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Hi, folks. We're back and we're talking more about

Â diffusion and, in particular, were going to start looking at one of the best

Â known models in the literature on diffusion, which is known as the Bass

Â Model. And it's been used quite extensively

Â especially in marketing and trying to understand things.

Â So, we've gone through some questions and background.

Â And were going to start with the Bass Model.

Â And one thing about the Bass Model is networks are not going to make on

Â explicit appearance here, were just going to have social interaction in the

Â background. And after we've looked at the Bass Model,

Â then we'll enrich the analysis by bringing interaction structure in

Â explicitly, and try to understand how things diffuse over time.

Â Okay, so why is the Bass Model interesting?

Â It's a benchmark model. It gives us very stark and simple

Â structure which can begin to produce things like the S shapes that we saw

Â before. And there's going to be two actions,

Â behaviors or states, so either you're not infected or you are, or you're not

Â adopting the new product or you are. and what the Bass Model will keep track

Â of is over time, what's the fraction of individuals who have adopted say, taken

Â action one by time t, okay? And in this model, you move from state 0

Â to state 1, you don't move back. So, everybody starts at state 0, then

Â people start moving towards state 1. Some fraction of the people can move to

Â state 1 and that will just move over time, and we'll just keep track over time

Â of how many people have now moved to state 1.

Â So, they have seen the movie, or they've adopted corn, or they caught the flu

Â etc., okay? So, it's a one-way street and we keep

Â track of that over time. Okay, two key parameters in the Bass

Â Model. p is going to be a rate of spontaneous

Â innovation or adoption. So, that's going to be a rate that's

Â independent of what else is going on in the economy or the world or the society.

Â So, some people will just go and decide to see a movie regardless of what's going

Â on, or they will adopt corn, regardless of whether other people are doing it, if

Â they've heard about it, etc. And there's some other individuals who

Â will actually do it as a function of having heard about it, or somehow

Â imitated other individuals who were doing it.

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So, there's these two parameters p and q. And the Bass Model basically boils down

Â then to a, a simple equation which keeps track of the differential of the fraction

Â of people who have adopted at time t as, as t varies, okay?

Â So, how is this changing with time? Well, there's two parts to it.

Â So, at any point in time, there the people who haven't yet adopted, so these

Â are the people who have not yet adopted yet moved to one, right, so they haven't

Â become a one yet so they're still zeros. So, 1 minus Ft are the people who are

Â still 0 and can possibly change. And then, the fraction of those people

Â who change, well, some of them change directly spontaneously, that's the p.

Â And the others change by imitating the existing population.

Â So, you also, you have a chance of, of just spontaneously deciding to become a

Â 1, or you also have a chance of imitating somebody in the population and that's

Â proportional to how many in the population are already ones.

Â So, this q parameter says that you, you have some rate of imitation.

Â These are the ones existing. This is your imitation and so this is the

Â probability that you end up adopting 1 due to some imitation.

Â This is the chance that you do it spontaneously and these are the fraction

Â of people who've not yet adopted and who could make that change.

Â So, that gives us an expression for the differential overtime.

Â Very simple and intuitive model in terms of its basic building blocks.

Â Okay. So, when we look at this if you want to

Â solve this expression, right, and then you can start with an initial condition

Â of F of 0 equals 0. If you solve that, you get a simple

Â equation for what F of t looks like. And it depends on, on p and q, obviously.

Â and so higher p's and q's are going to lead to faster diffusion so more people

Â adopting by any particular time, lower p's and ques are going to lead to lower

Â adoption rates. And so this thing will be increasing in p

Â and increasing in q at, at any point in time the number of people would adopt by

Â that time. Okay, so you get a simple solution.

Â And then, let's look at the aspects of this and, and why the model has been so

Â well used and, and is well-known. Okay, so first thing it's, it's going to

Â end up giving an s shape, if q is bigger than p is going to tend to 1 in the limit

Â as t becomes large. and basically what's going on in this

Â model is initially, only, the only way that people can adopt is, is mainly

Â going to be through p. And then eventually, q is going to become

Â the important parameter. and things will slow down as your, a

Â fraction of people that have adopted eventually reaches 1.

Â So, if we go back and look at exactly what this equation looks like, let's try

Â and analyze this in a little more detail and try and understand why we get

Â interesting dynamics out of this. Okay, so first of all when Ft gets close

Â to 1, this thing is going to have to slow down.

Â And so, what happens is that the grade at which you're gaining new people adopting

Â as F of t gets close to 1, this thing gets close to 0, right?

