Figure 1: Antisocial punishment is common and punishment does not promote cooperation.

Time-averaged frequencies of each strategy and transition rates between homogeneous populations, (a) without punishment, (b) when cooperators can punish defectors, and (c,d) with the full set of punishment strategies (for illustrative purposes, only selected strategies are shown in panel c). A strategy X-Y₁Y₂Y₃ is defined by a public goods game action (X: C, cooperator; D, defector; L, loner), a punishment decision taken towards cooperators (Y₁: N, no action, P, punish), defectors (Y₂: N or P) and loners (Y₃: N or P). Transition rates ρ are the probability that a new mutant goes to fixation multiplied by the population size. We indicate neutral drift (ρ=1, dotted lines), slow transitions (ρ=11.8, thin lines), intermediate transitions (ρ=25.7, medium lines) and fast transitions (ρ=38.7, thick lines). Transitions with rates less than 0.1 are not shown. Parameter values are N=100, n=5, r=3, c=1, γ=0.3, β=1 and σ=1. In panels c and d, strategies that punish others taking the same public goods action are included in the analysis, but not pictured, because they are strongly disfavoured by selection and virtually non-existent in the steady-state distribution. For clarity, transitions with ρ< 10 are not shown in panel d.