The Fisher-Wright and Moran Processes
Two models of how cooperators and defectors rise and fall in a finite population, one replacing every individual at once, the other one individual at a time.
We can begin by conceptualising a population of size N. Within the population, there are two types of individuals: cooperators, denoted c, and defectors, denoted d. The number of cooperators at any given time can be represented by i, while the number of defectors is necessarily N − i. The question in evolution is what happens to i over time. The Fisher-Wright process and the Moran process are useful models to help answer this question.
A central concept to both processes is fitness. Fitness can be understood as reproductive ability, such that an individual with higher fitness is more likely to produce offspring. Let the fitness of cooperators be Fc and the fitness of defectors be Fd. Within evolutionary game theory, fitness is derived from the payoffs. Both cooperators and defectors receive some average payoff from interacting with others, which translates into their respective likelihoods of reproducing successfully.
The Fisher-Wright process models the evolution of the population as one generation being entirely replaced by another. At a given time t, there are N individuals. At t + 1, there are N individuals who are the direct offspring of the previous N. A new generation replaces the previous one in discrete steps. Considering that there are i cooperators and N − i defectors, along with our definition of fitness, we can look at how the population evolves. The total reproductive weight of the cooperators is i · Fc, while that of the defectors is (N − i) · Fd. The probability p that any one offspring is a cooperator is the ratio of the reproductive weight of cooperators to the total reproductive weight of the population:
p = (i · Fc) / ( i · Fc + (N − i) · Fd )
The probability that any one offspring is a defector is simply 1 − p. Therefore, if the number of cooperators and defectors at time t is i and N − i, then the expected numbers at time t + 1 will be N · p and N · (1 − p) respectively.
Unlike the Fisher-Wright process, in which the entire population is replaced each time, the Moran process changes the population one individual at a time. At each step, one individual is born and one dies, so that the population N remains constant. As before, the number of cooperators is i and the number of defectors is N − i, and the probability that the reproducing individual is a cooperator is the same p as above, while the probability that it is a defector is 1 − p. The individual that dies is chosen uniformly at random, so the probabilities that it is a cooperator or a defector are i / N and (N − i) / N respectively. There are three possibilities: a cooperator reproduces and a defector dies, which increases the number of cooperators by one; a defector reproduces and a cooperator dies, which decreases it by one; or a cooperator reproduces and a cooperator dies, or a defector reproduces and a defector dies, resulting in no change.
Both the Fisher-Wright process and the Moran process serve as mathematical models for finite-population evolution over multiple generations.