Hi. In this next module, what we're going to do is we're going to focus on a particular topic. Known as aggregation. Now aggregation is really an interesting thing to think about because just think about basic mathematics, right? We learned early on that one+1=2, right? And we think we can just sort of add things up and the sum is the, sort of the whole of it's parts. Well when we start modeling more interesting phenomena, whether it's physical, the physical world, the biological world or the social world, we find that aggregation is actually really tricky and one of the reasons we model, right, is to get the logic correct. And we find the logic of aggregation is really, sort of incredibly surprising and novel, now we already saw that. Earlier, in the, in the previous section, we talked about Schelling segregation models. Right, remember, people had these rules that they followed in order to decide where to live based on their tolerance of other people, right, people who looked different than they did. And what we found is that reasonably tolerant people, sort of you know, individuals finding rules that were tolerant, could lead to macro level segregation like we see in a city like New York, or Philadelphia, or Detroit. So, what we want to do in this next lecture is just construct some very simple models, some toy models, and when I say toy models what I mean are models that have very few moving parts that help us kinda understand some very basic logic about how the world works. And we're gonna use these toy models. To understand the process of aggregation. So it's going to be simple but then also sort of [laugh] mind-boggling in a way. Okay, so one of the core ideas in aggregation goes back to a famous paper written by the physicist Phillip Anderson. And Anderson's a Nobel Prize winner in physics, famous physicist from Princeton. And Anderson wrote a paper called More is Different. And in this paper what he says is, look you can sort of take a reductionist approach and pull everything back and look at something, you know, in great detail and say this is a salt crystal or this is a water molecule, right? And, or, you know, this is a. A neuron, but there's something very different when you connect all those things together. And so you can't do purely reduction of science and look at its individual parts and understand the whole. So more is different. And that's really going to be the focus of this module of lectures, is how, how is it some of the ways in which more can be different. So what did Anderson mean exactly? So, the most famous example that people use is this. This is a picture of a single water molecule, right, two hydrogens, one oxygen. And we can understand all the properties of a single molecule. But. One water molecule can't be wet, right? Wetness, the fact that we can like put our hand through water and feel the slipperiness, comes about because the fact that those hydrogen-oxygen bonds are fairly weak and so our, the bonds in our hands are stronger, so we can just push through it and feel that wetness. So wetness is a property of a bunch of water molecules, not of a single water molecule. Right? But [inaudible] is sort of, child's play, compared to something like cognition, personality. So think of the amazing things our brain can do, right? But our brain consists of a bunch of little neurons. Right, there's neurons and there's axons and there's dendrites, and there's myelination and all that sort of stuff, right? It's very complicated. But if we break the brain down to its parts, we're never gonna understand where cognition comes from, where personality comes from, or where consciousness comes from. Those are all what we're gonna call emergent properties of the system. Now, whenever we're gonna at least, I just wanna again explain consciousness, or cognition, but we're gonna sort of at least work through how is it that at the macro level than merchant level, we can work stuff that's really far more interesting and surprising that we sell at the micro level, right? So how we're gonna do it? What's our plan, how we're gonna proceed with an human aggregation? We're gonna start out by thinking about aggregation of actions, I'm going to talk about something called the central limit theorem. But we're going to talk about how just some actions add up. And that will just get us thinking about this notion of aggregation in a simple way. Then we're going to look at a particular game called The Game of Life and we're going to look at a single rule, just sort of one set of rules and just see how that rule aggregates just to give us a sense of mystery and wonder about how amazing simple things can be when they add up. Right? Third thing we're gonna do is look at a whole family of rules. We're gonna look at a class of Models of one dimension or cellular automata models. These one dimensional models are extremely simple, almost can't imagine a simpler model. And yet, we're gonna find that these very simple models can do anything, literally anything. So we talked about those [inaudible], they can do anything. And then, last, just to pull this into social science a little bit, we're gonna talk about aggregation of preferences. So think about aggregation, you think about adding up, like one+1=2. You know, two+4=6, that sort of stuff. We're adding single numbers. But preferences aren't single numbers. But there is, they're, sort of, you know, I like bananas more than apples, or I like, you know, Fords more than BMWs or something like that, right? It's a different, you know, different preferences, and we can ask, how do you add up preferences? They might say why, why would we want to add our preferences. Well we want to add our preferences because if we have a small group, if we have an organization, if we have an entire society, often times we have to make collective choices. And so these collective choices have to depend on our aggregate preferences. So what does everybody want? So the way you have to do that, you have to add up, here's my preferences plus someone else's. What do we get? Right. Okay. So what I wanna do in this sort of brief opening lecture. Is in the next couple of minutes. Is unpack a little bit more of what we're gonna do when we talk about aggregation. So the first thing, in terms of aggregation of actions. Right.? Remember we talked about why you model. Right. Bunch of reasons. One Of them is to start of [inaudible] points and one is to understand data. So. When we talk about aggregation of action. [inaudible] Someone?s decision to go to a store. Someone's decision to go on a plane. Right. [inaudible] You know the [inaudible]. Think of the [inaudible]. 