Modelling the Prisoner’s Dilemma (II).
(a) Probability of defection (for a fixed value of λ = 10) as a function of α, and evaluated for different values of φ ∈ (0, 1) (see also Fig. 3-Panel (c) for complementary information). (b) Sequence of plots (also with λ = 10) for increasing values of α of the probabilities: PAction=D = PDD + PDC (orange), PBelief=D = PDD + PCD (black), PDD (blue, solid), PCD (blue, dashed), PDC (red, solid), and PCC (red, dashed), as a function of the parameter φ. (c) Relaxation time τ(α, φ) plotted as a function of φ for several values of α. For each curve, the dot indicates the minimum. The inset of the Panel corresponds to the highest values of α. (d) Minimum relaxation time τMin(α) with inset of φMin, both as a function of α. We define φMin(α) as the value of φ minimizing the relaxation time for each value of α. (e) Temporal evolution of PDD(t) starting from four different initial conditions ρ(0). Legend’s notation ρn(0) = (ρ11, ρ22, ρ33, ρ44) indicates the diagonal initial entries of the density matrix. We initialize the non-diagonal elements at zero value ρij(0) = 0 in the four cases. This plot illustrates the common convergence towards the stationary solution regardless of the initial values, as we show analytically in the Methods section. (f) Temporal evolution of the four components Pij(t) (i, j = C, D) for the initial condition PCC(0) = 1. Horizontal gray lines in panels (e,f) indicate the corresponding value of each component in the stationary solution. See further explanation of the graphs and the discussion regarding the behavioral aspects in the text.