In finite, well-mixed populations, replicator dynamics leads to oscillations with increasing amplitudes which eventually eliminate two of the three strategies. Global oscillations and synchronization can be suppressed by considering spatially extended systems such as lattice populations. It is important to note, however, that the stabilizing effects are intrinsically linked to the details of the population structure. In particular, the spatial separation of the oscillators prevents synchronization and leads to uncorrelated fluctuations. However, this can be compromised in spatial structures that include long range interactions such as small-world networks or random graphs where far reaching connections can induce global synchronization. To exemplify this, consider the extreme case of random regular graphs. Such graphs are generated by randomly assigning neighbors to each site under the constraint that every site ends up with the same number of neighbors while excluding self and double connections. In order to simplify comparisons, the number of neighbors of each individual and the updating procedure of the sample scenarios below are the same as in lattice populations.

Dynamical regimes in optional public goods games

Dynamical regimes for optional public goods games on random regular graphs

Frequency of cooperators (blue), defectors (red) and loners (yellow) in optional public goods interactions as a function of the multiplication factor of the common good. Individuals are arranged on random regular graphs where each node has eight neighbors and they interact in randomly formed groups of size N = 5. For small multiplication factors r < σ + 1 = 2 loners dominate. The reason is simple: in this case even in a group of cooperators the payoffs do not exceed the loner's income. As compared to lattice populations, all three strategies co-exist in dynamical equilibrium only for a very small range of r (σ + 1 < r < rL = 2.2). Above the threshold rL defectors reign. Only for much larger r > rC = 3.6 cooperators reappear and co-exist with defectors. Since loners are absent, the dynamics again reverts voluntary participation into compulsory interaction. Finally, for r > rD = 4.6 cooperators take over and manage to displace defectors.

Dynamical regimes in optional public goods games

Oscillations of strategies on random regular graphs

This figure clarifies the unexpected extinction of cooperators (and loners) for intermediate r. For r < σ + 1 = 2 the maximum (minimum) frequencies of cooperators quickly increase (decrease), which indicates global synchronization and diverging amplitudes of the oscillations. Above rL = 2.2 the oscillations eventually lead to the extinction of one strategy followed by another one. Once cooperators reappear (r > rC = 3.6), the fluctuations are very small. Note that the details of the population structure also affect the characteristics of strategy extinctions, i.e. of the type of phase transitions. For random regular graphs, the lack of spatial correlations results in a linear decrease of cooperators near rC (demonstrated for the prisoner's dilemma), which indicates a mean-field transition.

Frequencies of C, D and L on random regular graphs+Click to enlarge

Optional public goods games on random regular graphs

The dynamics of cooperators (blue), defectors (red) and loners (yellow) arranged on random regular graphs and engaging in optional public goods interactions with their neighbors depends on the multiplication factor r of the common good. Five qualitatively different dynamical regimes are observed: for low r only loners survive; increasing r leads to a narrow interval where all three strategies co-exist; for still higher r defectors dominate unchallenged; only for much higher r cooperators reappear and co-exist with defectors; finally, for very high r cooperators dominate.

Fluctuations of C on random regular graphs+Click to enlarge

Oscillations on random regular graphs

The sudden demise of cooperators and loners for intermediate values of r can be understood when considering not only the average strategy frequency but also their fluctuations. On the left, the average frequency of cooperators (blue) is shown together with their minima and maxima as a function of r. The large fluctuations hint at global oscillations in the system. Indeed, the long range connections in randon regular graphs enable global synchronization of strategy fluctuations. This contrast with lattice configurations where global synchronization cannot be achieved and the local fluctuations remain uncorrelated. For sufficiently high r, synchronization in random regular graphs results strategy fluctuations of increasing amplitude and at some point one strategy disappears. This breaks the cyclic dominance and hence a second strategy disappears leaving a homogeneous population behind. Which strategy vanishes first is random but loners are most likely, which results in a pure defector population.


Clicking either on one of the pictures below (or the corresponding link to the right) opens a new window with a running applet with all parameters preset to illustrate 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


For σ + 1 < r < rL = 2.2 cooperators, defectors and loners co-exist. The population structure leads to some synchronization which results in periodic oscillations of the strategy frequencies. With increasing r the amplitude quickly increases and thus increases the risk that one strategy gets eliminated. Note that the development of the oscillatory dynamics is rather sensitive to the initial configuration. Usually the system relaxes into a state with all loners, whenever the initial fraction of defectors is too high.

The figure (and simulation) on the left illustrates the periodic oscillations of the frequency of cooperators, defectors and loners as a function of time for r = 2.1. Loners dominate the population for most of the time, interrupted only by brief bursts of cooperation with an even briefer burst of defection in its wake.


Homogenous states

For r > rL = 2.2 the amplitude of the oscillations increases until one strategy is eliminated and thus a second one is doomed. Which strategy disappears first is random but depends on the population size and the initial configuration. In most cases defectors remain, which means that loners have the highest chance to vanish first.

The building up of the amplitude is nicely illustrated in the figure (and simulation) to the left, which depicts the frequency of cooperators, defectors and loners as a function of time for r = 2.4. Once the loners are extinct, cooperators follow swift and defectors eventually reach fixation.


Co-existence: cooperators and defectors

For sufficiently high r (r > rC = 3.6) cooperators reappear because under such favorable conditions they can thrive on their own, just as in lattice populations. Their evolutionary fate no longer hinges on the protection provided by loners. This leads to co-existence of cooperators and defectors, only, and hence to compulsory public goods interactions.

The figure (and simulation) to the left shows again the frequency of cooperators, defectors and loners as a function of time but for r = 4.3. Loners disappear quickly and then cooperators and defectors soon converge to an equilibrium state with very small fluctuations.