Competition and cooperation among different punishing strategies in the spatial public goods game

Inspired by the fact that people have diverse propensities to punish wrongdoers, we study a spatial public goods game with defectors and different types of punishing cooperators. During the game, cooperators punish defectors with class-specific probabilities and subsequently share the associated costs of sanctioning. We show that in the presence of different punishing cooperators the highest level of public cooperation is always attainable through a selection mechanism. Interestingly, the selection does not necessarily favor the evolution of punishers who would be able to prevail on their own against the defectors, nor does it always hinder the evolution of punishers who would be unable to prevail on their own. Instead, the evolutionary success of punishing strategies depends sensitively on their invasion velocities, which in turn reveals fascinating examples of both competition and cooperation among them. Furthermore, we show that under favorable conditions, when punishment is not strictly necessary for the maintenance of public cooperation, the less aggressive, mild form of sanctioning is the sole victor of the selection process. Our work reveals that natural strategy selection cannot only promote, but sometimes also hinders competition among prosocial strategies.

Figure 1

Fraction of cooperators as a function of the punishment fine $\alpha$ and the probability to punish $p$, as obtained for a low multiplication factor $r=3.5$ in the original model proposed in Ref. [27], where a uniform probability to punish was assumed for all cooperators. Note that both $\alpha$ and $p$ have a nonmonotonic impact on the fraction of cooperators.

Figure 2

Panel (a) shows the fraction of different cooperator classes in the final state in dependence of $\alpha$ when they start fighting with defectors simultaneously. Panel (c) shows an enlarged part of panel (a) at low $\alpha$ values, when cooperation becomes dominant over defection. To present the overall level of cooperation in the population, the cumulative fraction of $C_i$ strategies is also shown (denoted by $C$). For comparison, in panel (b) we have also plotted the resulting fraction of cooperator classes when they fight against defectors individually. As in panel (c), panel (d) shows an enlarged part of panel (b) at a specific interval of $\alpha$. The multiplication factor in all panels is $r=3.5$.

Figure 3

Evolution of typical spatial patterns, as obtained for four different prepared initial conditions when using $\alpha =0.42$ and $r=3.5$. The first row shows the case when just a few $C_5$ cooperators are initially present among defectors. It can be observed that even under such unfavorable initial conditions the $C_5$ strategy can successfully outperform defectors. The second row features a similar experiment with the $C_3$ strategy, which fails to survive among defectors even though the latter are initially in minority. The third row illustrates cooperation among strategies $C_3$ and $C_5$, which together dominate the whole population even though $C_3$ alone would fail under the same conditions (see second row). We note that a neutral drift starts when defectors die out, as explained in the main text. The fourth row demonstrates, however, that the cooperation among different punishing strategies illustrated in the third row is rather fragile. If initially the strategy $C_3$ is replaced by strategy $C_2$, then the latter simply die out and subsequently the whole evolution becomes identical to the one shown in the first row, where strategy $C_5$ alone outperforms all defectors. For clarity, here the employed system size is small with just $L\times L=100\times 100$ players.

Figure 4

Individual competition of three different punishing strategies, namely $C_2,C_3$, and $C_5$, against defectors in dependence on time. Note that initially only one cooperative strategy and defectors are present, using the same initial conditions as illustrated in Fig. 3. Positive value of ${\rho }_{C_i}-{\rho }_D$ indicates the invasion of cooperator strategy while its negative value suggests invasion to the reversed direction. Note that while both $C_2$ and $C_3$ strategies ultimately lose their battle, the latter is able to prevail significantly longer. This enables an effective help of strategy $C_5$ when they compete against defectors together, as illustrated in Figs. 3(c1)–3(c4).

Figure 5

Stationary fractions of different cooperator classes in dependence on $\alpha$ when they compete against defectors simultaneously (a) and individually (b). The cumulative fraction of all punishing strategies (denoted as $C$ in the legend) is also plotted. The multiplication factor in both panels is $r=4.0$, which enables pure cooperators $\left(C_0\right)$ to coexist with defectors even in the absence of punishment.