Department of Physics and Astronomy, Sejong University, Seoul 05006, Korea.
Department of Physics, Pukyong National University, Busan 48513, Korea.
Phys Rev E. 2019 Jan;99(1-1):012410. doi: 10.1103/PhysRevE.99.012410.
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}^{-α}. 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}^{-β}. In a limiting case of β→∞, 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 β, we conjecture a formula for c_{th} as a function of α and β. 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.
我们研究了长程相互作用的利他主义在循环上的进化动态。个体之间的相互作用通过囚徒困境（PD）游戏的简化版本来描述，其中收益由合作行动的成本 c 参数化。在我们的模型中，游戏交互和竞争的概率随玩家 A 和 B 之间的距离 r_{AB} 以代数方式衰减，但具有不同的指数：即，玩 PD 游戏的概率与 r_{AB}^{-α} 成正比。另一方面，如果选择玩家 A 死亡，那么玩家 B 占据空位置的概率与 r_{AB}^{-β} 成正比。在β→∞的极限情况下，只有最近邻之间才会发生对空位置的竞争，我们从理论上找到了以 c_{th}为条件的利他主义扩散的条件，其中 c 是低于此值则利他主义占主导地位的阈值。对于有限的β，我们推测了 c_{th}作为α和β的函数的公式。我们还提出了一种数值方法来定位 c_{th}，根据该方法，即使选择强度相当大，我们也可以观察到与该推测非常吻合。
