Curvature of Collective Returns Reshapes the Cooperation Transitions in Higher-Order Public Goods Games

In social and economic systems, collective returns often depend nonlinearly on group size, exhibiting a curvature that can be convex or concave, yet how this curvature reshapes cooperation transitions in higher-order social dilemmas is not fully understood. In this work, we study a controlled public goods game on hypergraphs with minimal ad hoc parameters, where linear pairwise interactions are supplemented by a three-body power-law payoff whose exponent γ controls the curvature of the collective return. Through systematic numerical simulations and microscopic diagnostics, we show that this curvature qualitatively reshapes cooperation transitions: for γ<1, the transition is continuous-like, similar to the linear benchmark γ=1, whereas for γ>1, the transition becomes abrupt and first-order-like. This distinction remains robust across different system sizes and multiple hypergraph topologies. We further show that cooperation in the γ>1 regime is primarily driven by a cascade-like proliferation of fully cooperative triads, accompanied by the depletion of partially cooperative configurations. Based on this mechanism, we predict and verify that a strict three-body product-form payoff—although motivated by a completely different social mechanism—produces the same discontinuous cooperation transition.