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Urn Models
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From the course by University of Michigan
Model Thinking
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From the lesson
Path Dependence & Networks
In this set of lectures, we cover path dependence. We do so using some very simple urn models. The most famous of which is the Polya Process. These models are very simple but they enable us to unpack the logic of what makes a process path dependent. We also relate path dependence to increasing returns and to tipping points. The reading for this lecture is a paper that I wrote that is published in the Quarterly Journal of Political Science
Meet the Instructors
Scott E. PageProfessor of Complex Systems, Political Science, and Economics
Center for the Study of Complex Systems
[cough] Hi. Welcome back. In this set of lectures we're talking about processes
that are path dependent. And to describe path dependence, what I want to do is use
a model. And this model is going to be based on a set of urns. These are very
simple urn models. And these urn models are going to help us distinguish between
processes that are path-dependent and processes that are what I call
"phat-dependent", where the order doesn't matter but the set of things that matters.
They're also going to help us distinguish between processes that have outcomes in a
given period that are path-dependent, and processes that have equilibriums, that have
long-run distributions over the set of outcomes, that are path-dependent. So the
outcome in a period might depend on what happened in the past, that's fine. But
that's different from saying, the long-run equilibrium, where the system is gonna go,
is gonna depend on the, on the process. So I'm gonna make a distinction between
path-dependent outcomes and path-dependent equilibrium. And I'm gonna do this using
this very simple class of urn models. So what is an urn model? An urn
model is always simple, it's got an urn that contains blue balls and it contains
red balls, and you pick balls out of the urn. So I might reach this urn, and I might
pick out a red ball. That's it. And then it's the probability or picking out balls of
different colors. So the simplest urn model is called the Bernoulli model, and that
works a follows: You just got a fix number of balls in the urn. So this urn has three
red balls and one blue ball. The probability of picking up a red ball is 3/4.
So if you think about this, this model can be used to explain a lot of
casino games like roulette and blackjack: what are the odds of getting a
face card? And you can think that there's, you know, 52 cards in the
deck, and 16 of those cards are face cards. So there's a 16 over 52 chance
that you're going to get a face card. So that's it, so the Bernoulli models are
really simple, probabilities stay fixed. And the probability in each draw in a
Bernoulli model is independent. So if I get a red ball this time, put that ball
back in the urn, that doesn't have any effect on the probability of getting a red
ball in the next period. So when you say something's independent, it means: what
happens now doesn't depend in any way on what happened in the past. What we want to
do is take this simple ideal of an urn model and use it to construct some
path-dependent processes. And the most famous path-dependent process is called the Polya
process. And here's how it works: Start with an urn with two balls, one red, one
blue. You pick a ball. So maybe I pick the red ball out. After you pick that ball,
you look at it, see its color, and then you put in another ball of that same color. Then
you do it again. So maybe the next time I pick another ball out, and again it's
red, so I add another red ball. Well, then maybe then next time, I pick a blue ball out,
and then add in a ball that's blue. Now if we think about this Polya process, what's
gonna happen is that over time, these probabilities are gonna change. So if we
think about the Polya process in action, what was, what's gonna happen is, we're
gonna get lots and lots of balls in here. Lots of red balls, lots of blue balls,
depending on what I pick out. And the probability of balls, of the different
colors in the balls can be changed; it's going to start out that the probability of
red balls is the half. Then it might go to two-thirds as I pick another one. Then it
might go to three-fourths if I pick another red ball, but then if I pick a
blue ball it's gonna go to three-fifths. So what we see is, the probability of
picking a red ball is going to change over time, and it's going to be path-dependent.
It's going to depend on the path of previous choices. What I'd like to do now
is state two results about the Polya process. I'm not going to prove them. In
the next lecture I'll prove why they're true. In between this lecture and the next
lecture what I'd like you to do is see if you can figure out why they're the case.
So do some examples, with, you know, just with picking three balls from the urn and
four balls from the urn, you know, four sequences along the path, and see if you
can see why these are true. So here's the first result, and it's sort of surprising.
