We live in a complex world with diverse people, firms, and governments whose behaviors aggregate to produce novel, unexpected phenomena. We see political uprisings, market crashes, and a never ending array of social trends. How do we make sense of it? Models. Evidence shows that people who think with models consistently outperform those who don't. And, moreover people who think with lots of models outperform people who use only one. Why do models make us better thinkers? Models help us to better organize information - to make sense of that fire hose or hairball of data (choose your metaphor) available on the Internet. Models improve our abilities to make accurate forecasts. They help us make better decisions and adopt more effective strategies. They even can improve our ability to design institutions and procedures. In this class, I present a starter kit of models: I start with models of tipping points. I move on to cover models explain the wisdom of crowds, models that show why some countries are rich and some are poor, and models that help unpack the strategic decisions of firm and politicians. The models covered in this class provide a foundation for future social science classes, whether they be in economics, political science, business, or sociology. Mastering this material will give you a huge leg up in advanced courses. They also help you in life. Here's how the course will work. For each model, I present a short, easily digestible overview lecture. Then, I'll dig deeper. I'll go into the technical details of the model. Those technical lectures won't require calculus but be prepared for some algebra. For all the lectures, I'll offer some questions and we'll have quizzes and even a final exam. If you decide to do the deep dive, and take all the quizzes and the exam, you'll receive a Course Certificate. If you just decide to follow along for the introductory lectures to gain some exposure that's fine too. It's all free. And it's all here to help make you a better thinker!
Random Walks
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From the course by University of Michigan
Model Thinking
1095 оценки
From the lesson
Randomness and Random Walks & Colonel Blotto
In this section, we first discuss randomness and its various sources. We then discuss how performance can depend on skill and luck, where luck is modeled as randomness. We then learn a basic random walk model, which we apply to the Efficient Market Hypothesis, the ideas that market prices contain all relevant information so that what's left is randomness. We conclude by discussing finite memory random walk model that can be used to model competition. The reading for this section is a paper on distinguishing skill from luck by Michael Mauboussin.
Meet the Instructors
Scott E. PageProfessor of Complex Systems, Political Science, and Economics
Center for the Study of Complex Systems
Hi, in this lecture I wanna talk about very simple [inaudible] the binary random
walk model now binary random walk model works as follows, assume that each period
I flip the coin. And it could either come up heads, or it could come up tails. All I
wanna do is turn that idea into a model. Here's how I'm gonna do it. I'm gonna
describe, define a variable called X. And that's gonna be my winnings. And then I'm
gonna have a fair coin, and if it flip it and if it comes up heads, half the time,
and I win a dollar, and if it comes up tails, I lose a dollar. And then what I'm
gonna do is keep track of how much money I've won. If it turns up I win the first
time, this goes to one, if I win again, it goes to two, if I lose, it falls back down
to one. That's all there is to the process. Now, you can think of this as a
decent model of gambling. And going to a casino and playing blackjack. Maybe
there's a 50 percent chance you win a game, 50 percent chance you lose a hand.
Suppose you bet a dollar on each hand. The random lock describes what your winnings
would look like. And here's sort of a picture of what that would be. You might
win the first hand and then lose, and lose, and then lose, and then win and then
lose a couple hands, and then win and then you'd be down let's say 2.00, in this
case. This is what we mean by a binary random lock. What I want to do, is I want
to start out with three mathematical results about these binary random logs
that are sort of surprising. The first one is this. If you play this N times, you
know 100 times, 1,000 times, a million times, and on again you're expected
winnings are going to be zero, Because it's equally likely to go up or down. So
that one makes a lot of sense, Right? Because half the time you go up, half the
time you go down. You should expect to break even. Now some surprising results.
Result two, pick any number K, can 5500. If you have a random walk that goes on
forever, you're gonna pass plus K and minus K an infinite number of times. So
what that means is if you're zero, and here's plus 500, and here's minus 500,
your random walk is gonna. Cross each one of these an infinite number of times. What
that means is, it's not gonna take off. There's no way it's just gonna take off
where you win billions and billions of dollars, and never go negative. So if
you're playing blackjack, and you played forever, If you went to a casino and
played blackjack forever, it's, an infinite number of times, you're gonna be
up 500 dollars, and an infinite number of times, you're gonna be down 500 dollars.
