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Hi. In this last lecture in aggregation,
we want to talk about the aggregation of

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preferences. So this is going to differ
from what we looked at before. Remember

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when we looked at the central limit
theorem, we looked at aggregating numbers

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or actions. And then we looked at the game
of life and cellular automata where talked

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about aggregating rules. Now we want to
talk about aggregating preferences. So

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preferences are going to be a different
structure, a different mathematical

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structure. Then, what we had with either
rules or numbers. So. To get a handle on

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this, to get a handle on how we aggregate
these things, first we gotta say, well,

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what are preferences? How do we, how do we
represent them? So, well, let's think

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about it. So let's suppose I'm just
asking, what do you prefer? Do you prefer

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apples or do you prefer bananas? So you
may say, well, you know, I prefer apples.

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Or someone else may say, no, I prefer.
Bananas. Or alternatively, I could say,

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how about bananas and coconuts? Do you
prefer bananas or do you prefer coconuts?

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And you might say, well, you know, I
prefer bananas to coconuts. So one way to

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write down preferences or think about
preferences is through revealed actions.

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So we can just give people sets of
choices, and ask them, which do you prefer

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over the other? So when you think about
overall preferences, what we'd like to do

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is we'd like to have a complete listing of
someone's preferences. So what we'll talk

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about often times are what are called
preference orderings, which are just a

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ranking of a whole set of alternatives.
Now, typically, those alternatives will be

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within a particular class. So I'll have a
preference ordering over fruit, and I'd

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have a preference ordering over
vegetables. I could have a preference

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ordering over houses, over cars, right? So
within a category, I can rank different

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things, alright? So, we can then ask,
well, how many preference orderings are

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there? Right, so what does this thing look
like. Well lets suppose it got, these

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three things. Apples, bananas, and
coconuts, and I could say okay well. On

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apples and bananas, there's two
possibilities, right? Either I prefer

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apples to bananas, right? Which I'm gonna
show you with a greater than sign. Or, I

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could prefer bananas to apples, so there's
two possibilities. Next, if I look at

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bananas and coconuts, right now I've got
that I could, I could prefer bananas to

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coconuts. Or alternatively, I could prefer
coconuts to bananas, so there's two

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possibilities there. And finally, with
apples and coconuts, I could either prefer

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coconuts to apples. Or apples to coconuts
and there's two possibilities there so two

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times two times two right I got to times
these. Is going to be eight. So there's

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going to be eight different ways, eight
different types of preferences I could

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have for these two types of, three types
of fruit. Right? So, that's a lot of

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different things, and each one of them I
can just represent by these sort of

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greater than signs. Like, which one I
liked first, you know, which one do I

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prefer? Now, there is a bit of a problem
though with this. Let me erase all of this

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for a second. There's a bit of a problem
with this, because, let's look at these

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particular preferences. These preferences
say, I prefer apples, right, to bananas. I

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prefer bananas to coconuts. And I prefer
coconuts. To apples. Now that doesn't make

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any sense. Because if I prefer apples to
bananas, and bananas to coconuts, then I

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should prefer apples to coconuts, right?
So this doesn't make any sense, and it

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should go like that. These are what we
would call transitive preferences. So they

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satisfy a relationship called
transitivity. So these are transitive.

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Preferences. And so we typically assume
that individuals, that people, have

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transitive preferences. Another way to
think about transitive preferences is that

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they're, they're rational. So it would be
irrational to say, oh, I like apples more

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than bananas, bananas more than coconuts,
but coconuts more than apples. That

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doesn't make any sense. So we think of
rational preferences as being preferences

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that are transitive. If I like A more than
B, and B more than C, then I also like A

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more than C. Okay? And then I can ask, if
this is true, right? If apple's bigger

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than bananas, banana's bigger than
coconuts. If that implies apples more than

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coconuts, that puts a restriction on how
many preferences I can have, I can no

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longer have anything. It rules certain
things out. So now we can ask. How many

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preferences can I get that way? Well, this
is actually also an easy calculation, and

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we've sort of done some of this math
before. Well, it means there's gonna be

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one thing I like best, right, that's
ranked first, one thing I like second

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best. And one thing ranked the third best.
Well, so, how many different things could

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I? Like, first, I could like the apple, I
could like the banana, or I could like the

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coconut. So I chose the apple, there was
three possibilities. Once I've chosen the

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apple first, I've got two things I can
choose next, the banana or the coconut. I

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choose the banana, but I could have chose
any one of two. But once I get to the

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third thing, I've only got one thing left.
So there's 3x2x1, which is six. So there's

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only six ways to be, sort of have rational
preferences over these three alternatives.

