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Hi. In this lecture we're again going to talk about replicator dynamics. And

Â Remember replicator dynamics have this idea in them that the proportion of people

Â playing a particular action at time T plus one depends on the proportion paid at a

Â time T And then the payoff for that action at time T. Now, in the previous lecture,

Â we talked about replicator dynamics in the context of people playing actions or

Â strategies, so populations of individuals. In this lecture, we're gonna talk about it

Â in an ecological context. So we're thinking about, we're thinking about

Â different phenotypes of a species, and those phenotypes having different

Â fitness?s. And think about replicator dynamics as a way to capture the dynamics

Â of that population, that, of a species. Let me explain what I mean. Remember, in

Â replicator dynamics, there's a set of types. But now, instead of assuming a

Â payoff to each type, I'm gonna assume there's a fitness to each type. That's

Â sort of how fit the species is. That, remember the species is two if particular

Â [inaudible]. A logical match and I'm also going to assume there's some proportion of

Â each type. Well, now what we can do is we can think of the exact same logic. How

Â many of each type are gonna get reproduced in the next population? Well, it's gonna

Â depend on the fitness of each type, and the proportion. Because the more birds

Â there are of a particular phenotype, the more offspring they're gonna have. But

Â it's also due to the more fit a particular type of a species is, the more offspring

Â it's gonna have. So the fitness and the proportion are gonna determine how many

Â there'll be in the next population. Now how I want to think of this is a fitness

Â wheel. So you can think of is, when you're choosing a mate, that there's this, this

Â giant wheel. And you sort of spin this wheel of fortune. And so there's a diff,

Â bunch of different types. There's type 1s, type 2s and type 3s. And you spin the

Â wheel and it stops on type two. Now the property [inaudible] on the type 2's

Â depend on two things, the number of 2's. So there's only two of them And the

Â thickness of 2's. The reason we call this the fitness wheel and not just a wheel is

Â the size of the pie here you can think of as being proportional to the fitness. So

Â the more fit you are the bigger your slices. So 2's are fit so they get really

Â big slices. 1's are not very fit so they get small slices. But then it's also the

Â case, the more of who they are, the more slices you get. So there's lots of ones,

Â there's four ones, so they get more slices. Well this fitness wheel

Â [inaudible] metaphor is the same thing as replicator dynamics. You can think of the

Â size of the slice. As being proportional to the fitness, is you're gonna get a

Â number of slices, representing the number of species of that type, and this will

Â give you exactly replicator dynamics. And what you can think of, and sometimes it's

Â bunching those all together, so putting all the 1's in one big slice, all the 2's

Â in one big slice, and all the 3's in one big slice, and then spin the wheel that

Â way. And that's another way to think of these replicator dynamics. Now, I'm going

Â to use replicator dynamics and maybe of the fitness field implicitly, to explain

Â something called Fisher's fundamental theorem. Fisher's theorem is going to be

Â really cool, because it is going to allow us to combine a bunch of models that we

Â have already used. So remember we had the model that said there is no cardinal that

Â meant there is a lot of variation within a species. Second, we had that model of

Â rugged landscapes, the idea being that like, when you encode a function, you

Â could think of it as a rugged landscape, that you are trying to climb hills. And

Â then third, we've got these model of replicator dynamics. What Fisher's

Â fundamental theorem is going to do, is it's going to combine all. All of these

Â models into one, and give us an insight about the role that variation plays in

Â adaptation. Okay, so hang on, a lot going on here. So remember our, there is no

Â cardinal. That meant that there's a population of things that we call

Â cardinal. Ther e's genetic and phenotypic variation in the population of cardinals.

Â And remember we also had the rugged landscape model, saying that if you think

Â of a cardinal, it could be, have a fitness, which is sort of, maybe,

Â somewhere here. This one here will have a fitness of this height. One down here is a

Â lower fitness. This has a fairly high fitness, and this has the highest fitness

Â of all. So we can place different cardinals on the landscape, and different

Â cardinals are gonna have different fitness?s. So we could think of, then.

Â Replicator dynamics is saying, what's gonna happen? You're gonna copy the

Â [inaudible] fit, and you're also gonna copy the people who exist in higher

Â proportion. So we can use, we can place all of those diverse cardinals on one

Â landscape. And then we can imagine that replic-, that replicator dynamics are

Â gonna help us choose the ones that are sort of higher up on the landscape. So

Â here's Fisher's theorem, the idea anyway, that higher variances, If you have more

Â variation, then you should be able to adapt faster. You should be able to climb

Â the landscape faster. Let's see why that's probably true. So suppose there's low

Â variation. There's very low variation. And now I apply some selective pressure. I can

Â only climb up a little bit. But if there's high variation, then I can climb a lot

Â faster. So the fast, the more variation, the more people I've got to copy, the more

Â likely there is to be someone good, the better I'm gonna do. So let's do an

Â example and see why this is the case. So let's start with the population that has

Â one third of people at fitness three, one third of the fitness four and one third of

Â the fitness five. So note that the average fitness here is just gonna equal four.

