  AIMS. This work addresses the AGN IR-selection dependency on intrinsic source
luminosity and obscuration, in order to identify and charaterise biases which
could affect conclusions in studies. METHODS. We study IR-selected AGN in the
Chandra Deep Field South (CDFS) survey and in the Cosmological Survey (COSMOS).
The AGN sample is divided into low and high X-ray luminosity classes, and into
unobscured (type-1) and obscured (type-2) classes by means of X-ray and optical
spectroscopy data. Specifically in the X-rays regime, we adopt the intrinsic
luminosity taking into account the estimated column density (N_H). We also take
the chance to highlight important differences resulting from adopting different
methods to assess AGN obscuration. RESULTS. In agreement with previous studies,
we also find that AGN IR-selection efficiency shows a decrease with decreasing
source AGN X-ray luminosity. For the intermediate-luminosity AGN population
(43.3<~log(L_X[erg/s])<~44$), the efficiency also worsens with increasing
obscuration. The same sample also shows an evolution with cosmic time of the
obscured fraction at the highest X-ray luminosities, independently of adopted
type-1/type-2 classification method. CONCLUSIONS. We confirm that AGN
IR-selection is genuinely biased toward unobscured AGN, but only at
intermediate luminosities. At the highest luminosities, where AGN IR-selection
is more efficient, there is no obscuration bias. We show that type-1 AGN are
intrinsically more luminous than type-2 AGN only at z<~1.6, thus resulting in
more type-1 AGN being selected the shallower the IR survey is. [ABRIDGED:
Conclusion on survey strategy. See manuscript.] Finally, when the James Webb
Space Telescope comes online, the broad-band filters 2.0um, 4.4um, 7.7um, and
18um will be key to disentangle AGN from non-AGN dominated SEDs at depths where
spectroscopy becomes impractical.
