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 $r_{AB}$, 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 $\beta\to\infty$, 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.