Fig. 6: Pool punishment can resolve the social dilemma of public goods game in structured populations.

a and b In the three-strategy system space, the state consistently converges to full D. However, along the DO edge, an unstable equilibrium point, x^(DO), creates a bi-stable space. In the D versus O dynamics, the final state, either D or O, is determined by the initial conditions. Under mild punishment (β = 1.2), a structured population tends to favor defection, reducing the basin leading to the O outcome. c and d Conversely, with strong punishment (β = 5), structured populations result in the cyclic dominance of the three strategies, thereby preventing the full D state in public goods games. In contrast, the state space in well-mixed populations remains two distinct basins on the DO edge, preventing cyclic dominance. e As the punishment strength β increases, \({x}_{D}^{(DO)}\) increases, expanding the initial space leading to the O outcome. In well-mixed populations, \({x}_{D}^{(DO)}\to 1\) as β → ∞, and the basin leading to defection cannot be completely eliminated. However, in structured populations, \({x}_{D}^{(DO)}\to [{(k+1)}^{1/k}{(k-1)}^{1-1/k}-1]/(k-2) \, > \, 1\) when β > β^⋆, invariably resulting in the cyclic dominance of the three strategies. f The diagram of the different effects of punishment in well-mixed versus structured populations. Structured populations are advantageous in promoting cooperation under strong punishment but are a bit less effective when the punishment is mild. Input parameters: r = 3, c = 1, α = 0.7, k = 4.