Fig. 1. The state space of the stochastic process (1) is a subset of the triangular area. Dashed lines show points of constant reputation, increasing from bottom to top. Dotted lines show points of constant certainty, increasing from left to right.

Fig. 2. Bifurcation plot showing the existence of fixed points for each d: as d increases from 0 to 1 (from left to right along the horizontal axis), the number of fixed points changes from 1 (the true fixed point at reputation value θ) to 2 (the true fixed point at θ and the false fixed point at π) and back to 1 (the false fixed point at π).

Fig. 3. 10⁵ steps of a typical sample path for Parameter set 1 with q=0.2 and R₀=0. We plot R_n against n, the upper line corresponding to θ and the lower line to π=(pθ)/(p+ωq) (cf. Fig. 2). R_n increases from 0 past π to θ and then remains close to θ. Thus, θ is confirmed as the unique fixed point reputation value as expected from the analytical results.

Fig. 4. 10×10⁵ steps of two honest users’ typical sample paths for Parameter set 2 with q=0.2, p=0.6 and R₀=0. We plot reputation values R_n of two users (one in black, one in grey). R_n increases from 0 past π to θ and then remains close to θ. Again, θ is confirmed as the unique fixed point reputation value.

Fig. 5. 10⁵ steps of a typical sample path for the same parameters as in Fig. 3 except u=0.999. The variability is smaller and it takes longer for the process to approach θ.

Fig. 6. 10×10⁵ steps of two honest users’ typical sample paths for the same parameters as in Fig. 4 except u=0.999. As in the case of two users in Fig. 5 the variability is smaller and it takes longer for the process to approach θ.

Fig. 7. 10⁵ steps of a typical sample path for Parameter set 1 except u=0.95 with q=0.8 and R₀=0. R_n increases from 0 to π=(pθ)/(p+ωq) and then settles down for some time in a neighbourhood of π and θ in an alternating fashion.

Fig. 8. 10×10⁵ steps of two honest users’ typical sample paths for Parameter set 2 except u=0.95 with p=0.2, and R₀=0. R_n increases from 0 to π and then settles down in a neighbourhood of π and θ in an alternating fashion.

Fig. 9. 10⁵ steps of a typical sample path for the same parameters as in Fig. 7 except R₀=1. R_n decreases from 1 to θ and then settles down for some time in a neighbourhood of π and θ in an alternating fashion as before (cf. Fig. 7). Thus, the starting value R₀ does not influence the process in the long run.

Fig. 10. 10×10⁵ steps of two honest users’ typical sample paths for the same parameters as in Fig. 8 except R₀=1. Again, the starting value R₀ does not influence the process in the long run.

Fig. 11. 10⁵ steps of a typical sample path for the same parameters as in Fig. 5 except now θ=0.4, d=0.8. Moreover, R₀=1. R_n decreases from 1 past θ to π and then remains close to π. Thus, π is confirmed as the unique fixed point reputation value as expected from the analytical results.

Fig. 12. 10×10⁵ steps of two honest users’ typical sample paths for the same parameters as in Fig. 6 except now θ=0.4, d=0.8. Moreover, R₀=1. As in the case of two users, π is confirmed as the unique fixed point reputation value.

Summary of notation