 The replicator equation is the first and most important game dynamic studied in connection with evolutionary game theory. The replicator equation and other deterministic game dynamics have become essential tools over the past 40 years in applying evolutionary game theory to behavioral models in biology and the social sciences. These models try to show the growth rate of a portion of agents using a certain strategy. As we'll illustrate in this video, this growth rate is equal to the difference between the average payoff of that strategy and the average payoff of the population as a whole. There are just three primary elements to a replicator model. Firstly, we have a set of agent types, each of which represents a particular strategy and each type of strategy has a payoff associated with it, which is how well they're doing. There is also a parameter associated with how many of each type there is in the overall population. Each type represents a certain percentage of the overall population. Now, in deciding what they might do, agents may adopt two approaches. They may simply copy what other people are doing. In such a case, the likelihood of an agent adopting any given strategy would be relative to the existing proportion of that strategy within the population. So if lots of people are doing some strategy, the agent would be more likely to adopt that strategy over some other strategy that few are doing. Alternatively, the agent might be more discerning and look to see which of other people's strategies is doing well and then adopt that one that is most successful, having the highest payoff. The replicator dynamic model is going to try and balance these two potential approaches that agents might adopt and hopefully give us a more realistic model than one where agents simply adopt either strategy solely. Given these rules, the replicator model is one way of trying to capture the dynamics of this evolutionary game to see which strategies become more prevalent over time or how the percentage mix of strategies changes. In a rational model, people simply adopt the strategy that they see as doing the best amongst those present. But equally, people may simply adopt a strategy of copying what others are doing. If 10% are using strategy 1 and 50% strategy 2 and 40% strategy 3, then the agent is more likely to adopt strategy 2 due to its prevalence. So the weight that captures how likely an agent will adopt a certain strategy in the next round of the game is a function of the probability of its prevalence times the payoff. If we wanted to think about this in a more intuitive way, we might think of having a bag of balls where a ball represents the strategy that will be played in the game. If a strategy has a bigger payoff, then it will be a bigger ball and we will be more likely to pick that ball. Equally, if there are more agents using that strategy in the population, there will be more balls in the bag representing that strategy, meaning again we will be more likely to choose it. The replicator model is simply computing which balls will get selected and thus what strategies will become more prevalent. One thing to note though is that the theory typically assumes large homogenous populations with random interactions. The replicator equation differs from other equations used to model replication in that it allows the fitness function to incorporate the distribution of the population types rather than setting the fitness of a particular type constant. This important property allows the replicator equation to capture the essence of selection. But unlike other models, the replicator equation does not incorporate mutation and so is not able to innovate new types of pure strategies. An interesting corollary to this is what is called Fischer's Fundamental Theorem which is a model that tries to capture the role that variation plays in adaptation. The basic intuition is that a higher variation of the population will give it greater capacity to evolve optimal strategies given the environments. Thus given a population of agents trying to adapt to their environments the rate of adaptation of a population will be proportional to the variation of types within that population. Fischer's Fundamental Theorem then works to incorporate this additional important parameter of the degree of variation among the population so as to better model the overall process of strategy evolution in the population. Static game theoretical solution concepts such as Nash Equilibrium play a central role in predicting the evolutionary outcomes of game dynamics. Conversely, game dynamics that arise naturally in analyzing behavioral evolution lead to a more thorough understanding of issues connected to the static concept of equilibrium. That is, both the classical and evolutionary approaches to game theory benefit through this interplay between them. Replicated dynamic models have become a primary method for studying the evolutionary dynamics in games both social, economic and ecological.