Phase diagram and criticality of the two-dimensional prisoner's dilemma model

The stationary states of the prisoner's dilemma model are studied on a square lattice taking into account the role of a noise parameter in the decision-making process. Only first neighboring players—defectors and cooperators—are considered in each step of the game. Through Monte Carlo simulations we determined the phase diagrams of the model in the plane noise versus the temptation to defect for a large range of values of the noise parameter. We observed three phases: cooperators and defectors absorbing phases, and a coexistence phase between them. The phase transitions as well as the critical exponents associated with them were determined using both static and dynamical scaling laws.

Figure 1

Normalized density of cooperators as a function of $b$ for several values of $K$ for (a) $a=0$ and (b) $a=1$. The inset in (a) corresponds to details of the transition for $K=0$.

Figure 2

Phase diagram in the plane $b$ vs $K$ for (a) $a=0$ and (b) $a=1$. Open circles represent the critical point $b_1$ from the absorbing $C$ to the mixed phase $(C+D)$, while open squares represent the critical points $b_2$ from the mixed phase to the absorbing $D$ phase.

Figure 3

Plots of the logarithm value (base 10) of the stationary order parameter as a function of the logarithm of the distance from the critical point for $a=0$ and $K=1.0$. Open circles represent the simulation data and the straight lines are the best fit to them. (a) Transition to the $C$ absorbing phase; (b) transition to the $D$ absorbing phase.

Figure 4

Same as in Fig. 3 for $a=1$.

Figure 5

Plots of the logarithm value (base 10) of the order parameter vs logarithm of time for $K=1.0$ and $a=0$ and several values of $b$ close to the critical points (a) $b_1=0.9765$ and (b) $b_2=1.0354$. The values of $b$ are indicated in the legend of the figures as well as the slope of the linear fit (bold straight line) at criticality.

Figure 6

Plots of the logarithm value (base 10) of the order parameter vs logarithm of time for $K=1.0$ and $a=1$ and several values of $b$ close to the critical points (a) best value $b_1=1.3110$ and (b) best value $b_2=1.6305$. The values of $b$ are indicated in the legend of the figures as well as the slope of the linear fit (bold straight line) at criticality.

Figure 7

Plot of the logarithm value (base 10) of the order parameter vs logarithm of time for $K=10.0$ and $a=0$ for several values of $b$ close to the critical point $b_1=0.9977$. The values used for $b$ are indicated in the legend of the figure as well as the slope of the linear fit (bold straight line) at criticality.

Figure 8

Plot of the logarithm value (base 10) of the order parameter vs logarithm of time for $K=100.0$ and $a=1$ for several values of $b$ close to the critical point $b_1=1.462$. The values used for $b$ are indicated in the legend of the figure as well as the slope of the linear fit (bold straight line) at criticality.

Figure 9

Dynamical scaling law for some values of $\mathrm{\Delta }b$ close to the critical points. (a) $a=0, K=1.0$, and $b_2=1.0354$. (b) $a=1, K=10.0$, and $b_1=1.430$.

Figure 10

Dynamical scaling law for some values of $\mathrm{\Delta }b$ close to the critical points for $a=0$ and $K=10.0$. (a) $b_1=0.9977$ and (b) $b_2=1.0015$.

Figure 11

Dynamical scaling law for some values of $\mathrm{\Delta }b$ close to critical points for $a=1$ and $K=100.0$. (a) $b_1=1.462$ and (b) $b_2=1.540$.

Figure 12

Critical exponents as a function of $K$ for (a) non-self-interaction and (b) self-interaction cases.