The κ-threshold Volunteer’s Dilemma on networks

The classical Volunteer’s Dilemma (VD) assumes that a single volunteer is sufficient to provide a collective benefit to all participants. However, there exist many real collective-action problems that instead have a threshold structure, i.e., a minimum number of contributors is required before any benefit is achieved for the group. We study this κ -threshold VD on both regular and Erdős–Rényi networks under pairwise comparison dynamics with a Fermi update rule. We vary the threshold κ and record the volunteering rate, the provision probability, and any over-contribution. We report three main findings. Firstly, on d -regular graphs the volunteering rate ρ rises monotonically with κ but the provision probability P prov falls; we find no evidence of a phase transition over the parameter ranges investigated κ = 1 and κ = 2 . Secondly, on Erdős–Rényi graphs, at matched mean degree k ̄ , P prov is uniformly lower than on regular graphs, but the degree-stratified volunteering rate ρ ( k ) is essentially flat: under uniform cost, strategies do not vary across degree classes even when payoffs depend on local degree. Thirdly, when the cost is made dependent on the degree ( c i = α k i γ , ), the volunteering rate falls monotonically with increasing γ . Taken together, these findings indicate that, for the ER networks and parameter ranges considered here, degree heterogeneity on its own is not sufficient to generate degree-stratified volunteering under pairwise comparison dynamics; such behaviour emerges when contribution costs are themselves degree dependent.