The first result is, any probability of red balls and blue balls is an equilibrium on
the long run. So you can end up with 60 percent red balls, you can end up with
4 percent red balls, you can end up with 99 percent red balls. Any one of
those things could happen, and they're all equally likely. So you're equally likely
to get 4 percent red balls or 85 percent red balls. That's amazing. And it
turns out, it's not very hard to prove. So I want you to see if you can figure out
why it's the case. Let me talk about why this is important though, in terms of the
Polya process, and also just sort of why the Polya process is important. So let's
think of the Polya process as a model of fashion, let's say. So suppose that
there's different two colored shirts you might buy. You might buy a red shirt or a
blue shirt. And so you look in the store window, and you see one red, one blue. And
you're not sure what to do, so maybe you choose red. Well, now your friend's going
to come buy a shirt. And she looks and she thinks well, I see one red and one blue in
the store window, but I see that my friend bought a red. So she sees two red and one
blue. And so maybe her probability of buying a red shirt is two-thirds because
she sees more red than blue. So then she buys a red. So now somebody else comes in and
they buy a shirt. And they're thinking, "okay, what's more popular, red or blue?" And they
see three people buying red and one person buying blue. And so they are three times as
likely to buy red or blue. What you're getting is a Polya-type process. Now,
what's interesting about this is that, if people make decisions in that way, what
this result tells us is: anything could happen, and anything is equally likely.
We're as likely to end up with 4 percent red shirts as 99 percent red
shirts. We could use this same model to think about people buying Macintosh
computers or IBM machines, right? Whether you use DOS or Mac? And the things that
you think about using, the probability that you buy a Mac machine is proportional
to the number of people who bought Mac machines in the past, you're going to get
a Polya-type process. So, what plays out, what happens in reality is gonna be very
contingent on the history, and anything could happen. So, that's result one.
See if you can figure out why it's true. Here's result two: Result two is that any
sequence of events that has R red balls and B blue balls is equally likely. So, if
I get red, red, red, blue, that has the same probability of happening as drawing
up blue, red, red, red. Why is that the case? Again, to use some examples, see if
you can figure it out. I'll prove it in the next lecture. So what this means, and
the reason this is interesting: It basically says, if you know the frequency
of red and blue balls. That, that doesn't tell you really anything about the order.
So if you just see the set of things that happened, you can't infer anything about
the order, because any order is equally likely. So, again, that's sort of
surprising. Let me move on to another process. This next process is called the
balancing process, and it's sort of the inverse of the Polya process. In the
balancing progress, again we have an urn. There's a red ball and a blue ball. But the
difference here is, when I pick a ball out, let's suppose I pick a red ball, what
I do is, I add a ball of the opposite color. So I pick out a red ball, I add a
blue ball. Well, think about what this process is going to do. Suppose I pick a
red ball. I add a blue ball. Suppose I pick another red ball. I add another blue
ball. Well now it becomes incredibly likely I'm going to pick a blue ball. So
if I pick a blue ball, then I add a red ball. We can see, is this process is gonna
balance itself out. So the balancing process in the long run is gonna end up
with 50 percent red balls and 50 percent blue balls. Now [inaudible] okay, this is
balls, where is this gonna apply to the real world? What it applies to, is situations
where you want to keep different constituencies happy, so let's suppose, put a
convention in the United States, you might be decide to put it in the northern
state or southern state. If you put it in the northern state, what you're doing is
increasing the probability that you're gonna put it in the southern state four years later. If you put it in
the northern state again then you're really increasing the probability that you'll put it in the southern
state, so it's like you're putting in balls of the other color. Or, think of the
International Olympic Committee deciding where they should put the Olympic
Games. Should they put it in Asia, should they put it in North America? Do they put it
in Europe? Do they put it in South America? Now you can think of the balls as being four
different colors. But as you pick balls that say Europe, as you put more, put the
Olympics in Europe twice in a row, in effect, you're putting balls of the other
continents into the urn, making it more likely you'll pick those. And what this
tells us is that you're gonna end up with equal probabilities of the different
continents. Or in the case of the northern and southern states, equal likelihood in
putting the convention in the north or in the south. So here's what we get, and this is sort of interesting: In
the Polya process, we could put in balls of the same color, what we got is that
anything could happen. We could get any probability between zero and one, and
they're all equally likely. So just as likely to get 99%, as 4%, as 60%, as
33%. In the balancing process, the only thing you can get is 55, 50%. You get an even
mix. So the Polya Process has, the equilibrium is incredibly path-dependent.