Â And so, this thing has to get close to 0. So, this thing is going to tend to 0 as

Â Ft gets close to 1. So eventually, it has to slow down just

Â because the fraction of people who haven't adopted yet becomes small, so

Â even if a lot of them are adopting, there's just not many of them left and

Â that's what gives you the last part of the curve.

Â So that's very intuitive. And it's coming directly out of the, the

Â limitation that this thing has to converge at most to 1.

Â When you're initially at 0, then this part is going to go to 0, right?

Â So, there's no imitation going on because there's nobody to imitate.

Â And everything is just happening from the spontaneous adopters.

Â So, initially, this thing is going to, this thing is going to look like 1,

Â everybodycan still adopt, but all of it's going to happen through the p.

Â So, what starts out is you start out with a slope of p, right, so you start out

Â with some initial adoption rate from 0, you're going to start out at a slope of

Â p, and then eventually, the q's going to start kicking in.

Â So, as you get more and more people adopting here, then the acute can kick

Â in, and that can begin to, to give you the S shape.

Â So, the idea can be that the S shape can start going up, as q begins to kick in,

Â as more people start imitating, okay? But there's, there's a competing factor,

Â which is, as more people are adopting, this thing is also going down, right, so

Â you have fewer people left to adopt. So, whether or not you get this S shape

Â is going to be a race between the increase in q and the decrease in the

Â population that's left. And eventually, we know it has to, to, to

Â asymptote and, and be concave. And so, the question is whether it's

Â going to be initially convex. And so, when we can, we can analyze that

Â by looking at this process close to 0. We know that the initial slope is, is p,

Â so let's look at, at what happens at some small epsilon.

Â So, we've, we've just started moving out. And we'll see whether we're going to

Â start accelerating or not. Is it accelerating or is it going to be

Â already decreasing in speed? So, what does dF dt look like at, at some

Â small epsilon? It's going to look like p plus q epsilon

Â times 1 minus epsilon. So, if you just plug in a small epsilon

Â for this, you get this. To a need to get the initial convexity,

Â you need this thing to be bigger than p, right?

Â We have to be accelerating, the, the slope started out at p, now we have to be

Â getting a slightly larger slope. So initially, to get that S shape, you're

Â going to have to have this be bigger than p, and what does that tell us?

Â that tells us basically that q is going to have to be bigger than p.

Â So, for very small epsilon, the only way that you can have this thing be bigger

Â than p is to have q be bigger than p. So, if q is bigger than p then you'll get

Â it effectively the q, q epsilon is going to be bigger than the p epsilon.

Â and so we end up with a situation where you get the convex initial condition.

Â So, q bigger than p gives us the initial growth where we get a convexity at the

Â beginning, okay? So, we get this S shape here if p is

Â bigger than q, we get initially the, the slope is p, then it starts accelerating.

Â And eventually, when F of t gets very large, it's going to have to slow down

Â because now there's just not many of the adopters left, and then things slow down

Â and we eventually get the asymptote towards to 1, okay?

Â So, that's the Bass Model. very compact, simple, easy model.

Â the the, this model has been used quite extensively.

Â Why it has been used so extensively? it's, it's very compact, so if you can

Â begin, you know, suppose that we've, we're just here at some time period,

Â that's already enough to estimate what p was and to begin to see, to estimate what

Â q is. So, as this process takes off, you don't

Â need much data to begin to analyze and form estimates of p and q.

Â Once you've got the estimates of p and q, you can get estimates of what the rest of

Â the process is going to look like. So, this model has been used extensively

Â in forecasting by trying to estimate from initial take up.

Â So, if you look, say, at the box office of a new movie, how many people go see

Â the movie in the first week? How many people go see the movie in the

Â second week? Based on that first week and second week,

Â you've already got a piece of two pieces of this curve.

Â You can begin to project how long is it going to take for this thing to keep you

Â know, to slow down and, and where will it eventually reach in terms of its

Â asymptote. So, the Bass Model has been enriched a

Â lot by adding in extra moving parts. Maybe some people wouldn't see the movie,

Â so here this assumes it goes up to 100%, maybe some people would never see a

Â certain movie. you can enrich it by, you know, adding in

Â different kinds of heterogeneity and so forth.

Â So, there's different ways to enrich this model.

Â And people use richer versions of the Bass Model for forecasting.

Â But this, this basic simple part gives us that S shape in a very simple way.

Â And we realize that, you know, it, it's this combination of social imitation,

Â which grows with time, which gives us that convexity, which is important here.

Â