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So when we think about rational
preferences, what we think of is these

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preference orderings, right, where one
thing is preferred to the next is

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preferred to the next. Now, in more
sophisticated models, we can also allow.

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Quality, right. So I could say I like, I'm
indifferent between bananas and coconuts.

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But here we're just going to assume that
like, you like one thing more than the

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next. So at, the first thing we get just
in thinking about these preferences is

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that if we impose some rationality
assumption like for people having

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preferences then there's fewer preference
than we'd get if we just sort of allowed

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just anything to go. So here's the game.
Here's sort of what we're going to play

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with in this particular model. We want to
think about suppose I've got a bunch of

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people who have rational preferences and
now suppose I want to ask how do their

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preferences add up? What I mean by that is
like that okay well think about it each

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person has preferences and now I can say
well what is the society's preference or

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even like in a family. I could say
everybody in our family has preferences

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over these fruits, right. Each member
does. Well can I say anything about the

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family's preferences. Well, first notice
if everybody has the same preferences it's

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pretty easy. If everybody in the family
likes apples and [laugh] then bananas, and

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then coconuts. Then we can say well the
family likes apples, and then bananas and

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then coconuts. It gets tricky. Right? If
different people like different stuff. So

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if one of us likes apples and then bananas
and then coconuts, then another person

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likes bananas and then coconuts and then
apples. So if we differ in our ordering,

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now becomes somewhat problematic to decide
well, what are our collective ordering?

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What are our collective preferences? So,
this is an aggregation problem, right? We

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get individuals with preferences and I
want to ask, "what's the collective

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preference?" Well, here's what's really
interesting, let's watch. So here's some

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preferences. Person one, right, here's
person one. They like apples, and then

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bananas and coconuts. Person two likes
bananas, and then apples and then

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coconuts. And person three like apples and
bananas, and then coconuts. Okay, so we

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think about this and we go okay, so what
are collective preferences? Well, there is

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some diversity here in what we want, but
it seems pretty clear that like, coconut

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should be last. Because everybody has
coconuts last. So we'll put the little

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coconut here. Alright, that's in last
place. Now it comes down to sort of apples

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versus bananas. Now, one thing we can do
is we can say, well let's treat people

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equally. Let's not suppose that person two
is somehow more important than person one

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or person two, three. So we treat people
equally and we can say, well let's just

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vote. And if we vote two people like
apples, and one person likes bananas, so

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then we can put the little apple here.
I'll do a really bad apple. That's gonna

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be better than the banana, [inaudible]
that's a horrible looking banana, and

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that's gonna be one of the coconuts.
[inaudible] these are collective

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preferences. Apple, banana, coconut, okay?
That's pretty easy. Well, now let's go for

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something where the preferences are even a
little bit more diverse. Now person one

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likes apples, bananas, coconuts. Person
two likes bananas, coconuts, apples. And

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person three prefers coc, coconuts to
apples to bananas. Now we gotta think,

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okay, what, what happens here? There's no,
doesn't seem to be any clear winner. But

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one thing we could do is, we could say,
well, let's, let's just do a pairwise

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vote. So let's just, you know, vote these
things through. So let's first compare

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coconuts to apples. So if you have
coconuts to apples, we notice that, again,

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let's number these people one, two, and
three. If we do coconuts versus apples, we

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see that person two and person three.
Right. [inaudible] Or coconuts. So

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coconuts is gonna win two To one. If we
get coconuts versus bananas. We see that

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well person one and person two both prefer
bananas to coconuts. So one and two prefer

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bananas. So bananas are gonna win. Let's
actually circle this. So coconuts win

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versus apples. And bananas won versus
coconuts. So therefore, it would stand a

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reason that bananas should win with
respect to apples, right? Because bananas

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are better than coconuts, coconuts are
better than apples. So therefore, bananas

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should be better than coconut. Well let's
check. Let's sort of check to be sure. So

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we compare apples to bananas. We see that
person one likes apples more than bananas.

