Long-range prisoner's dilemma game on a cycle.

We investigate evolutionary dynamics of altruism with long-range interaction on a cycle. The interaction between individuals is described by a simplified version of the prisoner's dilemma (PD) game in which the payoffs are parameterized by c, the cost of a cooperative action. In our model, the probabilities of the game interaction and competition decay algebraically with rAB, the distance between two players A and B, but with different exponents: That is, the probability to play the PD game is proportional to r(AB)(-alpha). If player A is chosen for death, on the other hand, the probability for B to occupy the empty site is proportional to r(AB)(-beta). In a limiting case of beta -> infinity, where the competition for an empty site occurs between its nearest neighbors only, we analytically find the condition for the proliferation of altruism in terms of c(th), a threshold of c below which altruism prevails. For finite beta, we conjecture a formula for c(th) as a function of alpha and beta. We also propose a numerical method to locate c(th), according to which we observe excellent agreement with the conjecture even when the selection strength is of considerable magnitude.