Evolution of cooperation under punishment

Punishment has been considered as an effective mechanism for promoting and sustaining cooperation. In most existing models, punishment always comes as a third strategy alongside cooperation and defection, and it is commonly assumed to be executed based on individual decision rules rather than collective decision rules. Differently from previous works, we employ a democratic procedure by which cooperators cast votes independently and simultaneously for whether to impose punishment on defectors, and we establish a relationship between the cooperators' willingness to punish defectors (WTPD) and whether the punishment is inflicted on defectors. The results illustrate that the population can evolve to full cooperation under consensual punishment. It is noteworthy that, compared with autonomous punishment, whether consensual punishment is more in favor of cooperation crucially depends on the minimum number of votes required for punishment execution as well as the cooperators' WTPD. Our findings highlight the importance of collective decision making in the evolution of cooperation and may provide a mathematical framework for explaining the prevalence of democracy in modern societies.

Figure 1

Schematic diagram of the model. In an infinite and well-mixed population, $N$ individuals are randomly selected to participate in a public goods game with strategies cooperation and defection. Cooperators each contribute $c$ ($c>0$) to the common pool and make a decision about whether to employ punishment for defectors simultaneously. However, defectors contribute nothing to the common pool. By $q$ we denote the average degree of cooperators' willingness to punish defectors (WTPD). Peer punishment is employed and imposed on defectors by all cooperators as long as a lowest degree of consensus $T$ ($T\in [0,1]$) is reached, i.e., $\frac{n_p}{n_c}\ge T$, and otherwise the punishment is not executed.

Figure 2

Gradient of selection $\dot{x}$ for the public goods game under consensual punishment. By $q$ we denote the degree of cooperators' willingness to punish defectors (WTPD), and by $T$ the lowest degree of consensus for employing punishment. Stable (unstable) equilibria are depicted with solid (open) circles, and the expected direction of evolution is depicted with arrows. Parameters: $r=3,c=1,\alpha =0.3,\beta =1,N=5$.

Figure 3

Bifurcation of equilibrium for varying parameters. There are some branch points depending on the degree of consensus required for employing punishment $T$. For instance, there exist two bifurcations which subdivide the range of parameter $q$ into three regions, ${\mathrm{I}}_0$, ${\mathrm{I}}_1$, and ${\mathrm{I}}_2$ for $T=2/3$. The branch-point parameters are 0.297 and 0.320, respectively. When $q$ is varied in ${\mathrm{I}}_0$ ($q<0.297$), there are two boundary equilibria with one stable equilibrium at $x=0$. As $q$ increases, two interior equilibria including one stable interior equilibrium occur for $0.297\le q\le 0.32$. The state $x=0$ is still stable. When $q$ is varied in ${\mathrm{I}}_1$, there exists only one unstable interior equilibrium. The state $x=1$ becomes stable. Stable (unstable) equilibria are depicted with solid (open) circles, and the expected direction of evolution is depicted with arrows. All parameters are identical to those in Fig. 2.

Figure 4

Effects of parameters on the gradient of selection $\dot{x}$. Stable (unstable) equilibria are depicted with solid (open) circles, and the expected direction of evolution is depicted with arrows. Parameters: $T=0.5$, $q=0.5$, and $c=1$. (a) $N=5$, $\alpha =0.3$, and $\beta =1$. (b) $r=3$, $\alpha =0.3$, and $\beta =1$. (c) $N=5$, $r=3$, and $\beta =1$. (d) $N=5$, $r=3$, and $\alpha =0.3$.

Figure 5

Comparison of autonomous and consensual punishments for varying parameters. The color in panels (a)–(c) shows the value of the gradient of selection $\dot{x}$. By $q$ we denote the degree of cooperators' willingness to punish defectors (WTPD), and by $T$ the lowest degree of consensus for employing punishment. The solid and open circles depict the stable and unstable equilibria, respectively. The expected direction of evolution is depicted with arrows. All parameters are identical to those in Fig. 2.