Markovian Analysis of a Small Population Evolutionary Prisoner’s Dilemma Game

In this work we consider an evolutionary game in which a small population repeatedly plays the Iterated Prisoner’s Dilemma (IPD) game using the pairwise proportional imitation (PPI) revision protocol. Since we are dealing with a small population, we can explicitly formulate a Markov chain (MC) model of the evolutionary game and study its properties. In particular, we identify the absorbing states and their basins of attraction. In addition, we briefly study the replicator dynamics of the same game for the case of a large population. We find that, as the size of the population increases, the Markovian dynamics approximate the replicator dynamics.