Evolutionary Games and Local Dynamics

Evolutionary games have been introduced by Maynard Smith and Price [1973] in the early 70’s with the aim to study the stability of populations in time. In the general model, a population consists of finitely many different types that interact randomly with each other, where each interaction leads to fitness payoffs for each of the types involved. Consequently, the population distribution over the different types is changing in time. When assuming that the fraction of a type changes in a Darwinian way proportionally to its current fraction and proportionally to the difference in fitness with average population members, then the population development can be represented by the so-called replicator dynamics, introduced by Taylor and Jonker [1978]. Although many other population dynamics have been studied for evolutionary games (see Hofbauer and Sigmund [1998] or Sandholm [2011] for a review of all kinds of dynamics), we would like to take the replicator dynamics as our starting point in view of its relation to the Darwinian theory. However, we want to switch from a global to a local perspective, because one of the underlying assumptions in the model of Maynard Smith and Price [1973] and in that of Taylor and Jonker