Figure 3. Responsible punishment can invade when rare.

Time-averaged frequencies of the 16 possible strategy combinations and typical transitions between homogeneous populations. Arrows with dashed lines indicate neutral drift between the two corresponding strategy combinations. Arrows with solid lines represent transitions where the target strategy has a fixation probability that exceeds the neutral probability 1/n. Populations marked with a colored ring can only be invaded through neutral drift. The Figure illustrates that unconditional defectors can be subverted by opportunistic defectors, which in turn can be swayed by responsible sanctions. However, once established, responsible sanctions can be replaced by unconditional defectors via the (unlikely) path via non-punishing cooperators, which can be invaded by non-punishing defector strategies. Parameter values are n = 80, b = 4, β = 3, c = γ = 1, s = 0.5, λ = 0.3 and frequencies are calculated for the limit of rare exploration. For clarity, we have only plotted arrows starting from strategies that are played in more than 0.5% of all cases.