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Hi, in this lecture I wanna talk about very simple [inaudible] the binary random

Â walk model now binary random walk model works as follows, assume that each period

Â I flip the coin. And it could either come up heads, or it could come up tails. All I

Â wanna do is turn that idea into a model. Here's how I'm gonna do it. I'm gonna

Â describe, define a variable called X. And that's gonna be my winnings. And then I'm

Â gonna have a fair coin, and if it flip it and if it comes up heads, half the time,

Â and I win a dollar, and if it comes up tails, I lose a dollar. And then what I'm

Â gonna do is keep track of how much money I've won. If it turns up I win the first

Â time, this goes to one, if I win again, it goes to two, if I lose, it falls back down

Â to one. That's all there is to the process. Now, you can think of this as a

Â decent model of gambling. And going to a casino and playing blackjack. Maybe

Â there's a 50 percent chance you win a game, 50 percent chance you lose a hand.

Â Suppose you bet a dollar on each hand. The random lock describes what your winnings

Â would look like. And here's sort of a picture of what that would be. You might

Â win the first hand and then lose, and lose, and then lose, and then win and then

Â lose a couple hands, and then win and then you'd be down let's say 2.00, in this

Â case. This is what we mean by a binary random lock. What I want to do, is I want

Â to start out with three mathematical results about these binary random logs

Â that are sort of surprising. The first one is this. If you play this N times, you

Â know 100 times, 1,000 times, a million times, and on again you're expected

Â winnings are going to be zero, Because it's equally likely to go up or down. So

Â that one makes a lot of sense, Right? Because half the time you go up, half the

Â time you go down. You should expect to break even. Now some surprising results.

Â Result two, pick any number K, can 5500. If you have a random walk that goes on

Â forever, you're gonna pass plus K and minus K an infinite number of times. So

Â what that means is if you're zero, and here's plus 500, and here's minus 500,

Â your random walk is gonna. Cross each one of these an infinite number of times. What

Â that means is, it's not gonna take off. There's no way it's just gonna take off

Â where you win billions and billions of dollars, and never go negative. So if

Â you're playing blackjack, and you played forever, If you went to a casino and

Â played blackjack forever, it's, an infinite number of times, you're gonna be

Â up 500 dollars, and an infinite number of times, you're gonna be down 500 dollars.

Â That's just a mathematical fact. Now here's the third mathematical fact that

Â would be even more surprising. Again, if you [inaudible] long enough, you're gonna

Â get a streak of K heads in a row and K tails in a row for any K. So let's say K

Â equals 30. If you sit down at a Y check table and played hand after hand after

Â hand for decades, you'd win 30 times in a row an infinite number of times. You'd

Â also lose 30 times in a row, eventually, an infinite number of times. So if you see

Â a streak, we often think, oh boy, that streak. I was really hot. I was really

Â winning. The cards were really falling my way. No, it's just going to happen. It's

Â going to happen because that's what the math tells us. Now let's see how the math

Â explains that. What are the odds that you win once? A half. What are the odds that

Â you win twice in a row? Well, one-half times one-half. Cuz you got two heads in a

Â row. What are the odds you win sixteen times in a row? That's one-half times

Â one-half times one-half times one-half, sixteen times, which is one-half to the

Â sixteenth. Well how big of a number is that? That's around 64,000. So the odds of

Â getting that are actually 1/64,532. So, that's not that likely. This here's the

Â point. Suppose there's a 100,000 people sitting at casinos. There's a 100,000

Â people sitting at casinos playing blackjack. You'd expect one of them to

Â get, even more than one, maybe one and a half, to get sixteen wins in a row. And

Â those people are gonna fly back home and they're gonna say, oh man I was so lucky .

Â Now, they just happen to have that one random luck that had sixteen hits in a

Â row. There's probably someone else that went home losing sixteen times in a row.

