Effects of strategy-migration direction and noise in the evolutionary spatial prisoner’s dilemma

Spatial games are crucial for understanding patterns of cooperation in nature (and to some extent society). They are known to be more sensitive to local symmetries than, e.g., spin models. This paper concerns the evolution of the prisoner’s dilemma game on regular lattices with three different types of neighborhoods—the von Neumann, Moore, and kagomé types. We investigate two kinds of dynamics for the players to update their strategies (that can be unconditional cooperator or defector). Depending on the payoff difference, an individual can adopt the strategy of a random neighbor [a voter-model-like dynamics (VMLD)] or impose its strategy on a random neighbor, i.e., invasion-process-like dynamics (IPLD). In particular, we focus on the effects of noise, in combination with the strategy dynamics, on the evolution of cooperation. We find that VMLD, compared to IPLD, better supports the spreading and sustaining of cooperation. We see that noise has nontrivial effects on the evolution of cooperation: maximum cooperation density can be realized either at a medium noise level, in the limit of zero noise or in both these regions. The temptation to defect and the local interaction structure determine the outcome. Especially, in the low noise limit, the local interaction plays a crucial role in determining the fate of cooperators. We elucidate these both by numerical simulations and mean-field cluster approximation methods.

Figure 1

(Color online) (a) Average fraction $f_c$ of cooperation as a function of the temptation to defect $b$ for a fixed noise intensity $\kappa =0.3$ on three types of lattices. Solid and open symbols correspond to the two strategy-migration dynamics VMLD and IPLD, respectively. (b) Theoretical estimations by the four-site cluster approximation on the von Neumann lattice and by the five-site cluster approximation on the kagomé lattice, correctly predict the evolving tendency of $f_c$ but do not differentiate between VMLD and IPLD dynamics. Note the different scaling of the $x$ axis in the two panels.

Figure 2

(Color online) Time series of $N_C/N_D$, the number of events of $D$ switching to $C$ divided by the reverse case, for $b=1.02$ and 1.045 on the von Neumann lattice.

Figure 3

Average fraction $f_c$ of cooperation as a function of the noise parameter $\kappa$ for fixed values of $b$ on (a) the von Neumann lattice, (b) the Moore lattice, and (c) the kagomé lattice. The solid and open symbols are as shown in Fig. 1. The lines are guides to the eyes.

Figure 4

(Color online) Replotting of the data in Fig. 3 for (a) $b=1.04$ on the Moore lattice with the VMLD and (b) $b=1.12$ on the kagomé lattice with the IPLD. The places where $f_c$ reaches its maximum value are indicated by arrows.

Figure 5

Average fraction $f_c$ of cooperation versus $\kappa$ in the case of “high” temptation to defect $b=1.2$ on the kagomé lattice. The solid line denotes the analytical results obtained by the five-site cluster approximation.

Figure 6

(Color online) Theoretical estimations for $f_c$ as a function of $\kappa$ by using (a) the four-site cluster approximation method on the von Neumann lattice and (b) the five-site cluster approximation on the kagomé lattice. The arrows mark the places where $f_c$ maximizes when $b=1.12$.

Figure 7

Average fraction $f_c$ of cooperation as a function of $b$ in the case of very weak noise $\kappa =0.01$ on the kagomé lattice (solid squares) and the Moore lattice (open circles).

Figure 8

(Color online) Local configuration stability analysis in the limit of zero noise on the kagomé lattice. The central five-site cluster of cooperators cannot be destroyed as long as $b<3/2$ even if no neighbor or next-nearest neighbor is a cooperator. See the text for details.

Figure 9

Illustration of a small system with $4\times 4$ players. (a) Both VMLD and IPLD give out the same transformation probability for $j$ to change from a defector to a cooperator. (b) The VMLD favors better the diffusion of cooperation than the IPLD when the interface is rugged.