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!
Preference Aggregation
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From the course by University of Michigan
Model Thinking
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From the lesson
Aggregation & Decision Models
In this section, we explore the mysteries of aggregation, i.e. adding things up. We start by considering how numbers aggregate, focusing on the Central Limit Theorem. We then turn to adding up rules. We consider the Game of Life and one dimensional cellular automata models. Both models show how simple rules can combine to produce interesting phenomena. Last, we consider aggregating preferences. Here we see how individual preferences can be rational, but the aggregates need not be.There exist many great places on the web to read more about the Central Limit Theorem, the Binomial Distribution, Six Sigma, The Game of Life, and so on. I've included some links to get you started. The readings for cellular automata and for diverse preferences are short excerpts from my books Complex Adaptive Social Systems and The Difference Respectively.
Meet the Instructors
Scott E. PageProfessor of Complex Systems, Political Science, and Economics
Center for the Study of Complex Systems
Hi. In this last lecture in aggregation, we want to talk about the aggregation of
preferences. So this is going to differ from what we looked at before. Remember
when we looked at the central limit theorem, we looked at aggregating numbers
or actions. And then we looked at the game of life and cellular automata where talked
about aggregating rules. Now we want to talk about aggregating preferences. So
preferences are going to be a different structure, a different mathematical
structure. Then, what we had with either rules or numbers. So. To get a handle on
this, to get a handle on how we aggregate these things, first we gotta say, well,
what are preferences? How do we, how do we represent them? So, well, let's think
about it. So let's suppose I'm just asking, what do you prefer? Do you prefer
apples or do you prefer bananas? So you may say, well, you know, I prefer apples.
Or someone else may say, no, I prefer. Bananas. Or alternatively, I could say,
how about bananas and coconuts? Do you prefer bananas or do you prefer coconuts?
And you might say, well, you know, I prefer bananas to coconuts. So one way to
write down preferences or think about preferences is through revealed actions.
So we can just give people sets of choices, and ask them, which do you prefer
over the other? So when you think about overall preferences, what we'd like to do
is we'd like to have a complete listing of someone's preferences. So what we'll talk
about often times are what are called preference orderings, which are just a
ranking of a whole set of alternatives. Now, typically, those alternatives will be
within a particular class. So I'll have a preference ordering over fruit, and I'd
have a preference ordering over vegetables. I could have a preference
ordering over houses, over cars, right? So within a category, I can rank different
things, alright? So, we can then ask, well, how many preference orderings are
there? Right, so what does this thing look like. Well lets suppose it got, these
three things. Apples, bananas, and coconuts, and I could say okay well. On
apples and bananas, there's two possibilities, right? Either I prefer
apples to bananas, right? Which I'm gonna show you with a greater than sign. Or, I
could prefer bananas to apples, so there's two possibilities. Next, if I look at
bananas and coconuts, right now I've got that I could, I could prefer bananas to
coconuts. Or alternatively, I could prefer coconuts to bananas, so there's two
possibilities there. And finally, with apples and coconuts, I could either prefer
coconuts to apples. Or apples to coconuts and there's two possibilities there so two
times two times two right I got to times these. Is going to be eight. So there's
going to be eight different ways, eight different types of preferences I could
have for these two types of, three types of fruit. Right? So, that's a lot of
different things, and each one of them I can just represent by these sort of
greater than signs. Like, which one I liked first, you know, which one do I
prefer? Now, there is a bit of a problem though with this. Let me erase all of this
for a second. There's a bit of a problem with this, because, let's look at these
particular preferences. These preferences say, I prefer apples, right, to bananas. I
prefer bananas to coconuts. And I prefer coconuts. To apples. Now that doesn't make
any sense. Because if I prefer apples to bananas, and bananas to coconuts, then I
should prefer apples to coconuts, right? So this doesn't make any sense, and it
should go like that. These are what we would call transitive preferences. So they
satisfy a relationship called transitivity. So these are transitive.