Â He probably took the bus. So what you get is, with a random luck. You, your likely

Â to get these long sequences, your also gonna end up with zero. And you're gonna

Â pass above lines and you're gonna fall below lines. So there's gonna be big

Â winners, big losers. And it's just random. And there's another phenomenon, what that

Â means in simplicity is in these results and that's a progression to the mean. So,

Â the first result said you're going to be. Expect to be at zero. The second result

Â said you're going to be above 500 an infinite number of times and you're going

Â to be below 500 a negative 500 an infinite number of times. Another way to think

Â about that is if you have a random y it's just going to on average go back to zero.

Â So if you think about it suppose I'm up 50, well if I think what's going to happen

Â the next twenty periods, I should expect to be right back at 50. I shouldn't expect

Â to keep winning, even though I've won before because there's no bias upward.

Â [sound]. So. If you think about things like free throw shooting, these, these are

Â examples of when you think like, oh man, people get hot. So there's a famous study

Â called the hot hand study. When you look at the 1981, 1980 Boston Celtics announced

Â the thing of a random walk where it's 50 percent likely to go up and 50 percent

Â likely to go down. Here, the Celtics made 75 percent of the free throws. So it's 75

Â percent chance it's gonna go up. And a 25 percent chance in some sense of the walk

Â going down. So, if people really did get hot. If there really were hot hands, you'd

Â expect there to be after you miss a free throw you're less likely to make it then

Â after you've made a free throw. Well they looked at an entire season worth of data

Â and they found out it if you miss you first free throw, you make the second one

Â 75 percent of the time. But if you make you first free throw, you're feeling it.

Â Then . [sound] 75 percent of the time. So here's the point, we may think there's all

Â sorts of hot streaks out there, hot hands and thinks like that. If you actually look

Â at the data. Many of them turn out to be just simple random walks. Now the

Â probability is almost 50%, and in this case there's 75%. But again, this process,

Â the free throw shooting of the Boston Celtics in 1980 was just a straight

Â forward binary random walk, with the probability going up 1.75, the probability

Â going down 1.25. This doesn't stop us that whenever we see streaks to think, oh boy,

Â this person must have. Really some great skill, or they must be hot, or things must

Â be going that way. But this is a problem because if you try to infer from someone's

Â success that what they're doing made them successful that can be wrong. It could be

Â that you just happen to be seeing a sequence of sixteen heads in a row. So

Â there's a famous book by Jim Collins called Good to Great. And what he did is

Â he looked at a whole bunch of companies that have been successful, then he said,

Â let's see what the characteristics of those companies are. And what he found was

Â that these are the companies that are good to great. These are what they have. They

Â have humble leaders. They have the right people on the bus, they have the right

Â employees. Not necessarily the smartest employees, but the right employees. They

Â confront facts, they don't delude themselves. They look out there and they

Â say, this is a fact. This is where the market's moving, this is what our

Â competitor's doing. They, they don't delude themselves in any way. They

Â focus... They focus on whatever it is that they're trying to do. They also rinse

Â their cottage cheese. What does this mean? This refers to a tri-athlete named Dave

Â Scott who ran, who trained like an unbelievable amount of time. He ran and

Â swam so much every day it would just blow your mind. And he had, this same guy,

Â rinsed his cottage cheese so that he wouldn't have too much fat in his diet. So

Â it's this total commitment to what you're doing. They a lso embraced technology. So

Â they were people who weren't afraid of technology, and they sought out what we

Â called super adaptively, this idea that. You got one good idea in one place and

Â another good idea in another place, you can combine them. So these were the

Â characteristics of companies that were Good to Great, and this was the

Â best-selling business book of all time. Now here's the list of the great companies

Â in 2001. Places like Abott Laboratories, Circuit City, Kroger, Phillip Morris,

Â Pitney Bowes. We can then ask, how did those companies do over the next decade?

