We begin by presenting colour maps encoding the stationary fraction of cooperators ρ c in dependence on both the multiplication factor α and the initial amount of common resources R for three different values of the maximal possible endowment b. Several interesting conclusions can be drawn from the results presented in Fig. 1. First, it can be observed at a glance that increasing b (from left to right) increases ρ c over wide regions of α and R. This suggests that if common resources in the population abound, they should be distributed rather than held back. Although this seems to go against cautionary usage and conservation, holding back has, in the long run, several unintended consequences. If the common resources are not distributed right away, defectors can exploit the accumulated stock long after cooperators have disappeared from the neighbourhood. This creates an evolutionary niche for free-riders by means of which they can rise to complete dominance.

Figure 1 Socially responsible actions are viable even if the common resources are initially scarce, as long as the common pool is subsequently kept properly filled. Either too low or too abundant contributions, or failure to distribute them in time, can lead to the tragedy of the commons. Colour maps encode the fraction of cooperators ρ c in dependence on the multiplication factor α and the initial amount of common resources available to each group R, for three different values of the maximal endowment b: (a) 5, (b) 10 and (c) 20. Full size image

The impact of α and R is not as straightforward. As can be inferred from Fig. 1, only intermediate values of α ensure ρ c > 0. However, the span of the optimal interval depends on the maximal endowment b. The larger the maximal endowment b, the broader the interval of suitable values of α. Moreover, there exists an upper bound on R, beyond which cooperators cannot survive. The maximal R increases slightly with increasing b, but the effect is rather small. Conversely, even if initially the common resources are very scarce, cooperators are not negatively affected provided α and b are from within the limits that ensure ρ c > 0. This suggests that cooperative behaviour may develop even under adverse conditions and it is in fact more likely to do so than under abundance. The extinction of cooperators at both too large R and too large α indicates that an excessive abundance of common resources acts detrimental on the evolution of cooperation and that thus it deters social responsibility.

To further support our conclusions, we show in Fig. 2(a) the fraction of groups where the cumulative common goods can be sustained at equilibrium [i.e., R i (∞) > 0] and in Fig. 2(b) the fraction of groups where the cumulative common goods can provide enough endowments [i.e., R i (∞) ≥ Gb] for all involved. It can be observed that, in comparison to Fig. 1, the fraction of sustainable groups is larger than zero in a broader region of parameter values. It is much higher than the corresponding fraction of cooperators for large α. In combination with Fig. 1, we thus find that there exists an intermediate region of α that enables cooperators to dominate the population, as well as maintains a sufficient level of common goods in each group for individuals to be fully satisfied. Although the region for such a complete win-win outcome is not broad, it can be broadened by increasing the value of b.

Figure 2 Sustainability of common resources is achieved by socially responsible actions. Only an intermediate contribution strength, combined with initially scarce resources, leads to sustainable common resources. In panel (a) the colour map encodes the fraction of groups where the resources can be sustained [i.e., R i (∞) > 0], while in panel (b) the colour map encodes the fraction of groups where the cumulative common goods can provide enough endowments [i.e., R i (∞) ≥ Gb] for all involved. For results in both panels we use the maximal endowment b = 10. Full size image

The series of snapshots presented in Fig. 3 offers an insight as to what causes the described evolutionary outcomes. We use different colours not just for cooperators and defectors, but also depending on the available amount of common resources. More precisely, blue (yellow) colour denotes cooperators (defectors) that are central to groups where R i (t) ≥ Gb. On the other hand, green (red) colour denotes cooperators (defectors) where R i (t) < Gb. Grey are defectors where there are no more common resources left (note that R i (t) is always larger than zero if cooperators are present). For clarity, we always begin with R i (0) = Gb. Accordingly, blue cooperators and yellow defectors are initially distributed uniformly at random (leftmost panels of Fig. 3).