Preferences. And so we typically assume that individuals, that people, have
transitive preferences. Another way to think about transitive preferences is that
they're, they're rational. So it would be irrational to say, oh, I like apples more
than bananas, bananas more than coconuts, but coconuts more than apples. That
doesn't make any sense. So we think of rational preferences as being preferences
that are transitive. If I like A more than B, and B more than C, then I also like A
more than C. Okay? And then I can ask, if this is true, right? If apple's bigger
than bananas, banana's bigger than coconuts. If that implies apples more than
coconuts, that puts a restriction on how many preferences I can have, I can no
longer have anything. It rules certain things out. So now we can ask. How many
preferences can I get that way? Well, this is actually also an easy calculation, and
we've sort of done some of this math before. Well, it means there's gonna be
one thing I like best, right, that's ranked first, one thing I like second
best. And one thing ranked the third best. Well, so, how many different things could
I? Like, first, I could like the apple, I could like the banana, or I could like the
coconut. So I chose the apple, there was three possibilities. Once I've chosen the
apple first, I've got two things I can choose next, the banana or the coconut. I
choose the banana, but I could have chose any one of two. But once I get to the
third thing, I've only got one thing left. So there's 3x2x1, which is six. So there's
only six ways to be, sort of have rational preferences over these three alternatives.
So when we think about rational preferences, what we think of is these
preference orderings, right, where one thing is preferred to the next is
preferred to the next. Now, in more sophisticated models, we can also allow.
Quality, right. So I could say I like, I'm indifferent between bananas and coconuts.
But here we're just going to assume that like, you like one thing more than the
next. So at, the first thing we get just in thinking about these preferences is
that if we impose some rationality assumption like for people having
preferences then there's fewer preference than we'd get if we just sort of allowed
just anything to go. So here's the game. Here's sort of what we're going to play
with in this particular model. We want to think about suppose I've got a bunch of
people who have rational preferences and now suppose I want to ask how do their
preferences add up? What I mean by that is like that okay well think about it each
person has preferences and now I can say well what is the society's preference or
even like in a family. I could say everybody in our family has preferences
over these fruits, right. Each member does. Well can I say anything about the
family's preferences. Well, first notice if everybody has the same preferences it's
pretty easy. If everybody in the family likes apples and [laugh] then bananas, and
then coconuts. Then we can say well the family likes apples, and then bananas and
then coconuts. It gets tricky. Right? If different people like different stuff. So
if one of us likes apples and then bananas and then coconuts, then another person
likes bananas and then coconuts and then apples. So if we differ in our ordering,
now becomes somewhat problematic to decide well, what are our collective ordering?
What are our collective preferences? So, this is an aggregation problem, right? We
get individuals with preferences and I want to ask, "what's the collective
preference?" Well, here's what's really interesting, let's watch. So here's some
preferences. Person one, right, here's person one. They like apples, and then
bananas and coconuts. Person two likes bananas, and then apples and then
coconuts. And person three like apples and bananas, and then coconuts. Okay, so we
think about this and we go okay, so what are collective preferences? Well, there is
some diversity here in what we want, but it seems pretty clear that like, coconut
should be last. Because everybody has coconuts last. So we'll put the little
coconut here. Alright, that's in last place. Now it comes down to sort of apples
versus bananas. Now, one thing we can do is we can say, well let's treat people
equally. Let's not suppose that person two is somehow more important than person one
or person two, three. So we treat people equally and we can say, well let's just
vote. And if we vote two people like apples, and one person likes bananas, so
then we can put the little apple here. I'll do a really bad apple. That's gonna
be better than the banana, [inaudible] that's a horrible looking banana, and
that's gonna be one of the coconuts. [inaudible] these are collective
preferences. Apple, banana, coconut, okay? That's pretty easy. Well, now let's go for
something where the preferences are even a little bit more diverse. Now person one
likes apples, bananas, coconuts. Person two likes bananas, coconuts, apples. And
person three prefers coc, coconuts to apples to bananas. Now we gotta think,
okay, what, what happens here? There's no, doesn't seem to be any clear winner. But
one thing we could do is, we could say, well, let's, let's just do a pairwise
vote. So let's just, you know, vote these things through. So let's first compare
coconuts to apples. So if you have coconuts to apples, we notice that, again,
let's number these people one, two, and three. If we do coconuts versus apples, we
see that person two and person three. Right. [inaudible] Or coconuts. So
coconuts is gonna win two To one. If we get coconuts versus bananas. We see that
well person one and person two both prefer bananas to coconuts. So one and two prefer
bananas. So bananas are gonna win. Let's actually circle this. So coconuts win
versus apples. And bananas won versus coconuts. So therefore, it would stand a
reason that bananas should win with respect to apples, right? Because bananas
are better than coconuts, coconuts are better than apples. So therefore, bananas
should be better than coconut. Well let's check. Let's sort of check to be sure. So
we compare apples to bananas. We see that person one likes apples more than bananas.
