Sample path analysis of contribution and reward in cooperative groups

Explaining cooperative behavior is one of the major challenges in both biology and human society. The individual reward in cooperative group depends on how we share the rewards in the group. Thus, the group size dynamics in a cooperative group and reward-allocation rule seem essential to evaluate the emergence of cooperative groups. We apply a sample path-based analysis called an extension of Little's formula to general cooperative group. We show that the expected reward is insensitive to the specific reward-allocation rule and probabilistic structure of group dynamics, and the simple productivity condition guarantees the expected reward to be larger than the average contribution. As an example, we take social queues to see the insensitivity result in detail.