Â Well let's look at the weights, let's use [inaudible] dynamics and let's figure out

Â the weights for each of these different strategies. The weight on strategy one Is

Â going to be one third times three, which is one. The wait on strategy two is one

Â third times four, which is four thirds. And the wait on strategy t hree is one

Â third times five, Which is five thirds. Well now, let's compute the proportion we

Â are gonna have in the next period of each type is of type one. Proportion of type

Â one is just gonna equal one over one plus four thirds plues five thirds, which is

Â gonna be. Three over twelve, to proportion of type two, is gonna be four thirds over

Â one, plus four thirds, plus five thirds which is gonna be four over twelve. And a

Â proportion of type three, is just gonna be five thirds over one, plus four thirds,

Â plus five thirds, which is five over twelve. So we're gonna have 3/12's fitness

Â three, 4/12's fitness four and 5/12's fitness five. Now if we figure out what's

Â are new average fitness gonna be, that's gonna be three Times three 12's, plus

Â four, times four 12's, plus five, Times five 12's. So what we're going to get is

Â nine plus sixteen which is 25, [inaudible] 50 over twelve. >> Which if we divide here

Â is gonna be four and a sixth. So what we get is we started on an average fitness of

Â four, we end up with an average fitness of a sixth. Let's now do a case where we've

Â got medium variants, so before the paths were three four, and five, now the

Â variances, the fitness?s are two, four, and six. Let's do the same thing. So

Â what's the weight on strategy one, that's gonna be one third times two, Which is

Â 2/3's. The weight on strategy two is going to be one-third times four which is 4/3's

Â and the weight on strategy three is going to be one-third times six, Which is six

Â thirds. And again, the average fitness here, as before, was equal to four, So now

Â if we want to complete the probability that someone?s going to be of type one in

Â the next period. That's just going two-thirds over two-thirds + 4/3 + 6/3, so

Â that's going to be two over twelve. The probability of someone?s, of type two Is

Â gonna be, and notice we can get rid of all the thirds here. So that's just gonna be

Â four over two plus four plus six, so that's 4/12. And the probability

Â [inaudible] of type three is gonna be just six over two plus four plus six, whic h is

Â 6/12. So now, if we wanna [inaudible] the new average fitness in this new

Â population, 'cause before, it went from one-third, one-third, one-third, to 2/12,

Â 4/12, 6/12, We're gonna get that it's two. X(2x12)+ 4x(4x12) + 6x(6x12). So that's

Â gonna be four+16 which is twenty+36 which is 56/12. All right, And so that's gonna

Â to be Four, and 4/6ths, So before we had four and six. Now we're going to get four

Â and 4/6ths. Last let's do a population where we have really high variance. So

Â 1/3rd of a population of zero, 1/3rd of a fitness of four, and 1/3rd of a fitness of

Â eight. Again let's do all the math. Here's what we get. For Wayern strategy one, this

Â will be easy. It's going to be zero, because the fitness is zero. Latent

Â strategy two, is gonna be four thirds. And latent strategy three is gonna be eight

Â thirds. So the probability of someone's strategy one next time is gonna be zero.

Â The probability of someone's strategy two is just gonna be four thirds over four

Â thirds plus eight thirds. So that's a third, which means a probability of

Â someone's a strategy three is gonna be two thirds. So when we compute the new average

Â fitness, number will be four there. The average fitness was four. The new average

Â fitness is gonna be one third Times four plus two-thirds times eight. So that's 4/3

Â plus sixteen, which is 20/3, right? So what we get is we get the average fitness

Â is then gonna be six and two-thirds. So in the first case, what we get is we got a

Â gain. We need to finish this where at three, four, and five, fitness increased

Â by one-sixth. In the second case, remember we had a little bit more variation, the

Â fitness increased by 4/6. And in the third case when there had been greater

Â variation, the fitness increased by two and 4/6. Remember because the average came

Â up from four to six and two-thirds. So what we see, we see the, the amount of

Â gain seems to be increasing in the variation. So the more variation, the

Â faster the population can adapt. Well, let's compute the variation in each of

Â these populations. Remember va riation is just the difference from the mean. So in

Â the first case, the variation will be three minus four, squared, plus four minus

Â four squared, plus five minus four squared, so that's just gonna be two. And

Â now just to the last, in the last case you're gonna get zero minus four squared

Â which is sixteen, plus four minus four squared which is zero. Plus eight minus

Â four squared, which is also sixteen, we're gonna get 32. So this gives us the

Â variation within each population. So what we had before was the gain, and if we put

Â this in terms of six, the gains are one sixth, four sixths, and sixteen sixths.

Â And if we look at the variation, it's two, eight, and 32. We'll notice, this is gis

Â one goes to two, this is gis times two, this is gis times two, and this is gis

Â times two. So the gain is exactly half the variation, in each case. In effect, this

Â is Fisher's fundamental theorem. The change in average fitness due to

Â selection, if we have replicator dynamics, is gonna proportional to the variation. So

Â more variation, More adaptation and they're proportional, And we saw this by

Â combining three models. There is no cardinal, there's a rugged landscape in

Â replicator dynamics. And we get this really interesting result. That the change

Â in average fitness due to selection, due to replicator dynamics, is gonna be

Â proportional to the variation. And again, we got it by combining three different

Â models. So this one of the powers of being a many model thinker, is you can then

Â combine them to ask much more deeper scientific questions. But there's a rub

Â here, and this is what we're gonna come to in the next lecture. We just got this idea

Â that says more variation is better. But this one's counter to something we learned

Â very early on in the course, which is that you wanna reduce variation because of six

Â sigma. So what I wanna do in the last lecture mission is contrast these two

Â models. Because when we think about becoming model thinkers, what you'd like

Â to do is have lots of models in your head, and use those to adjudicate differ ent

Â intuitions. To figure out which logic applies in which situation. So that's

Â where we'll go next. Alright, thank you.

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