That's just a mathematical fact. Now here's the third mathematical fact that
would be even more surprising. Again, if you [inaudible] long enough, you're gonna
get a streak of K heads in a row and K tails in a row for any K. So let's say K
equals 30. If you sit down at a Y check table and played hand after hand after
hand for decades, you'd win 30 times in a row an infinite number of times. You'd
also lose 30 times in a row, eventually, an infinite number of times. So if you see
a streak, we often think, oh boy, that streak. I was really hot. I was really
winning. The cards were really falling my way. No, it's just going to happen. It's
going to happen because that's what the math tells us. Now let's see how the math
explains that. What are the odds that you win once? A half. What are the odds that
you win twice in a row? Well, one-half times one-half. Cuz you got two heads in a
row. What are the odds you win sixteen times in a row? That's one-half times
one-half times one-half times one-half, sixteen times, which is one-half to the
sixteenth. Well how big of a number is that? That's around 64,000. So the odds of
getting that are actually 1/64,532. So, that's not that likely. This here's the
point. Suppose there's a 100,000 people sitting at casinos. There's a 100,000
people sitting at casinos playing blackjack. You'd expect one of them to
get, even more than one, maybe one and a half, to get sixteen wins in a row. And
those people are gonna fly back home and they're gonna say, oh man I was so lucky .
Now, they just happen to have that one random luck that had sixteen hits in a
row. There's probably someone else that went home losing sixteen times in a row.
He probably took the bus. So what you get is, with a random luck. You, your likely
to get these long sequences, your also gonna end up with zero. And you're gonna
pass above lines and you're gonna fall below lines. So there's gonna be big
winners, big losers. And it's just random. And there's another phenomenon, what that
means in simplicity is in these results and that's a progression to the mean. So,
the first result said you're going to be. Expect to be at zero. The second result
said you're going to be above 500 an infinite number of times and you're going
to be below 500 a negative 500 an infinite number of times. Another way to think
about that is if you have a random y it's just going to on average go back to zero.
So if you think about it suppose I'm up 50, well if I think what's going to happen
the next twenty periods, I should expect to be right back at 50. I shouldn't expect
to keep winning, even though I've won before because there's no bias upward.
[sound]. So. If you think about things like free throw shooting, these, these are
examples of when you think like, oh man, people get hot. So there's a famous study
called the hot hand study. When you look at the 1981, 1980 Boston Celtics announced
the thing of a random walk where it's 50 percent likely to go up and 50 percent
likely to go down. Here, the Celtics made 75 percent of the free throws. So it's 75
percent chance it's gonna go up. And a 25 percent chance in some sense of the walk
going down. So, if people really did get hot. If there really were hot hands, you'd
expect there to be after you miss a free throw you're less likely to make it then
after you've made a free throw. Well they looked at an entire season worth of data
and they found out it if you miss you first free throw, you make the second one
75 percent of the time. But if you make you first free throw, you're feeling it.
Then . [sound] 75 percent of the time. So here's the point, we may think there's all
sorts of hot streaks out there, hot hands and thinks like that. If you actually look
at the data. Many of them turn out to be just simple random walks. Now the
probability is almost 50%, and in this case there's 75%. But again, this process,
the free throw shooting of the Boston Celtics in 1980 was just a straight
forward binary random walk, with the probability going up 1.75, the probability
going down 1.25. This doesn't stop us that whenever we see streaks to think, oh boy,
this person must have. Really some great skill, or they must be hot, or things must
be going that way. But this is a problem because if you try to infer from someone's
success that what they're doing made them successful that can be wrong. It could be
that you just happen to be seeing a sequence of sixteen heads in a row. So
there's a famous book by Jim Collins called Good to Great. And what he did is
he looked at a whole bunch of companies that have been successful, then he said,
let's see what the characteristics of those companies are. And what he found was
that these are the companies that are good to great. These are what they have. They
have humble leaders. They have the right people on the bus, they have the right
employees. Not necessarily the smartest employees, but the right employees. They
confront facts, they don't delude themselves. They look out there and they
say, this is a fact. This is where the market's moving, this is what our
competitor's doing. They, they don't delude themselves in any way. They
focus... They focus on whatever it is that they're trying to do. They also rinse
their cottage cheese. What does this mean? This refers to a tri-athlete named Dave
Scott who ran, who trained like an unbelievable amount of time. He ran and
swam so much every day it would just blow your mind. And he had, this same guy,
rinsed his cottage cheese so that he wouldn't have too much fat in his diet. So
it's this total commitment to what you're doing. They a lso embraced technology. So
they were people who weren't afraid of technology, and they sought out what we
called super adaptively, this idea that. You got one good idea in one place and
another good idea in another place, you can combine them. So these were the
characteristics of companies that were Good to Great, and this was the
best-selling business book of all time. Now here's the list of the great companies
in 2001. Places like Abott Laboratories, Circuit City, Kroger, Phillip Morris,
Pitney Bowes. We can then ask, how did those companies do over the next decade?
