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!
Blotto: No Best Strategy
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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.
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Scott E. PageProfessor of Complex Systems, Political Science, and Economics
Center for the Study of Complex Systems
Hi, in this lecture what I want to do is just describe the basic Colonel Blotto
game, and then give you a sense of some of the main results from the game. So here's
how the game works. Colonel Blotto again is this game of strategic mismatch. I've
got a bunch of troops, you've got a bunch of troops. You've got to decide where to
allocate them across fronts. Whoever's got the most troops on the front is going to
win. So how do we formalize that? Do it as follows, we construct a game, the game has
the following assumptions. There's two players, each has T troops, so we're going
to start out by assuming we've got the same number of troops. There's N fronts,
so these are the places that we can put our troops. And we're going to assume that
T is large relative to N, so you see these two greater than signs, that means T is a
lot bigger than N. So if we have ten fronts, we may have thousands of troops.
Now the action I'm going to take, my strategy in this game, is going to be to
allocate my troops across those fronts. So how many troops do I put on each of those
fronts? And we determine the winner of the game by just sort of counting up who wins
each front. So it's the number of fronts, won. So if I have more troops on a front,
I win. If you have more troops on a front, you win. That's it, Really simple game. It
turns out, it's sort of a complicated game, It's complicated for the following
reason, It's zero sum. So zero sum games mean that there's one person gets in this
case, plus one, the winner. The other person gets minus one. So when I add those
up they get zero. So there's someone who wins, someone who loses. If you're in a
game where there's someone who wins, someone who loses, it's naturally gonna be
really competitive. Because suppose you're at some sort of equilibrium. In that
equilibrium, one player's gonna be winning, one player's gonna be losing.
Which means the player who's losing is gonna try and do something different so
that they don't lose. If you talk to successful business people, one thing that
you' [laugh] frequently hear from them is they'll say, stay out of zero sum games.
The reason you wanna stay out of zero sum games is pretty straight forward. You've
gotta work really, really hard, and at best, you're gonna break even. Because
somebody's gonna win, somebody's gonna lose. You'd rather be in a game that sort
of positive sum where everybody can win. So zero sum games by their nature, tend to
pretty darn competitive. We'll see that's gonna definitely be the case of Blotto. So
again, here's the example. Let's suppose we got T's 100. So each person's got 100
troops and there's three fronts. So I'll be player one. I'm player one. I say,
well, the obvious thing to do is to evenly allocate my troops across the three
fronts. So I'll do 33, 33, 33. I've got one left over so I'll put that on front
one. Now you're sitting there, you're thinking okay, I'm player two. You're
player two and you're thinking, okay what should I do. Well, a lot of us thought
that Scott did the obvious thing, which was to put a third, a third, a third. So
what I'm going to do is, I'm going to sacrifice one front. I'm gonna sacrifice
this third front and only put twenty troops there And I'm going to put forty
troops on each of these other two fronts. And by putting forty troops on each of
these other two fronts, I'm gonna win. Each of those fronts. That's what you were
thinking. So you're gonna win those fronts. Whereas I'm only gonna win front
number three. So the point is you're gonna win two, I'll win one so you're the winner
of the game. Player two wins. So if you think about Colonel Blotto, what you see
is equally allocating your troops doesn't make a lot of sense because somebody can
beat it. So you need a better strategy. But here's the problem [laugh]. Any
strategy can be defeated. Let's see why, let's go back. So here was the first case.
I did 34, 33, 33. You did 40, 40, twenty, so you beat me. So we know that 34, 33, 33
can get beat. What about 40, 40, twenty? So suppose now it's, we're sitting, we're
gonna play again. And I know, b oy, you beat me with 40, 40, twenty. And I think
that you think that, well, you're just gonna keep the same strategies that worked
before. Well, what I could do, is I could do, I could put 30 here so I win there,
and 50 here, and then put twenty there. So I do twenty, 50, 30. So I beat you. Well
how could beat my twenty, 50, 30? Well you could say, well hmm, why don't I just go
50, zero, 50 and you'd beat me? So you can see is any position can be defeated and
there is no real obvious beat strategy to play. Here's another insight about Blotto.
You don't need all of your troops to win. So let's go and look at this example
again. You're at 40, 40, twenty. I could beat you with 62 troops. I could have 41,
21, and zero. So even though we both have 100 troops, I wouldn't need a full 100
troops to beat you. That's another sort of interesting insight [laugh]. So the thing
about this Blotto game was, it's fairly complicated. And it's all about these
strategic mismatches, and where you allocate your troops. Anything can be
defeated, and you can beat something without even using all your troops. So
what happens in this, This game, Is it an equilibrium? Is it a cycle, are we going
to alternate? Is it going to be complex? Or is it going to be completely random.
Who wins? Well, what you get in this is actually a mixture of these two things.
It's gonna be sort of an equilibrium and random. What do I mean by that? What
there's gonna be is there's gonna be equilibrium where we choose strategies
randomly. And so therefore, the winner is gonna be random. So what you can think of
is this, Here's the fronts, Here's front number one, here's front number two,
here's front number three. If I put all my troops on front one, then I'd be taking a
strategy here. Now, that's a horrible strategy, because that can be beat by just
about anything. Right here in the center would mean I'm doing 33, 33, and let's say
34. So I'm sort of evenly allocating my troops. What you can show in Colonel
Blotto is if I make a little. Like hexagon here . That you should just sort of
randomly pick a strategy in here. If you randomly pick a strategy in here, there's
nothing the other person can do to beat you. What they can do is basically have an
expected value of zero. So what's gonna happen is if you play this game, both
players will randomly choose strategies in that space, and they're equally likely to
win. So what we're gonna get is we're gonna get equilibrium. Equilibrium is
gonna be with both randomly [inaudible] strategies, but the winner is gonna be
random. So if we think about Blotto, then, we can ask, is winning at Blotto skill or
is it luck? Because that's something we've talked about [laugh] in the past. Well,
depends on how you think of it. If you think of the number of troops as your
ability, Then if you got more ability, you should have a better chance of winning
because you have more troops to move around. So, if we're playing lotto with
different numbers of troops and troops just represent our ability then there's,
there's definitely some skill involved. Now it's also true that allocating those
troops is strategic. If I could figure out how you're going to allocate your troops
for some reason, then my increased in strategic ability over you, my advantage,
may enable me to beat you sometimes. But if you're playing a mixed strategy, If
you're playing a equilibrium strategy, then there isn't any strategic ability. We
can just cross this out and it's all just going to come down to luck. So against
really smart players, Blotto may end up equal numbers of troops, Blotto's probably
luck. We have maybe one player who is smarter than the other, or one player with
more troops than the other. Then Blotto starts becoming more skilled. But again,
the interesting thing about Blotto. Anything can be beaten. You don't need all
your troops, and it really comes down to, Where is that other person gonna put their
troops. So, what you want to do is not be understood. You want to be confusing to
the other person so you want to random off. So, it's an interesting g ame.
Alright so, that's the basic Blotto. Where we're gonna go next is we're going to
think a little bit more deeply about this idea of one side having an advantage and
see what that means for the nature of competition. We're also gonna talk about
why Blotto has become interesting again. Why there's been a renewed interest in
Blotto. Remember initially Blotto was about moving troops on battle fields, and
that sorta fell out of favor. We're gonna see why it's generally. Insight about
strategic mismatch has a lot to say about competition generally, okay, thanks.
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