Group size dynamics for a group following game with shared rewards

We consider a population game in which a group of mobile individuals must repeatedly decide between two strategies: following other neighbors or moving on their own. When following others there is an improvement due to directed motion but a sharing of the credit for any rewards obtained. Conversely, the individual strategy provides for a full share of the credit for any rewards obtained at the cost of less-directed motion. This process leads to a dilemma with dynamically adjusting strategies and a resulting dynamic change in the number of group followers. The process is modeled as an evolutionary game with simple parameters that describe the differences in the motion opportunities and the sharing of rewards. A detailed analysis of the dynamics of the game is presented to show how the parameters affect the resulting equilibria solutions. Numerical methods are used to validate the analysis conclusions and simulations are performed to validate the analytical model of the evolutionary game.