Figure 1

The fraction of defectors ${\rho }_D$ as a function of the synergy factor $r$, as obtained for different combinations of strategies that compete for space on the square lattice. In addition to pure cooperators ($C$) and defectors ($D$), one-third of the lattice is initially occupied by one conditionally cooperative strategy $C_i$, as marked by each depicted curve. It can be observed that the higher the value of $i$ is, the earlier (at lower $r$) the downfall of ${\rho }_D$ is. Remarkably, if the most cautious conditional cooperators are introduced ($C_4$), defectors are completely defeated irrespective of $r$. We note that the results obtained with the related three-strategy strict rule model are identical.Reuse & Permissions

Figure 2

Schematic presentation supporting the interface stability analysis of competing domains. The leading process, which modifies the interface between the ordered domains more intensively, is the invasion across the border marked by the dashed line.Reuse & Permissions

Figure 3

Time evolution of the complete six-strategy public goods game with unconditional cooperators ($C_0$) and defectors ($C_5$) as well as the four conditionally cooperative strategies ($C_1$, $C_2$, $C_3$, and $C_4$), as obtained for $r=1.05$. As can be deduced from the results presented in Fig. 1, at such low synergy factors all but the $C_4$ strategy are outperformed by defectors. However, after the defector-induced extinction of $C_{0,...,3}$, the most cautious conditional cooperators ($C_4$) are able to completely invade the defectors ($C_5$). The inset shows the same evolution as obtained if using the strict rule model, and it can be observed that the final outcome is the same.Reuse & Permissions

Figure 4

Time evolution of the complete six-strategy public goods game with unconditional cooperators ($C_0$) and defectors ($C_5$) as well as the four conditionally cooperative strategies ($C_1$, $C_2$, $C_3$, and $C_4$), as obtained for $r=4.5$. At such a high synergy factor all five cooperative strategies $C_{0,...,4}$ are able to withstand being wiped out by defectors. In fact, the latter are forced to extinction primarily by $C_4$ and, to a much lesser extent, by $C_3$. As soon as defectors die out, however, all cooperative strategies become equivalent, and their evolution becomes identical to that of the voter model. As in Fig. 3, the inset shows the same outcome for the strict rule model.Reuse & Permissions

Figure 5

Fixation probabilities $P_f$ of the complete six-strategy game to eventually arrive at a pure $C_i$ phase depending on the synergy factor $r$. It can be observed that while at $r\to 1$ the fixation at $C_4$ is practically unavoidable, in the high $r$ limit all five cooperative strategies ($C_{0,...,4}$) become equally probable as the victors of the evolutionary process. Importantly, regardless of $r$, defectors are unable to survive, let alone dominate.Reuse & Permissions

Figure 6

Characteristic snapshots depicting the competition between conditional cooperators $C_4$ and unconditional defectors $C_5=D$ when starting from a random initial state, as obtained for $r=1.05$. Black shows $C_5$, white shows $C_4$ when they cooperate in at least three out of five groups, and light blue (gray) shows $C_4$ that cooperate in less than three of the five groups in which they are members. The snapshots were taken at (a) 0, (b) 10, (c) 30, and (d) 100 full Monte Carlo steps. The final state is a pure $C_4$ phase (all players are depicted in white), which is not shown. It can be observed that, initially, all the cooperators are practically inactive or hidden [light blue (gray) and black dominate in (a)]. Only after the spatial reciprocity takes effect and the first cooperative clusters are formed do the conditional cooperators actually start cooperating, although they do so only in the interior of the clusters [see (b)]. At the borders separating the two competing strategies, however, virtually all $C_4$ remain inactive as they are unable to gather the required number of positive signals from their neighbors [see (c)]. This thin intermediate layer of hidden and inactive $C_4$ then acts as a shield that makes it incredibly difficult for defectors to invade. In fact, defectors become effectively quarantined into bubbles from which they are unable to exploit cooperators [see (d) and Fig. 7 for a close-up]. Ultimately, this mechanism results in the tragedy of the defectors, irrespective of the value of $r$. The linear system size used here is $L=200$.Reuse & Permissions

Figure 7

An $80\times 80$ close-up of Fig. 6d, demonstrating clearly the spontaneous emergence of convex isolated bubbles of defectors (depicted in black) that are contained by inactive conditional cooperators of type $C_4$ [depicted in light blue (gray)]. While the latter will predominantly cooperate with the bulk of active conditional cooperators of the same type (depicted in white), they will certainly defect in the opposite direction, where there are unconditional defectors. Consequently, defectors cannot exploit $C_4$-type players, which leads to a gradual but unavoidable shrinkage of the defector quarantines.Reuse & Permissions