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Skill and Luck
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來自 University of Michigan 的課程
模型思维
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從本節課中
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.
與講師見面
Scott E. PageProfessor of Complex Systems, Political Science, and Economics
Center for the Study of Complex Systems
Hi. In this lecture, what I wanna do is, I wanna talk about the role randomness plays
in performance. In particular, I want to think about the role, the relative role of
skill and luck. And we can think of luck as randomness. Now, a simple way to write
this down would just be to say that someone's outcome depends on two things.
It depends on their skill, their ability, and it also depend on luck. It depends on,
you know, did things go the right way? Did the [inaudible] fall in their direction,
did complexity unfold in a way that was beneficial to them, that sort of thing.
You can take any sort of domain whether it's sports, law, politics and you can
say, well that person's really good, they've got a lot of skill, or
[inaudible], alternative we say, oh, boy she has been so lucky, everything's
falling her way. So just think about normal language we do this, what will be
nice to do is construct a model that helps us understand how much skill is there and
how much luck is there. Now, there's a book coming out by Michael Mobenson who
works at Legmason. And he did a financial analyst. And what Michael does is that he
has taken a whole bunch of different domains and try to figure out exactly how
much in this domain is skill and how much in this domain is luck. Now the way to
write this in a model, you can say look, let's write the outcome not just as luck
plus skill. But as A times luck plus one minus A times skill. Now how would you
unpack this? How would you figure out how big A is. What you can do is you need sort
of a sequence of outcomes. And what you can see is if this same team or the same
firm continues to perform well. Over time, then you've got to figure that A is fairly
small. But if what you see is that there's huge jumps from period to period then a is
probably pretty big. So let me show you an example, suppose the outcome Equals
one-half luck plus one-half skill. And suppose someone had skill equal to a half.
Then their outcome is gonna equal one-half luck, which is a random variable, plus
one-half. So wh at I'm gonna see is I'm gonna see, sometimes they're one,
sometimes they're one-half. Sometimes they're a quarter. I'm gonna see huge
jumps in their outcome Alternatively, if the payoff were different, If I could
write the outcome. As .1L plus .9 skill, and their skill was equal to one. And then
I'd have their outcomes equal to .1L plus .9. And now, I'd see .95, .85, .87, .92,
and I'd see a much tighter distribution. So what you can do is you can figure out
the relative amounts of skill and luck in a domain by looking at lots of data, and
looking at the performance of lots of athletes, firms, and those sorts of
things. So you can figure out some things are high skill, like the 100 meter dash.
And other things involve maybe a little bit more luck, possibly darts. Now why
would I care about this? Well what Mobinson says, is there's just a lot
reasons why you care. One is you want to fairly assess outcomes. You don't want to
be out, going around saying boy this person is so lucky, when in fact, they're
actually skillful. And another important thing, we'll talk about this later in the,
these set of lectures, is you wanna recognize if results, if outcomes depend a
lot on luck, there's gonna be a lot of reversion to the mean. So if somebody's
won three times in a row, but it happens to be lucky, you shouldn't expect them to
win the fourth time. If it's been mostly skill, you should expect them to do it the
fourth time, and the fifth time, and the sixth time. So understanding whether it's
luck or skill is gonna let you figure out whether there's gonna be reversion to the
mean or not. Another thing, Giving good feedback, You're a manager. You see
someone do really well. If you know this depends entirely on skill, you wanna say,
hey boy, that's just great, and we really appreciate all the effort you're putting
in and you're doing a great job. If it's mostly luck, you wanna say to that person.
We know things are going well, but don't be disappointed if things don't go as well
next time because a lot of fluctuations, a lot of randomness in the situation. So
again, giving good advice, giving good guidance, is gonna be, it's gonna be
helpful for you to know whether someone has basically succeeded because of skill
or if it's because of luck. And then finally, fair allocation of resources.
