Spatial prisoner's dilemma games with zealous cooperators

The existence of a zealot who stays a cooperator irrespective of the result of an interaction has been reported to add “social viscosity” to a population and thereby helps increase the cooperation level in prisoner's dilemma games, which premises the so-called well-mixed situation of a population. We found that this is not always true when a spatial structure, i.e., connecting agent, is introduced. Deploying zealots is counterproductive, especially when the underlying topology is homogenous, similar to that of a lattice. Our simulation reveals how the existence of never-converting cooperators destroys rather than boosts cooperation. We explain detailed mechanisms behind this interesting finding by referring to our previously presented concepts with respect to evolutionary dynamic processes for spatial games under the names enduring and expanding periods.

Figure 1

Cooperation fraction $P_c$ in the entire PD area for different ratio $f_{\mathrm{zc}}$ of ZCs. $f_{\mathrm{zc}}=\left(0,0.05,0.1,0.2\right)$ from the left. (A1)–(A4): Complete graphs correspond to well-mixed population, (B1)–(B4): BA-SF, and (C1)–(C4): lattice. $N={10}^4$ is assumed.

Figure 2

Snapshots of a certain trial in an SPD game on BA-SF when $f_{\mathrm{zc}}=0.1$ and $N=500$. Dilemma strength is set as $\left(D_g,D_r\right)=\left(0.2,0.2\right)$. A node in black denotes cooperators (Cs), that in white denotes defectors (D), and that in gray denotes zealous cooperators (ZCs). Each size of nodes corresponds to the degree. A line in gray indicates a C–C link or a ZC–ZC link, and that in black indicates a C–ZC link, where D links (D–C, D–ZC, and D–D) are removed.

Figure 3

Time series in an SPD game on BA-SF when $f_{\mathrm{zc}}=0.1$. Dilemma strength $(D_g,D_r)$ and population size ($N=500$) correspond to the snapshot in Fig. 2. This time evolution indicates the same trial as presented in Fig. 2.

Figure 4

Time evolution of a certain trial in an SPD game on BA-SF for each $f_{\mathrm{zc}}$. Population size of $N={10}^4$ is assumed.

Figure 5

Snapshots of a certain trial in an SPD game on a lattice. Red denotes cooperators (Cs), blue denotes defectors (Ds), and green denotes zealous cooperators (ZCs). Dilemma strength is set as $\left(Dg,Dr\right)=\left(0.4,0\right)$. (a) $f_{\mathrm{zc}}=0$ (default), (b) $f_{\mathrm{zc}}=0.1$.

Figure 6

Time evolution of a certain trial in an SPD game on a lattice for each $f_{\mathrm{zc}}$. Setting conditions such as dilemma strength and population size are the same as those in Fig. 5.

Figure 7

Cooperation fraction $P_c$ in the entire PD area for different ratio $f_{\mathrm{zc}}$ of ZCs. $f_{\mathrm{zc}}=\left(0,0.05,0.1,0.2\right)$ from the left. Panels (a), (b), and (c) respectively show Synchronous IM, Synchronous Pairwise Fermi, and Asynchronous. We assumed $P_{s_i\leftarrow s_j}=\frac{1}{1+exp\left[\frac{{\pi }_i-{\pi }_j}{\kappa }\right]}$ with presuming $\kappa =0.1$ for Synchronous Pairwise Fermi. In each panel, top, middle and bottom respectively show the result of lattice, BA-SF, and Erdős–Rényi random graph (E-R random). $N={10}^4$ is assumed.