Evolutionary dynamics of continuous strategy games on social networks under weak selection: A preliminary study

Cooperation is a fundamental principle of all biological systems. Most previous studies presumed that the interactions between individuals are discrete, namely, each individual offers either cooperation or defection. This discrete strategy seems unrealistic in real systems and cooperative behavior in nature should be viewed as a continuous trait. Existing research work on games with a continuous strategy mainly focuses on infinite well-mixed populations. Additionally, our previous work showed that there is a considerable difference in terms of equilibria between continuous and discrete strategy games on graphs under strong selection. This paper studies the game dynamics in finite structured populations under weak selection using the stochastic dynamics based on respectively the mutant fixation probability (ργ) and the fixation probability ratio of mutant to resident (ργ/ρχ). For three update rules, called 'birth death' (BD), 'death-birth' (DB) and 'imitation' (IM), we derive exact conditions for natural selection favoring one strategy over another. Comparing discrete strategy games, we find that for continuous ones (i) the rule, b/c >; k, is also valid; (ii) the same selection conditions are also derived using ργ/ρχ; however, (iii) the selection conditions obtained using ργ and ργ/ρχ are the same instead of different; and (iv) interestingly, the '1/3' rule is not observed for DB and IM updating.