Table 2.

Simulation results in the hierarchical case.

(a) Lifespan $\mathit{\ell }=3$ Steps
Step $\mathit{t}$	Malicious Bots				White-Hat Bots
	$\overline{{\#}_{\mathbf{mal}}^{\mathbf{obs}}}$	$\overline{{\#}_{\mathbf{mal}}^{\mathbf{in}-\mathbf{obs}}}$	$\overline{{\#}_{\mathbf{mal}}^{\mathbf{unobs}}}$	$\overline{{\mathit{R}}_{\mathbf{mal}}}$	$\overline{{\#}_{\mathbf{wh}}^{\mathbf{obs}}}$	$\overline{{\#}_{\mathbf{wh}}^{\mathbf{in}-\mathbf{obs}}}$	$\overline{{\#}_{\mathbf{wh}}^{\mathbf{unobs}}}$	$\overline{{\mathit{R}}_{\mathbf{wh}}}$
$0^{-}$	5.79	0.00	14.21	20.00%	0.00	0.00	0.00	0.00%
$0^{+}$	5.79	0.26	13.95	20.00%	10.00	0.00	0.00	10.00%
${\mathrm{10,000}}^{-}$	23.48	0.51	43.40	67.39%	7.55	6.17	3.08	16.80%
${\mathrm{10,000}}^{+}$	23.48	0.51	43.40	67.39%	0.00	0.00	3.08	3.08%
(b) Lifespan $\ell =\mathbf{4}$ Steps
Step $\mathit{t}$	Malicious Bots				White-Hat Bots
	$\overline{{\#}_{\mathbf{mal}}^{\mathbf{obs}}}$	$\overline{{\#}_{\mathbf{mal}}^{\mathbf{in}-\mathbf{obs}}}$	$\overline{{\#}_{\mathbf{mal}}^{\mathbf{unobs}}}$	$\overline{{\mathit{R}}_{\mathbf{mal}}}$	$\overline{{\#}_{\mathbf{wh}}^{\mathbf{obs}}}$	$\overline{{\#}_{\mathbf{wh}}^{\mathbf{in}-\mathbf{obs}}}$	$\overline{{\#}_{\mathbf{wh}}^{\mathbf{unobs}}}$	$\overline{{\mathit{R}}_{\mathbf{wh}}}$
$0^{-}$	5.79	0.00	14.21	20.00%	0.00	0.00	0.00	0.00%
$0^{+}$	5.79	0.26	13.95	20.00%	10.00	0.00	0.00	10.00%
${\mathrm{10,000}}^{-}$	0.71	0.38	1.72	2.81%	29.81	33.06	10.41	73.28%
${\mathrm{10,000}}^{+}$	0.71	0.38	1.72	2.81%	0.00	0.00	10.41	10.41%
(c) Lifespan $\ell =\mathbf{5}$ Steps
Step $\mathit{t}$	Malicious Bots				White-Hat Bots
	$\overline{{\#}_{\mathbf{mal}}^{\mathbf{obs}}}$	$\overline{{\#}_{\mathbf{mal}}^{\mathbf{in}-\mathbf{obs}}}$	$\overline{{\#}_{\mathbf{mal}}^{\mathbf{unobs}}}$	$\overline{{\mathit{R}}_{\mathbf{mal}}}$	$\overline{{\#}_{\mathbf{wh}}^{\mathbf{obs}}}$	$\overline{{\#}_{\mathbf{wh}}^{\mathbf{in}-\mathbf{obs}}}$	$\overline{{\#}_{\mathbf{wh}}^{\mathbf{unobs}}}$	$\overline{{\mathit{R}}_{\mathbf{wh}}}$
$0^{-}$	5.79	0.00	14.21	20.00%	0.00	0.00	0.00	0.00%
$0^{+}$	5.79	0.26	13.95	20.00%	10.00	0.00	0.00	10.00%
${\mathrm{10,000}}^{-}$	0.00	0.00	0.00	0.00%	32.72	43.62	8.18	84.52%
${\mathrm{10,000}}^{+}$	0.00	0.00	0.00	0.00%	0.00	0.00	8.18	8.18%