Hi, in this set of lectures we're going to talk about a particular model known as Replicator Dynamics. Now replicator dynamics are interesting, because they are used in three very different disciplines. They're used in Psychology to model learning. They're used in economics to model populations of people learning. And they're used in ecology to model evolution. So when you think about replicator dynamics, here's the basic idea. You imagine there's some set of types out there, so there's some set of types that become lists from one to N. Now each type has a payoff, which is sort of how well that type is doing, and then, there's a proportion of each type. So when you think of this, there's a population out there of types and those populations are succeeding at different levels. Some are doing well, some are doing less well. And what we want to think of, we want to think of, sort of the evolution or the dynamics of that process. In terms of how the distribution across types change, because what's going to happen? Think about it. If you got a distribution type, and some of them are doing better, and some of them are doing worse. The worse type should start copying the better types because the better types are doing better. So it's, that's the process that replica dynamics are going to help us try to understand. So if you think about learning, you can imagine there's a whole population of people out there, and they're, they're trying different strategies. So some people are using strategy one, some are using strategy two, some are using strategy three. So there's different proportions that people using these different strategies. When you look out there, you see those proportions. Maybe you see maybe 40% are using strategy one, 40% are using strategy two, and only 20% are using strategy three. So that might lead you to believe that why strategy one and two are better. But then you can maybe also look at the payoffs of these. Maybe the people using strategy one are getting a payoff of five, strategy two are using the payoff of four, and strategy three maybe there's a payoff of six. So now, do you think about well, what strategy should I use? It depends, if you're thinking in terms of which other strategies are more common. Which we thought of in our, sort of like, standing ovation models and our contagion models, and our conformity models. You might think strategy one and strategy two, because that's what other people are doing. But, if you think in terms of a rational act in terms of payoff, you'd say, well, I'm going to use strategy three, because look, that's got the highest payoff. So, Replicator Dynamics are going to do, is they're going to allow us to model situations where both of those dynamics are in play. Where people are both sort of copying more prominent strategies, and they're also copying strategies that tend to do better in the real world. Now, that's a model of learning. But we can use that same model, the same replicator dynamics model, to think about evolution. So now, instead of strategies, you can think of there being phenotypes. So now there's Phenotype one, Phenotype two, Phenotype three. What do I mean by phenotype? So these could be, let's suppose these are frogs. And these frogs have different lengths of tongues, so that, this one, two, three, is just giving us different lengths. And maybe P1 is the phenotype, is the longest tongue. And we give it a portion and say, 50% are this type, 30% are this type, and 20% are that type. So you could look around and say this is the distribution of types. Now if you think about reproduction of the species, the more there is of a type, the more likely they'll reproduce. So type one should be more likely to reproduce. But suppose we throw fitness in here, and we say type ones have a fitness of five, type twos have a fitness of four, and type threes have a fitness of six. Now again, you'd say, well, type ones have an advantage, because there's more of 'em, but type threes have an advantage, because they have higher fitness. So evolution is playing the same sort of trade off. Things that exist in higher proportion are more likely to get reproduced, just because there's more of them. But it's also the case that things that are more fit, are a lot more likely to reproduce, because well, they're more fit. So what replicator dynamics allows us to do, is model both learning in evolution using the exact same model. And to look at the dynamics over a population. So again sort of think of it as this, this, this distribution across these types. And that that distribution is changing, in response to the payoffs that those different types get. In the case of learning, or to the fitness that those types get, in the case of evolution. Now, once we've studied Replicator Dynamics, we're going to go to a really interesting called Fisher's Fundamental Theorem. Fisher's Fundamental Theorem talks about how the rate of adaption of a population, so [UNKNOWN] population is adapting to its surroundings. The rate of adaption is actually proportional to the variation of that population, [INAUDIBLE] the variations and the types. So, we'll write a formal theorem that says that. Now after we do that, what we're going to see is that that sort of runs counter to something we learned much earlier in the course, which was six sigma. Remember six sigma that you want to reduce variation. So Fisher's Fundamental Theorem says more variation is great cause [INAUDIBLE] adapt faster. Six sigma said no, more variation is worse, because it means more errors. So what we'll do is we'll try and make sense of that apparent contradiction in the final lecture in this sequence. Alright, let's get started.