That's okay. Person two likes bananas more than apples but person three. Likes apples
more than bananas. So we get that one and three right. Both prefer apples than
bananas. So apples win. Well, look at this. The group. Here's the collected.
Here's where the collected preferences are. The collected likes coconuts more
than apples. Apples more than bananas. And bananas. [inaudible] coconuts. That's
irrational. That's not transitive. So here's the really funky thing. We've got
individuals, every single individual is completely rational. They've got nice
transitive preferences. There's no inconsistencies. But then when we vote
when we try to aggregate these preferences we get something which is not consistent.
So this is a paradox of aggregation. You know, so before we talk about aggregation,
we've got things, like, you know, simple rules could create complex phenomena. Here
[inaudible] aggregation of preferences. You know, aggregation of some structure
can give us something that's not, it doesn't have one of the properties of the
parts. So each part was rational. But the collective isn't rational. So this is
sometimes called, this is formally called Condersay paradox. So each person is
rational. Each person has rational preferences. But then when we vote, the
collective is not rational. The collective says they go back. The collective says
"okay Coconuts vs Apples - Coconuts". "Apples vs. Bananas, Bananas." So then you
think that for sure, sure they must like Coconuts more than Bananas. But, in fact, if you
have people vote they pick Bananas over Coconuts. So this is the Condersay
paradox. Each person is rational, but the collective is irrational. So this has some
pretty severe implications. The implications are gonna be, that when we
think about voting, that now, suddenly, we're not necessarily gonna get a good
outcome. We could get almost a random outcome. And then it means that people
might wanna vote strategically. So later on in the course, when we start
constructing models of how people vote, this'll be near the end of the course.
We'll see that the fact that aggregation doesn't work. Right? That there's a
problem with aggregation? With these over-preferences? But that's going to
create incentives, opportunities, conditions under witch people might want to
manipulate agenda, lie about their preferences or misrepresent their
preferences in order to get outcomes that they want to get. So here's a case right?
Where aggregation doesn't give us something we want. Okay? So just to drive
this home. Each person has totally sane rational transitive preferences. Exactly
what you expect. But when you look at the collective. The collective has this
irrationality, so aggregation is sort of a funny thing. That's why social science,
right, in particular in this case, politics, is so darn interesting. Right?
Because. What happens at the macro level is, sorta, logically inconsistent, even
though its going out to the micro level makes a lot of sense. [cough] okay so now
we see an aggregation in several forms, right. We've seen aggregation of numbers
in the central limit theorem. We've seen aggregation of rules, right, in both the
game of life and the one-dimensional cellular automaton. We saw we could get really
complicated [inaudible] from simple parts. And now I'm looking at preferences, we've
sort of said what's a good aggregation of some other mathematical structure, namely
these orderings. And we found that orderings that are you know, in the
mathematical sense transitive, sort of in the social sense rational. Don't
necessarily aggregate into orderings that are transitive and rational. So we get
that they're sort of, we can lose [inaudible] consistency as we go up. So,
these sort of interesting aspects of aggregation are ideas we're gonna play
with throughout the course. Now, none of these particular models, models any real
thing, per se, [inaudible] but they're building blocks. They're giving us a basis
for how to think. If nothing else, I hope these lectures are giving you some sense
of, like, the mysteries and the intricacies of adding things up. And why
the social world is very, very different than the parts that comprise it. Thank you
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