An asymptotic analysis of the evolutionary spatial prisoner’s dilemma on a path

In this paper, we consider the Evolutionary Spatial Prisoner’s Dilemma (ESPD) in which players are modelled by the vertices of an underlying graph  GG representing some spatial organisational structure amongst the players. During each round of the ESPD every pair of adjacent players in GG play a classical prisoner’s dilemma against each other, and they update their strategies from one round to the next based on the perceived success achieved by the strategies of neighbouring players during the previous round. In this way, players are able to adapt and learn from each other’s strategies as the game progresses without being able to rationalise good strategies. We characterise all steady states of the ESPD for the case where GG is a path, and we also characterise the structures of those initial states that lead to the emergence of persistent substates of cooperation over time. We finally determine analytically (i.e. without using simulation) the probability that the game’s states will evolve from a randomly generated initial state towards a steady state which accommodates some form of persistent cooperation. More specifically, we show that there exists a range of game parameters for which the likelihood of the emergence of persistent cooperation increases to almost certainty as the length of the path increases.