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That's okay. Person two likes bananas more
than apples but person three. Likes apples

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more than bananas. So we get that one and
three right. Both prefer apples than

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bananas. So apples win. Well, look at
this. The group. Here's the collected.

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Here's where the collected preferences
are. The collected likes coconuts more

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than apples. Apples more than bananas. And
bananas. [inaudible] coconuts. That's

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irrational. That's not transitive. So
here's the really funky thing. We've got

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individuals, every single individual is
completely rational. They've got nice

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transitive preferences. There's no
inconsistencies. But then when we vote

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when we try to aggregate these preferences
we get something which is not consistent.

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So this is a paradox of aggregation. You
know, so before we talk about aggregation,

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we've got things, like, you know, simple
rules could create complex phenomena. Here

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[inaudible] aggregation of preferences.
You know, aggregation of some structure

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can give us something that's not, it
doesn't have one of the properties of the

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parts. So each part was rational. But the
collective isn't rational. So this is

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sometimes called, this is formally called
Condersay paradox. So each person is

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rational. Each person has rational
preferences. But then when we vote, the

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collective is not rational. The collective
says they go back. The collective says

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"okay Coconuts vs Apples - Coconuts".
"Apples vs. Bananas, Bananas." So then you

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think that for sure, sure they must like
Coconuts more than Bananas. But, in fact, if you

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have people vote they pick Bananas over
Coconuts. So this is the Condersay

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paradox. Each person is rational, but the
collective is irrational. So this has some

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pretty severe implications. The
implications are gonna be, that when we

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think about voting, that now, suddenly,
we're not necessarily gonna get a good

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outcome. We could get almost a random
outcome. And then it means that people

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might wanna vote strategically. So later
on in the course, when we start

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constructing models of how people vote,
this'll be near the end of the course.

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We'll see that the fact that aggregation
doesn't work. Right? That there's a

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problem with aggregation? With these
over-preferences? But that's going to

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create incentives, opportunities,
conditions under witch  people might want to

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manipulate agenda, lie about their
preferences or misrepresent their

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preferences in order to get outcomes that
they want to get. So here's a case right?

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Where aggregation doesn't give us
something we want. Okay? So just to drive

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this home. Each person has totally sane
rational transitive preferences. Exactly

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what you expect. But when you look at the
collective. The collective has this

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irrationality, so aggregation is sort of a
funny thing. That's why social science,

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right, in particular in this case,
politics, is so darn interesting. Right?

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Because. What happens at the macro level
is, sorta, logically inconsistent, even

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though its going out to the micro level
makes a lot of sense. [cough] okay so now

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we see an aggregation in several forms,
right. We've seen aggregation of numbers

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in the central limit theorem. We've seen
aggregation of rules, right, in both the

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game of life and the one-dimensional
cellular automaton. We saw we could get really

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complicated [inaudible] from simple parts.
And now I'm looking at preferences, we've

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sort of said what's a good aggregation of
some other mathematical structure, namely

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these orderings. And we found that
orderings that are you know, in the

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mathematical sense transitive, sort of in
the social sense rational. Don't

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necessarily aggregate into orderings that
are transitive and rational. So we get

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that they're sort of, we can lose
[inaudible] consistency as we go up. So,

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these sort of interesting aspects of
aggregation are ideas we're gonna play

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with throughout the course. Now, none of
these particular models, models any real

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thing, per se, [inaudible] but they're
building blocks. They're giving us a basis

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for how to think. If nothing else, I hope
these lectures are giving you some sense

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of, like, the mysteries and the
intricacies of adding things up. And why

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the social world is very, very different
than the parts that comprise it. Thank you