Phase Transitions and Volunteering in Spatial Public Goods Games

We present a simple yet effective mechanism promoting cooperation under full anonymity by allowing for voluntary participation in public goods games. This natural extension leads to “rock-scissors-paper”–type cyclic dominance of the three strategies, cooperate, defect, and loner. In spatial settings with players arranged on a regular lattice, this results in interesting dynamical properties and intriguing spatiotemporal patterns. In particular, variations of the value of the public good leads to transitions between one-, two-, and three-strategy states which either are in the class of directed percolation or show interesting analogies to Ising-type models. Although volunteering is incapable of stabilizing cooperation, it efficiently prevents successful spreading of selfish behavior.

Figure 1

Frequencies of cooperators $C$ (open squares) and defectors $D$ (open diamonds) as a function of the multiplication rate $r$ for $\sigma =1$ and $\tau =K=0.1$. The solid line shows the frequency of defectors in pair approximation. The arrow indicates $r_C$ where cooperators vanish.

Figure 2

Frequencies of cooperators $C$ (open squares), defectors $D$ (open diamonds), and loners $L$ (open triangles) as a function of $r$ for $\sigma =1$ and $\tau =K=0.1$. The results of pair approximation are shown as dotted ($C$), solid ($D$), and dashed ($L$) lines. The values of $r_D$ and $r_L$ are indicated by arrows.

Figure 3

Distribution of cooperators (white), defectors (black), and loners (gray) on a $200\times 200$ portion of a larger lattice. The parameters are $\sigma =1$, $\tau =K=0.1$, and $r=2.035$, i.e., slightly above the transition to homogeneous states of loners.

Figure 4

Log-log plot of the frequency fluctuations $\chi$ vs $r-r_D$ (symbols and parameters as in Fig. 2). $\chi$ denotes the square of average fluctuation amplitudes produced by the system of size $M^2$. The solid line shows the fitted power law with $\gamma =1.72$.

Figure 5

Log-log plot of the correlation length $\xi$ vs $r-r_D$ for the three strategies (parameters as in Fig. 2). The solid line denotes the fitted power law with an exponent $-0.99$.