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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â satisfy a relationship called transitivity. So these are transitive.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â that if we impose some rationality assumption like for people having

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â manipulate agenda, lie about their preferences or misrepresent their

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â