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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.