Emergence of social cooperation in threshold public goods games with collective risk

In real situations, people are often faced with the option of voluntary contribution to achieve a collective goal, for example, building a dam or a fence, in order to avoid an unfavorable loss. Those who do not donate, however, can free ride on others' sacrifices. As a result, cooperation is difficult to maintain, leading to an enduring collective-risk social dilemma. To address this issue, here we propose a simple yet effective theoretical model of threshold public goods game with collective risk and focus on the effect of risk on the emergence of social cooperation. To do this, we consider the population dynamics represented by replicator equation for two simplifying scenarios, respectively: one with fair sharers, who contribute the minimum average amount versus defectors and the other with altruists contributing more than average versus defectors. For both cases, we find that the dilemma is relieved in high-risk situations where cooperation is likely to persist and dominate defection in the population. Large initial endowment to individuals also encourages the risk-averse action, which means that, as compared to poor players (with small initial endowment), wealthy individuals (with large initial endowment) are more likely to cooperate in order to protect their private accounts. In addition, we show that small donation amount and small threshold (collective target) can encourage and sustain cooperation. Furthermore, for other parameters fixed, the impacts of group size act differently on the two scenarios because of distinct mechanisms: in the former case where the cost of cooperation depends on the group size, large size of group readily results in defection, while easily maintains cooperation in the latter case where the cost of cooperation is fixed irrespective of the group size. Our theoretical results of the replicator dynamics are in excellent agreement with the individual based simulation results.