  Galaxy clusters observed through the thermal Sunyaev-Zeldovich (tSZ) effect
are a recent cosmological probe. The precision on the cosmological constraints
is affected mainly by the current knowledge of cluster physics, which enters
the analysis through the scaling relations. Here we aim to study one of the
most important sources of systematic uncertainties, the mass bias, $b$. We have
analysed the effects of a mass-redshift dependence, adopting a power-law
parametrisation. We applied this parametrisation to the combination of tSZ
number counts and power spectrum, finding a hint of redshift dependence that
leads to a decreasing value of the mass bias for higher redshift. We tested the
robustness of our results for different mass bias calibrations and a discrete
redshift dependence. We find our results to be dependent on the clusters sample
that we are considering, in particular obtaining an inverse (decreasing)
redshift dependence when neglecting $z<0.2$ clusters. We analysed the effects
of this parametrisation on the combination of cosmic microwave background (CMB)
primary anisotropies and tSZ galaxy clusters. We find a preferred constant
value of mass bias, having $(1-b) =0.62 \pm 0.05$. The corresponding value of
$b$ is too high with respect to weak lensing and numerical simulations
estimations. Therefore we conclude that this mass-redshift parametrisation does
not help in solving the remaining discrepancy between CMB and tSZ clusters
observations.