Suppose again, you're a manager. You've got two employees, and you're trying to
figure out, who should I give the big raise to? This person who had a lot of
sales, or the person who had little sales? If it's mostly skill, yeah, you should
give it almost all to the person who had a lot of sales. If there's a huge luck
component, you should go for more equal distribution of payout. Because the fact
is, is that one person probably just got lucky. So there's a lot of managerial
reasons, a lot of predictive reasons why you'd like to understand whether or not
[inaudible] skill whether it's luck. So having even a, even a fairly basic model
like this would be really useful. Now I wanna get to an [inaudible] in his book
that's really profound. So we can think about some domains being high skill, like
the 100 meter dash, right. To win the 100 meter dash in the Olympics you've gotta be
really, really fast. We think of these as some of the greatest athletes in the
world. We think of Jesse Owens, Usain Bolt, as incredibly skillful people. And
then there's other domains, like playing rock paper scissors, where you think,
okay, whoever wins that is lucky because you just randomly picking rock, paper, or
scissors. So 100 meter dash is skill. Rock paper scissors is luck. But what
Mauboussin points out is there's something called the paradox of skill. And that is,
when you get the very best people competing against each other, they tend to
actually have fairly similar skills, And so therefore, the winner is likely
determined by quite a bit of luck. So, paradoxically, when you get high-skilled
people competing, less variation in skill, more luck. Let me show this through a
couple of examples. So suppose first I have a fairly In a wide set of skill
levels. So notice o ne persons 60, one persons 50, one persons 40, Now this isn't
a man where the component of skill is roughly ten times the component of luck.
Right? So skill is 60, luck is only six. So the outcomes in this case, after I add
in the luck component go right in order 66, 55, 49; so the most skillful person
wins, the second most skillful person takes second. And, the third most skillful
person takes third. So what we get is outcomes that align exactly with what we
would expect based on skill. Luck doesn't play any role. But now let's suppose I
move to the Olympic Games and I take the best people in the world competing in this
event. Now their skill levels are 61, 60 and 59. Yeah, there's a little bit of luck
that comes into play, and what we see is the winner actually is the third best
person. Because even though their skill is only 59, they happen to get a luck of
nine, which gets them to 68, which is better than the other two people. So the
paradox of skill is once you get a bunch of people who have fairly similar skill
levels and even if there is only a small luck component, luck is gonna play a large
role in determining the winner and we see this actually in sports. So in baseball
which is American sport, one of the big awards to win is to win the batting title,
the best hitter. So last year, Miguel Cabrera from Detroit, my buddy [inaudible]
here, Miguel won the American League batting title, which was great. But notice
he got hits in 34.4 percent of his at-bats and behind him were two people at 33.8%.
Just take away a few hits from him and add a few hits to either of these other people
and he doesn't win the batting title. So was it skill that got him in the top four?
Absolutely. But it was luck that won him the batting title. That's the paradox of
skill. It takes skill to get there, but once you're there, the winner, especially
when it's this close, is probably gonna be determined by some luck. And luckily for
Detroit, McGill won it. Well here's another, even more famous example. This is
a picture of Michael Phelp S. This is from CNN Sports Illustrated. This is a picture
of Michael Phelps out touching Michael Cavic in the 100 meter butterfly. In the
previous Olympics. And you can see, if you look really close at this picture that
Phelp's hands are bent a tiny bit and Cavic's hands have not yet touched it. So
literally Michael Phelp beat him by a fingertip. Now Phelp is the most
[inaudible] Olympic athlete of all time. So there's no doubt that he has a lot of
skills. He has a ton of skills but so does Cavic. [laugh] a lot of skill as well. And
so the winner in this particular race happened to come down to just a little bit
of luck, maybe by the tip of a finger. Okay, so what have we learned in this
lecture? We learned that in some cases you can think of outcomes as being
combinations of skill and luck, and you can determine how much skill and how much
luck by looking at variations in outcomes. Is there a lot of flipping, or is there
sort of consistent winners? We also then got from this very simple model, a
paradoxical result. And a paradoxical result is, is that when you get all high
skill people competing against one another, even when it's a low luck
environment, luck will play a large role because of the paradoxical skill. Alright,
so that's a luck and skill model, now we're going to move on to a model of
random walks. Alright, thank you.
