Cooperation in the face of crisis: effect of demographic noise in collective-risk social dilemmas

In deciding whether to contribute to a public good, people often face a social dilemma known as the tragedy of the commons: either bear the cost of promoting the collective welfare, or free-ride on the efforts of others. Here, we study the dynamics of cooperation in the context of the threshold public goods games, in which groups must reach a cumulative target contribution to prevent a potential disaster, such as an environmental crisis or social unrest, that could result in the loss of all private wealth. The catch is that the crisis may never materialize, and the investment in the public good is lost. Overall, higher risk of loss promotes cooperation, while larger group size tends to undermine it. For most parameter settings, free-riders (defectors) cannot be eliminated from the population, leading to a coexistence equilibrium between cooperators and defectors for infinite populations. However, this equilibrium is unstable under the effect of demographic noise (finite population), since the cooperator-only and defector-only states are the only absorbing states of the stochastic dynamics. We use simulations and finite-size scaling to show that cooperators eventually die off and derive scaling laws for the transient lifetimes or half-lives of the coexistence metastable state. We find that for high risk, the half-life of cooperators increases exponentially with population size, while for low risk, it decreases exponentially with population size. At the risk threshold, where the coexistence regime appears in a discontinuous manner, the half-life increases with a power of the population size.