Â And here's the answer. Not very well. So the S and P 500 had a zero percent in that

Â time. The ones in red didn't do very well. So only two companies went well. So that

Â which was IPNG and Newcore which sopped up four fold. But many others didn't do well

Â at all. Including Fanny-Mae who went into receivership. So not explicitly

Â criticizing Collins, what I'm saying is that, if you have success in the past, a

Â lot of that could just be that things fell your way. Maybe it was a complex process

Â and luck went your way and your head came up, your coin came up heads ten periods in

Â a row. And what you often get is regression to the mean. So we see here, in

Â the great companies, is just standard regression to the mean. Now some of you

Â may [inaudible] thinking at this point, wait a minute, this seems like one of our

Â earlier models, and it should. Remember the no free lunch theorem? The no free

Â lunch theorem said, no algorithm is better than any other. So big rocks first, little

Â rocks first, it depends on the context. Well, that same thing may be true here.

Â So, if we go back and look at. Collin's identified humble leaders, right upon the

Â bust. Technology [inaudible]. It may well be these were all really good

Â characteristics for the time. For the period 1990 to 2001. And hence he's able

Â to identify these companies that were great. The ones that really ascribed to

Â those particular techniques. However in 2001 and 2010 those same characteristics

Â may not have worked v ery well. And, so, we know Norphy Lunch theorem. Any

Â heuristic, humble leaders rinsing your cottage cheese, whatever it is, may work

Â in one setting, may not work in another setting. And so it could be, if its luck

Â whether your particular heuristic is working at any given moment in time, you'd

Â expect to see this sort of regression to the mean. As a related idea. To this

Â notion of sort of, streaks, and that's clusters. Now you'll, you've probably

Â noticed if you look on the internet, there's all sorts of spatial graphs. So we

Â can see all sorts of graphs where things are situated in space. So you might see

Â data that looks like this. Now, when we see data that looks like this, we might

Â say, oh boy, it looks like there's a cluster here, and a cluster here, and a

Â cluster here. And if this were, let's say, data on cancer or crime or something like

Â that, you might think, boy, we better go identify what's going on in that cluster.

Â Well here's the problem, if I just randomly throw things down on a graph, I'm

Â gonna get clusters. Let's suppose I have a 1,000 by 1,000 checkerboard. And I fill in

Â each square at probably one / tenth so I fill in 100,000 squares this is a million

Â squares it's only a tenth of them. And I wanna ask how many clusters are there that

Â are this big that are five by five where there's nine or 96 times 996 clusters that

Â size so the point is there's just going to be a whole bunch of places where I might

Â have a lot of these dots filled in. And then I might think oh my gosh that square

Â right there that square something horrible is happening or if I want to ask how many

Â rows of ten are there well there's 990 times 1,000 rows of ten so it's going to

Â be very likely just like it's likely for someone to throw sixteen heads in a row

Â then I'm going to see some strip of length ten where a whole bunch of them are filled

Â in and then I'm gonna think wow something must have happened here. But the reality

Â is, something necessarily didn't happen there. It could just be that it's random.

Â So when you look at graphs and you see these clusters, you have to be aware that

Â they may just be random. Just like when you see somebody win fifteen times at

Â blackjack, or when you see some company be incredibly successful. So what have we

Â learned? What we've learned is, we can often think of. An outcome is being a

Â sequence of random events. And if an outcome of sequence of random events, what

Â we're gonna expect to see, is we're gonna expect to see an expect value of zero. But

Â we're gonna see some big winners and some big losers. And we can't then necessarily

Â infer just because someone?s been successful in the past, But fairly

Â successful in the future. So we start with two random walkers and one who happened to

Â go up and one who happen to goes down, and then we think, right, who in heaven's sake

Â are we gonna place our bets on. Well, this one's just as likely to go down as this

Â one is to go up. You don't know anything. So what we really want try and figure out

Â in these situations is, is something a random walk? Or is it not? Is there some

Â reason to believe that there is, that this person's going up for a reason. And this

Â person's going down for a reason, or is the data consistent with things being

Â purely random? And if it is, we should expect some regression in the mean, we

Â should expect the two of them to perform about the same. Alright. Thank you.

Â