Figure 3: Inverse relationship between cooperation and antisocial punishment across parameters.

(a) The steady state frequency of cooperation and antisocial punishment from 5,000 random parameter sets is shown. An inverse relationship is clearly visible: when antisocial punishment is rare, cooperation (and pro-social punishment) are common. (b) To explore this relationship, we vary r from (σ–c) to 5 for various values of σ, γ, and β, fixing N=100, n=5, and c=1. We see that increasing r always increases cooperation while decreasing antisocial punishment. We also see that when β is small, the range of cooperation and antisocial punishment values is large, whereas values are tightly constrained when β is larger. Achieving high levels of cooperation and low levels of antisocial punishment requires both large expected returns on public investment (large r) and mostly symbolic punishments (small β).