In the natural world, performing a given task which is beneficial to an entire group requires the cooperation of several individuals of that group who often share the workload required to perform the task. The mathematical framework to study the dynamics of collective action is game theory. We study the evolutionary dynamics of cooperators and defectors in a population in which groups of individuals engage in N-person, non-excludable public goods games. We analyze the N-person Prisoner's dilemma (NPD), where the collective benefit increases proportional to the cost invested, and the N-person Snowdrift game (NSG), where the benefit is fixed but the cost is shared among those who contribute. We impose the existence of a threshold which must be surpassed before collective action becomes successful, and discuss the evolutionary dynamics in infinite and finite populations. In infinite populations, the introduction of a threshold leads, in both dilemmas, to a unified behavior, characterized by two interior fixed points. The fingerprints of the interior fixed points are still traceable in finite populations, despite evolution remaining active until the population reaches a monomorphic end-state. As the group size and population size become comparable, we find that spite dominates, making cooperation unfeasible.
Pacheco, J. M., Santos, F. C., Souza, M. O., & Skyrms, B. (2011). Evolutionary Dynamics of Collective Action. In The Mathematics of Darwin’s Legacy (pp. 119–138). Springer Basel. https://doi.org/10.1007/978-3-0348-0122-5_7