In the balancing process, it's not at all. So that's the distinction I want to
highlight. I want to make a distinction between the period outcomes. What happens in a
particular period, versus: in the long run, what does the distribution of balls in the
urn look like. So when we think about path dependence, we can think of it either in
terms of these outcomes, or we can think of it in terms of the equilibria that are
generated. So that's gonna be an important distinction. Path-dependent outcomes just
means that what happens in a period depends on what happens in the past.
That's gonna be true of both the Polya process, and the balancing process.
Path-dependent equilibrium means, what happens in the long run depends on the process
along the way. So if I think of the Polya process and the balancing process, the
Polya process has both path-dependent equilibria and path-dependent outcomes.
But the balancing process only has path- dependent outcomes. The equilibrium is
always one half. It doesn't depend on what happened along the way. It's always going
to end up with equal amounts. Now, this is an incredibly important insight, because it
affects how we think about identifying path dependence in the real world. I'll
give you an example. In the balancing process, history matters at each step in
time. So when you think about, what's the odds of getting blue balls in period seven, well that
hap, depends on how many blue balls I picked in periods one through six. So each
period's outcome depends on the past. So if I'm writing a narrative, I'd say, "oh boy,
this really was contingent on what happened previously". However, that doesn't
mean that what happens in the long run depends on the path. What happens in the long run
could be completely independent of the path, like in the balancing process. So
history can matter at each moment in time, but it can't matter in the long run. And
so if I just tell a story that says, you know, period seven dependent on six, eight
dependent on seven, nine dependent on eight, seven, and six, that in no way
means that history was contingent. It could very well be that the equilibrium
was set in stone beforehand, that the process was naturally going to go to one
place. Now, what are the examples of that? Well, two examples from history might be
what we call in America "Manifest Destiny", the idea that the United States was likely
to be a continent that stretched from sea to shining sea. So, the history played out
in particular ways, but there would be some people that argue that it didn't
matter what that path was, we were destined to be a country that stretched
"from sea to shining sea". So, even though, the Louisiana Purchase, particular wars,
the gold rush, all those things happened in a particular sequence, some people
argued it didn't matter, eventually we were gonna become one nation, stretching
from sea to sea. Another example is the railroads. People argue, once the
railroads are invented, that they sort of built themselves. So sure, there were
these, you know, people like the Carnegies and the Stanfords that laid the tracks.
But the thing is, it didn't matter who those people were, and it didn't matter
what order they were laid. The tracks were gonna be laid connecting the cities one
way or another. And it really doesn't, you know, particular sequences, the fact
that this track update, you know, track such-and-nineteen is going to be connected
to track such-and-eighteen. So yes, each lay of each track was path-dependent,
but the long-run outcome may not have been. The tracks are gonna be what they're
gonna be, because that's where it's economically efficient for those tracks to
be laid. So again, at each moment in time, the event depends on past history. But the
long-run equilibrium doesn't depend on past history. Now remember when we talked
about path dependence, we were saying the outcome probabilities depend on the
sequence of past ev events, past outcomes. I want to make a distinction between that,
remember we had that path dependence and phat dependence where outcome
probabilities depend on past outcomes, but not their order. So I call this "phat"
because remember I can think of path and rearrange the letters to get phat, I'm
just switching the h over here. Now I also do this because "phat" is a little bit
of, you know, joke about thick description. Right? So fat means thick.