Figure 3 Spatial patterns explain why an excessive abundance of common resources deters social responsibility. Blue (yellow) are cooperators (defectors) that are central to groups where the common resources abound, while green (red) are cooperators (defectors) that are central to groups where the common resources are scarce. Grey denotes defectors where the common resources are completely depleted. Top row show the time evolution (from left to right) for α = 1, b = 10 and R = 50. Due to the low multiplication factor the common resources vanish fast. Middle row shows the time evolution for α = 10, b = 10 and R = 50. Here only cooperative groups succeed in keeping the pool from emptying. Groups with defectors quickly become unsustainable and hence pave the way towards cooperator dominance. Bottom row show the time evolution for α = 20, b = 10 and R = 50. Due to the high value of α common resources start to abound excessively, making even predominantly defective groups sustainable and thus fit to invade cooperators. Full size image

For low α (top row of Fig. 3), the common resources are depleted fast. Defectors turn to red and cooperators turn to green and widespread grey patches occur only after a few iterations of the game. Soon all is left are isolated islands of defectors who exploit the few remaining cooperators, until eventually all common resources vanish. Consequently, grey defectors come to dominate the entire population. This scenario is characteristic for the case when short-term benefits and ineffective cooperative efforts prevent sustainable management of common resources.

For intermediate α (middle row of Fig. 3), the scenario is very different. Grouped cooperators are able to preserve and enrich their resources, while groups with defectors fail to do so. Blue cooperative domains, where the common resources abound, become separated from red defectors by strips of green cooperators, which essentially protect the blue domains from being exploited further. The interfaces where green cooperators and red defectors meet become the shield that protects blue cooperative domains. In fact, blue cooperators are able to spread by means of an indirect territorial battle. It is important to note that yellow defectors are practically non-existent, i.e., a defector cannot sustain a profitable group and accordingly areas of grey soon emerge. These defectors become easy targets once being exposed to blue cooperators.

For high α (bottom row of Fig. 3), the situation changes again. Here the effectiveness of cooperators is so high that even a few in each group are able to provide more than enough resources for defectors. Accordingly, yellow defectors emerge, which can prevail even against blue domains of cooperators. Note that defectors still have an evolutionary advantage stemming from their refusal to sacrifice a fraction of personal benefits for the conservation of common resources. The stationary state is thus a diverse mix of all possible states, where defectors are more widespread since they don't contribute to the common pool. Nevertheless, if α is not too large some cooperators can still prevail by forming clusters, which as for intermediate values of α are shielded by green cooperators. The “shield”, however, is not very effective and accordingly has many holes, manifesting rather as isolated green cooperators which signal loss of the blue status rather than forming a compact chain that would prevent the invasion of defectors. If α is larger still (not shown), the utter abundance of common resources leads to the complete dominance of defectors and ultimately to the tragedy of the commons as for low values of α. The evolutionary path is significantly different though, given that for large α the tragedy is preceded by widespread yellow (rich) rather than red (poor) defectors. It is also worth noting that large initial values of R result in an identical demise of cooperation as large values of α.

To verify the robustness of the presented results, we conclude this section by considering several variations of the proposed collective-risk social dilemma game. In particular, we have studied the effects of (i) the population size, (ii) the topology of the population structure, (iii) different uncertainties by strategy adoptions, (iv) the delay in individual strategy updating, (v) the birth-death update rule36, as well as (vi) the effects of cooperator's priority towards limited endowments32. Since the obtained results are not central to the main message of this study, we present all the details and the obtained results in the Supplementary Information. Most importantly, we find that on structured populations our conclusions remain intact under all considered circumstances, thus indicating a high degree of universality. Nevertheless, we emphasize that our conclusions could be challenged under well-mixed conditions. Well-mixing will break up the clusters that we have described in the preceding paragraphs and this may change the results in favour of non-cooperation. This may be particularly relevant for human cooperation6,37, where the movements of and between groups could introduce well-mixed conditions.