Continuous game dynamics on populations with a cycle structure under weak selection

Understanding the emergence of cooperation among selfish individuals is an enduring conundrum in evolutionary biology, which has been studied using a variety of game theoretical models. Most of the previous studies presumed that interactions between individuals are discrete, but behavior in real systems can hardly be expected to have this dramatically discrete nature. In addition, existing research on continuous strategy games mostly focus on infinite well-mixed populations. Especially, there is few theoretical work on their evolutionary dynamics in structured populations. In the previous work [1], we theoretically studied the game dynamics of continuous strategies in a spatially structured population with its average degree k ≥ 3 under weak selection. Here, we study their evolutionary dynamics under weak selection on a cycle (k = 2), where each individual only interacts with its two immediate neighbors. Using the concept of fixation probability, we derive exact conditions for natural selection favoring one strategy over another for three update rules, called `birth-death´, `death-birth´, and `imitation´. It shows that for continuous strategy games, the same conditions are derived; especially, the simple rule b/c >; k is valid as well, where b/c is the benefit-to-cost ratio of an altruistic act. In addition, we present a network gain decomposition of the game equilibrium, which might provide a new view of network reciprocity, one of five mechanisms for evolution of cooperation.