300,000,000 People each day. People get up and make choices. And what we see at that [inaudible] level is sort of the average of those choices. And what we could show at a very simple model. Is why often times, those [inaudible] choices have a lot of structure to them. A lot of [inaudible]. And we're gonna get things that look like this picture. This is called a normal distribution or bell curve. And this bell curve, implies with it a certain amount of predictability and understand ability. So, very simple model lets us explain a whole bunch of things that happen in the real world. Alright. Next thing we're gonna wanna do. We use models to understand patterns. So a lot of what we see isn't just points, but distributions of things. It's things flowing. Now this is true in the physical world, the biological world. It's true inside our heads with neurons. It's also true sort of in the social world. So we're gonna construct a toy model, a fun model called the game of life and this game of life is gonna be very simple rules and we're gonna start out with patterns. So, here's a pattern right here, right? And time moves in this direction. Right. And we can see as this time moves this weird configuration keeps changing its shape. And then eventually down here, notice that in the exact same configuration it was that it started out with, but it sort of moved one down to the right. Now this is what we're going to call a glider and this is going to be a recurrent pattern in this model. And we're going to see how this thing which looks like it's living, hence The Game of Life, is really. Comes from this simple rules. Comes from one simple rule building on itself, right? Then, 'kay once we've done that sort of simple rule thing we're gonna even go to a simpler model called the one dimensional cellular automaton models, and these models we're gonna show how, a very, very simple model, works as follows. Imagine along string of lights, and each light can be on or off. And each like that has a rule whether or not to be on or off. Based on just two things. Whether it's on and off. And what it's two neighbors are doing. So it could be just says, if I'm on and my two neighbors are off. I'm gonna switch to off. So each light can use the same rule. And we're gonna see what sort of behavior we can get. What we're gonna find is we can get everything. [inaudible]. Remember what we talked about? What could the world do? Well, we could see equilibria. Right. We could see patterns, we could see complete randomness and chaos or we could see complexity. But we're gonna show how very, very simple rules can generate all four of those. And this is amazing, right? I mean, it's sort of, if you're in the mood to be amazed this will be an amazing result. So what do I mean exactly. I mean look at this incredibly complex pattern. Now you might look at something like this and say, wow, to produce something that complex there must be some really interesting complicated underlying dynamics. The answer is going to be no. You can compute, you can get things this interesting, right? With very, very simple rules. Alright. Then the last thing we're gonna do, the last lecture in this module, is gonna be about aggregating preferences. So what do I mean by preferences? Well, let's suppose there's apples, bananas and coconuts. And this might be me right here, so let's put a little S here. And it might be that I like apples better than bananas, better than coconuts. So these little greater than signs mean which one I like more than, [inaudible] each other. So this is apple is greater than bananas is greater than coconuts for me. Now, for someone else, like my soon Cooper, he might prefer bananas, right, to apples. And apples. To coconuts. So different people can have different preferences, and what we wanna talk about is how those aggregate. Now what we'll see is aggregation of preferences introduces all sorts of interesting paradoxes and creates all sorts of problems. Which is why, or at least one reason why, politics is so interesting, because the aggregation of just these simple preferences creates difficulties that don't arise when we think of just adding up numbers. Okay. So, big picture here. A lot of what interests me, as a social scientist, is groups of people. Aggregations of people. Now. How do we understand that? How do we understand how societies work, economies work, political systems work, organizations work? Well. You've gotta do two things. Sometimes, you've gotta understand how the parts work and then you've gotta understand how you add ?em up. So we're gonna sort of do that in the opposite order. We're first gonna talk about some of the complicatedness of adding things up, that's this module. And then the next module, we'll talk about the parts. Like, all these individual people in here, right? So, to understand the world, we're gonna have a twofold approach. First, understand, sort of, how things can add up. Second thing, add, understand the parts that do add up. And then the models that follow in this course, right? What we'll do is sort of put all those things together to make sense of things. Okay so. That's the outline this module. We're gonna you know. Play with some very, very simple play models that help us understand some of the mysterious phenomena we see in the world. And also just some of the sort of. [inaudible] Amazing results in here and some simple things add up to create very complex holes. Thank you.
