really anything about the order.

Â So if you just see the set of things that happened, you can't infer anything about

Â the order, because any order is equally likely. So, again, that's sort of

Â surprising. Let me move on to another process. This next process is called the

Â balancing process, and it's sort of the inverse of the Polya process. In the

Â balancing progress, again we have an urn. There's a red ball and a blue ball. But the

Â difference here is, when I pick a ball out, let's suppose I pick a red ball, what

Â I do is, I add a ball of the opposite color. So I pick out a red ball, I add a

Â blue ball. Well, think about what this process is going to do. Suppose I pick a

Â red ball. I add a blue ball. Suppose I pick another red ball. I add another blue

Â ball. Well now it becomes incredibly likely I'm going to pick a blue ball. So

Â if I pick a blue ball, then I add a red ball. We can see, is this process is gonna

Â balance itself out. So the balancing process in the long run is gonna end up

Â with 50 percent red balls and 50 percent blue balls. Now [inaudible] okay, this is

Â balls, where is this gonna apply to the real world? What it applies to, is situations

Â where you want to keep different constituencies happy, so let's suppose, put a

Â convention in the United States, you might be decide to put it in the northern

Â state or southern state. If you put it in the northern state, what you're doing is

Â increasing the probability that you're gonna put it in the southern state four years later. If you put it in

Â the northern state again then you're really increasing the probability that you'll put it in the southern

Â state, so it's like you're putting in balls of the other color. Or, think of the

Â International Olympic Committee deciding where they should put the Olympic

Â Games. Should they put it in Asia, should they put it in North America? Do they put it

Â in Europe? Do they put it in South America? Now you can think of the balls as being four

Â different colors. But as you pick balls that say Europe, as you put more, put the

Â Olympics in Europe twice in a row, in effect, you're putting balls of the other

Â continents into the urn, making it more likely you'll pick those. And what this

Â tells us is that you're gonna end up with equal probabilities of the different

Â continents. Or in the case of the northern and southern states, equal likelihood in

Â putting the convention in the north or in the south. So here's what we get, and this is sort of interesting: In

Â the Polya process, we could put in balls of the same color, what we got is that

Â anything could happen. We could get any probability between zero and one, and

Â they're all equally likely. So just as likely to get 99%, as 4%, as 60%, as

Â 33%. In the balancing process, the only thing you can get is 55, 50%. You get an even

Â mix. So the Polya Process has, the equilibrium is incredibly path-dependent.

Â In the balancing process, it's not at all. So that's the distinction I want to

Â highlight. I want to make a distinction between the period outcomes. What happens in a

Â particular period, versus: in the long run, what does the distribution of balls in the

Â urn look like. So when we think about path dependence, we can think of it either in

Â terms of these outcomes, or we can think of it in terms of the equilibria that are

Â generated. So that's gonna be an important distinction. Path-dependent outcomes just

Â means that what happens in a period depends on what happens in the past.

Â That's gonna be true of both the Polya process, and the balancing process.

Â Path-dependent equilibrium means, what happens in the long run depends on the process

Â along the way. So if I think of the Polya process and the balancing process, the

Â Polya process has both path-dependent equilibria and path-dependent outcomes.

Â But the balancing process only has path- dependent outcomes. The equilibrium is

Â always one half. It doesn't depend on what happened along the way. It's always going

Â to end up with equal amounts. Now, this is an incredibly important insight, because it

Â affects how we think about identifying path dependence in the real world. I'll

Â give you an example. In the balancing process, history matters at each step in

Â time. So when you think about, what's the odds of getting blue balls in period seven, well that

Â hap, depends on how many blue balls I picked in periods one through six. So each

Â period's outcome depends on the past. So if I'm writing a narrative, I'd say, "oh boy,

Â this really was contingent on what happened previously". However, that doesn't

Â mean that what happens in the long run depends on the path. What happens in the long run

Â could be completely independent of the path, like in the balancing process. So

Â history can matter at each moment in time, but it can't matter in the long run. And

Â so if I just tell a story that says, you know, period seven dependent on six, eight

Â dependent on seven, nine dependent on eight, seven, and six, that in no way

Â means that history was contingent. It could very well be that the equilibrium

Â was set in stone beforehand, that the process was naturally going to go to one

Â place. Now, what are the examples of that? Well, two examples from history might be

Â what we call in America "Manifest Destiny", the idea that the United States was likely

Â to be a continent that stretched from sea to shining sea. So, the history played out

Â in particular ways, but there would be some people that argue that it didn't

Â matter what that path was, we were destined to be a country that stretched

Â "from sea to shining sea". So, even though, the Louisiana Purchase, particular wars,

Â the gold rush, all those things happened in a particular sequence, some people

Â argued it didn't matter, eventually we were gonna become one nation, stretching

Â from sea to sea. Another example is the railroads. People argue, once the

Â railroads are invented, that they sort of built themselves. So sure, there were

Â these, you know, people like the Carnegies and the Stanfords that laid the tracks.

