When starting from a setting where cooperators, defectors and loners can co-exist, several interesting transitions are observed upon variation of the multiplication factor of the coomon good r. Lowering r leads to a transition to a homogenous state of loners, whereas increasing r reduces and eventually eliminates loners such that only cooperators and defectors co-exist. Further increases in r drive defectors to extinction and cooperators reach fixation. Each transition can be characterized and classified through different physical properties.

Phase transitions and diverging fluctuations

Phase transitions and diverging fluctuations

Upon decreasing the multiplication factor r and approaching the extinction threshold of defectors (and cooperators) rD, the fraction of defectors decreases linearly (mean-field transition). However, even at vanishing frequencies, the defectors give rise to interesting pattern formation in the cooperators and loner distribution. The frequency of cooperators and loners seems to approach 1/2 and typical domain size of cooperators and loners increases as r approaches rD. In this limit, equal frequencies of cooperators and loners are maintained by the vanishing fraction of defectors. This transition is accompanied by a power law divergence of the fluctuations in the frequency of loners (yellow) and cooperators (blue, mostly hidden by loner data). The fluctuations of defectors (red) remain constant. The solid line shows the fitted power law Χ ∝ (r - rD)-γ with γ = 1.72. Interestingly, this exponent is characteristic for the order parameter fluctuations in the Ising model (γ = 7/4) when approaching the critical point from above.

Diverging fluctuations+Click to enlarge

Diverging fluctuations

The extinction of defectors for high r as well as the extinction of cooperators in compulsory public goods interactions at lower r both exhibit critical phase transitions. Another critical phase transition occurs in optional public goods games when loners disappear. In this case, loners vanish leaving a fluctuating state of cooperators and defectors behind but the transition remains in the universaility class of directed percolation. All these transitions are accompanied by diverging fluctuations which are a hallmark of critical transitions. Another interesting transition with diverging fluctuations occurs when defectors (and cooperators) disappear at rD = σ + 1 and leave a state with all loners behind. Constant fluctuations are observed in the frequency of defectors (red) whereas fluctuations of loners (yellow) and cooperators (blue) diverge as r approaches the extinction threshold of defectors rD.

The tutorial on (compulsory) public goods games includes a separate section with a more detailed introduction and discussion of critical phase transitions in spatial games. This also includes a brief discussion of the maybe surprising but certainly substantial relevance of phase transitions for biological systems.


Clicking either on one of the pictures below or the corresponding link to the right will open a new window with a running applet illustrating the respective scenario. You can use this as a starting point to study effects of variations of the parameters.


Time evolution of cooperators, defectors and loners in structured populations where individuals engage in voluntary public goods interactions.

Color code:CooperatorsDefectorsLoners
 New cooperatorsNew defectorsNew loners

Transition to all loners

The transition to a state with all loners exhibits a linear decrease in the frequency of defectors as r approaches the threshold rD from above. Defectors form small islands invading the territory of cooperators but at the same time they are invaded by loners, which paves the way for the return of cooperators. Near the extinction threshold, defectors easily die out if the system size is not large enough. To avoid accidental extinctions, sufficiently large systems are required such that the average number of defectors is much larger than the root-mean-square of their fluctuations. Nevertheless, the vanishing defectors maintain the domain dynamics of cooperators and loners. If defectors are lost due to fluctuations, cooperators will take over.


Extinction of loners

The extinction of loners for r approaching the extinction threshold rL from below exhibits a critical phase transition in the universality class of directed percolation. The frequency of loners decreases with a power law proportional to (r - rL)-β with β = 0.58. The increase of fluctuations in the frequency of loners is consistent with a power law divergence predicted by scaling hypothesis.

Once loners disappear the voluntary participation reverts into compulsory public goods interactions. In finite systems, the accidential loss of loners can occur due to random fluctuations already for r < rL. If this happens, the clustering advantage of cooperators is often sufficient to result in the co-existence of cooperators and defectors but the equilibrium fraction of cooperators is substantially lower than with the support of even a tiny fraction of loners.


Extinction of defectors

Defectors vanish only after loners have disappeared. Thus, voluntary participation in the public goods game was reverted to compulsory interactions long before defectors go extinct. The extinction of defectors reflects another critical phase transition that also belongs into the universality class of directed percolation. This demonstrates the robustness of directed percolation transitions because the extinction of defectors leaves a homogenous state behind, whereas the extinction of loners occurs on an inhomogeneous and fluctuating background of cooperators and defectors.

This link points to a separate section on phase transitions in the context of traditional, compulsory public goods games in spatial settings.