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