And thick descriptive accounts take into account the full sequence. The Polya
process is phat. It's not path. And the reason why is, all that matters is the set
of outcomes. If I had red, red, blue, that means I've added two red balls and one
blue ball. And if I had blue, red, red, that also means I've added one blue ball and
two red balls. So all that matters is the set of outcomes, not the sequence. Why is
this so important? Why do we care about it being path-dependent versus
phat-dependent? Well, let's do just a tiny bit of math and see why. How many different
paths are there? Let's suppose I pick, I have 30 periods, and I wanna know, how
many different paths are there? Well, there's two things I could have picked the
first period, red or blue. Two I could have picked the next period. Two I could
have picked the third period, and so on. So there's two times two times two ... 30
times, different histories. That creates over a billion different histories,
different paths. If I wanna know how many sets there are, well, I could have zero, I
could have added zero blue balls. I could have added one, I could have added two, I
could have added three, up to 30. So there's 31 sets. So we got a billion
paths, and only 31 sets. So if something's set dependent, that means that, in round
31, that means there's only 31 different possibilities that matter. If its path
dependent, that means there's a billion possibilities that matter. And this raises
two questions. First up, could something actually be path-dependent, depend on all this going
different paths? And in second, if there is one, can I construct it using my simple
urn model? Well, the answer to both questions is: yes, you can. So here's the
simple process using the urn model: It's called the Sway process, and it's gonna
give us some really interesting insight. So the Sway process works as follows: I start
with one blue and one red ball. When I pick a ball out, I add a ball of that same
color, just like the Polya process. But I also add 2^(t-s) minus 2^(t-s-1) balls
of the ball chosen in each period s less than t. What does
that mean? That's complicated. Let me explain it with a picture, that makes a lot
more sense. So in period one, I pick a blue ball, I add a blue ball. In period
two, I pick a red ball, so I add a red ball, and I also add a blue ball for the blue
ball that I pick in period one. Period three, let's suppose I pick a blue ball,
so I add a red ball again for the red ball I picked in period two. And now I add two
blue balls, so I'm multiplying times two the blue ball that I picked in period one.
Period four, suppose I get a red ball, now I add a red ball. I add a blue ball
for period three. I add two red balls for period two. And I add four, or two times
two, blue balls for period one. So what happens is, for each ball that I pick, I'm
going, I'm adding one, then two, then four, then eight, then sixteen, then 32, as I
move through time. The reason that's why I call this "the sway", because as decisions
go, as you go back in time, decisions take on more and more weight. This is really
interesting, because what's creating this full path dependence? It's the path taking
on more and more influence. So, when people talk about path dependence, when
they give examples of the law, when they talk about institutional choices, when
they talk about technological adoptions, they all think about early movers having a
bigger effect and the past really sort of having increasing weight. What turns out,
one way you can get full path dependence, is by having exactly that sort of process.
By having the past take on exponentially more weight over time. So that's really
interesting. Very simple urn model has told us how something that we had some loose
intuition about, for something really to depend on full history, not just on the
set, you've got to have the early path take on more and more weight over time. Or
at least that's one way in which you can get full path dependence. I realize
there's a lot going on here, but here's what's really interesting and it's really
useful. If you take a really simple class of models based on urns, just these really
simple urn models, from those urn models we learned some interesting things. We
learned how to construct a very simple path-dependent process, called the Polya
process; we constructed a balancing process, which showed us you can have
path-dependent outcomes in each period, but not have path-dependent equilibrium.
You can go to a particular thing each time. And we even constructed something
called the sway process, that gives us full path dependence and that showed us
one way you can get full path dependence and not just phat dependence is by having
the weight of history increase over time, which is something intuitively I think
that a lot of historians and a lot of scholars of technology have felt was true.
And this sort of shows us why, in fact, that may be the case. Why, when we see
some of these things really contingent, it might be the case that what happened early
on had a larger effect. Okay, thank you.
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