And here's the answer. Not very well. So the S and P 500 had a zero percent in that
time. The ones in red didn't do very well. So only two companies went well. So that
which was IPNG and Newcore which sopped up four fold. But many others didn't do well
at all. Including Fanny-Mae who went into receivership. So not explicitly
criticizing Collins, what I'm saying is that, if you have success in the past, a
lot of that could just be that things fell your way. Maybe it was a complex process
and luck went your way and your head came up, your coin came up heads ten periods in
a row. And what you often get is regression to the mean. So we see here, in
the great companies, is just standard regression to the mean. Now some of you
may [inaudible] thinking at this point, wait a minute, this seems like one of our
earlier models, and it should. Remember the no free lunch theorem? The no free
lunch theorem said, no algorithm is better than any other. So big rocks first, little
rocks first, it depends on the context. Well, that same thing may be true here.
So, if we go back and look at. Collin's identified humble leaders, right upon the
bust. Technology [inaudible]. It may well be these were all really good
characteristics for the time. For the period 1990 to 2001. And hence he's able
to identify these companies that were great. The ones that really ascribed to
those particular techniques. However in 2001 and 2010 those same characteristics
may not have worked v ery well. And, so, we know Norphy Lunch theorem. Any
heuristic, humble leaders rinsing your cottage cheese, whatever it is, may work
in one setting, may not work in another setting. And so it could be, if its luck
whether your particular heuristic is working at any given moment in time, you'd
expect to see this sort of regression to the mean. As a related idea. To this
notion of sort of, streaks, and that's clusters. Now you'll, you've probably
noticed if you look on the internet, there's all sorts of spatial graphs. So we
can see all sorts of graphs where things are situated in space. So you might see
data that looks like this. Now, when we see data that looks like this, we might
say, oh boy, it looks like there's a cluster here, and a cluster here, and a
cluster here. And if this were, let's say, data on cancer or crime or something like
that, you might think, boy, we better go identify what's going on in that cluster.
Well here's the problem, if I just randomly throw things down on a graph, I'm
gonna get clusters. Let's suppose I have a 1,000 by 1,000 checkerboard. And I fill in
each square at probably one / tenth so I fill in 100,000 squares this is a million
squares it's only a tenth of them. And I wanna ask how many clusters are there that
are this big that are five by five where there's nine or 96 times 996 clusters that
size so the point is there's just going to be a whole bunch of places where I might
have a lot of these dots filled in. And then I might think oh my gosh that square
right there that square something horrible is happening or if I want to ask how many
rows of ten are there well there's 990 times 1,000 rows of ten so it's going to
be very likely just like it's likely for someone to throw sixteen heads in a row
then I'm going to see some strip of length ten where a whole bunch of them are filled
in and then I'm gonna think wow something must have happened here. But the reality
is, something necessarily didn't happen there. It could just be that it's random.
So when you look at graphs and you see these clusters, you have to be aware that
they may just be random. Just like when you see somebody win fifteen times at
blackjack, or when you see some company be incredibly successful. So what have we
learned? What we've learned is, we can often think of. An outcome is being a
sequence of random events. And if an outcome of sequence of random events, what
we're gonna expect to see, is we're gonna expect to see an expect value of zero. But
we're gonna see some big winners and some big losers. And we can't then necessarily
infer just because someone?s been successful in the past, But fairly
successful in the future. So we start with two random walkers and one who happened to
go up and one who happen to goes down, and then we think, right, who in heaven's sake
are we gonna place our bets on. Well, this one's just as likely to go down as this
one is to go up. You don't know anything. So what we really want try and figure out
in these situations is, is something a random walk? Or is it not? Is there some
reason to believe that there is, that this person's going up for a reason. And this
person's going down for a reason, or is the data consistent with things being
purely random? And if it is, we should expect some regression in the mean, we
should expect the two of them to perform about the same. Alright. Thank you.
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