Â But the thing is, it didn't matter who those people were, and it didn't matter

Â what order they were laid. The tracks were gonna be laid connecting the cities one

Â way or another. And it really doesn't, you know, particular sequences, the fact

Â that this track update, you know, track such-and-nineteen is going to be connected

Â to track such-and-eighteen. So yes, each lay of each track was path-dependent,

Â but the long-run outcome may not have been. The tracks are gonna be what they're

Â gonna be, because that's where it's economically efficient for those tracks to

Â be laid. So again, at each moment in time, the event depends on past history. But the

Â long-run equilibrium doesn't depend on past history. Now remember when we talked

Â about path dependence, we were saying the outcome probabilities depend on the

Â sequence of past ev events, past outcomes. I want to make a distinction between that,

Â remember we had that path dependence and phat dependence where outcome

Â probabilities depend on past outcomes, but not their order. So I call this "phat"

Â because remember I can think of path and rearrange the letters to get phat, I'm

Â just switching the h over here. Now I also do this because "phat" is a little bit

Â of, you know, joke about thick description. Right? So fat means thick.

Â And thick descriptive accounts take into account the full sequence. The Polya

Â process is phat. It's not path. And the reason why is, all that matters is the set

Â of outcomes. If I had red, red, blue, that means I've added two red balls and one

Â blue ball. And if I had blue, red, red, that also means I've added one blue ball and

Â two red balls. So all that matters is the set of outcomes, not the sequence. Why is

Â this so important? Why do we care about it being path-dependent versus

Â phat-dependent? Well, let's do just a tiny bit of math and see why. How many different

Â paths are there? Let's suppose I pick, I have 30 periods, and I wanna know, how

Â many different paths are there? Well, there's two things I could have picked the

Â first period, red or blue. Two I could have picked the next period. Two I could

Â have picked the third period, and so on. So there's two times two times two ... 30

Â times, different histories. That creates over a billion different histories,

Â different paths. If I wanna know how many sets there are, well, I could have zero, I

Â could have added zero blue balls. I could have added one, I could have added two, I

Â could have added three, up to 30. So there's 31 sets. So we got a billion

Â paths, and only 31 sets. So if something's set dependent, that means that, in round

Â 31, that means there's only 31 different possibilities that matter. If its path

Â dependent, that means there's a billion possibilities that matter. And this raises

Â two questions. First up, could something actually be path-dependent, depend on all this going

Â different paths? And in second, if there is one, can I construct it using my simple

Â urn model? Well, the answer to both questions is: yes, you can. So here's the

Â simple process using the urn model: It's called the Sway process, and it's gonna

Â give us some really interesting insight. So the Sway process works as follows: I start

Â with one blue and one red ball. When I pick a ball out, I add a ball of that same

Â color, just like the Polya process. But I also add 2^(t-s) minus 2^(t-s-1) balls

Â of the ball chosen in each period s less than t. What does

Â that mean? That's complicated. Let me explain it with a picture, that makes a lot

Â more sense. So in period one, I pick a blue ball, I add a blue ball. In period

Â two, I pick a red ball, so I add a red ball, and I also add a blue ball for the blue

Â ball that I pick in period one. Period three, let's suppose I pick a blue ball,

Â so I add a red ball again for the red ball I picked in period two. And now I add two

Â blue balls, so I'm multiplying times two the blue ball that I picked in period one.

Â Period four, suppose I get a red ball, now I add a red ball. I add a blue ball

Â for period three. I add two red balls for period two. And I add four, or two times

Â two, blue balls for period one. So what happens is, for each ball that I pick, I'm

Â going, I'm adding one, then two, then four, then eight, then sixteen, then 32, as I

Â move through time. The reason that's why I call this "the sway", because as decisions

Â go, as you go back in time, decisions take on more and more weight. This is really

Â interesting, because what's creating this full path dependence? It's the path taking

Â on more and more influence. So, when people talk about path dependence, when

Â they give examples of the law, when they talk about institutional choices, when

Â they talk about technological adoptions, they all think about early movers having a

Â bigger effect and the past really sort of having increasing weight. What turns out,

Â one way you can get full path dependence, is by having exactly that sort of process.

Â By having the past take on exponentially more weight over time. So that's really

Â interesting. Very simple urn model has told us how something that we had some loose

Â intuition about, for something really to depend on full history, not just on the

Â set, you've got to have the early path take on more and more weight over time. Or

Â at least that's one way in which you can get full path dependence. I realize

Â there's a lot going on here, but here's what's really interesting and it's really

Â useful. If you take a really simple class of models based on urns, just these really

Â simple urn models, from those urn models we learned some interesting things. We

Â learned how to construct a very simple path-dependent process, called the Polya

Â process; we constructed a balancing process, which showed us you can have

Â path-dependent outcomes in each period, but not have path-dependent equilibrium.

Â You can go to a particular thing each time. And we even constructed something

Â called the sway process, that gives us full path dependence and that showed us

Â one way you can get full path dependence and not just phat dependence is by having

Â the weight of history increase over time, which is something intuitively I think

Â that a lot of historians and a lot of scholars of technology have felt was true.

Â And this sort of shows us why, in fact, that may be the case. Why, when we see

Â some of these things really contingent, it might be the case that what happened early

Â on had a larger effect. Okay, thank you.

Â