## Volunteering in Public Goods games:

# Oscillations

#### by Christoph Hauert, Version 1.0, September 2005.

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*Oscillations*

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 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

`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 = 2`(`

*r*`). Above the threshold`

*σ*+ 1 <*r*<*r*= 2.2_{L}`defectors reign. Only for much larger`

*r*_{L}`cooperators reappear and co-exist with defectors. Since loners are absent, the dynamics again reverts voluntary participation into compulsory interaction. Finally, for`

*r*>*r*= 3.6_{C}`cooperators take over and manage to displace defectors.`

*r*>*r*= 4.6_{D}### Oscillations of strategies on random regular graphs

This figure clarifies the unexpected extinction of cooperators (and loners) for intermediate ` r`. For

`the maximum (minimum) frequencies of cooperators quickly increase (decrease), which indicates global synchronization and diverging amplitudes of the oscillations. Above`

*r*<*σ*+ 1 = 2`the oscillations eventually lead to the extinction of one strategy followed by another one. Once cooperators reappear (`

*r*= 2.2_{L}`), 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`

*r*>*r*= 3.6_{C}`(demonstrated for the prisoner's dilemma), which indicates a mean-field transition.`

*r*_{C}Click to enlarge | ## Optional public goods games on random regular graphsThe 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 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.r |
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Click to enlarge | ## Oscillations on random regular graphsThe sudden demise of cooperators and loners for intermediate values of . 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.r |

## Scenarios

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.

## Legend | Time evolution of cooperators, defectors and loners in structured populations where individuals engage in voluntary public goods interactions.
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## OscillationsFor 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.rThe figure (and simulation) on the left illustrates the periodic oscillations of the frequency of cooperators, defectors and loners as a function of time for | |||||

## Homogenous statesFor 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 | |||||

## Co-existence: cooperators and defectorsFor sufficiently high ) 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.r > r = 3.6_{C}The figure (and simulation) to the left shows again the frequency of cooperators, defectors and loners as a function of time but for |

## References

- Szabó, G. & Hauert, Ch. (2002)
*Evolutionary prisoner's dilemma games with voluntary participation*, Phys. Rev. E.**66**, 062903. - Hauert, Ch. & Szabó, G. (2005)
*Game theory and physics*, Am. J. Phys.**73**, 405-414. - Hauert, Ch. (2006)
*Cooperation, Collectives Formation and Specialization*, Advances in Complex Systems**9**, 315-335.