clusters. Consequently, the redMaPPer algorithm gives higher preference to galaxies in denser regions than those in less dense regions. A less luminous galaxy located in a denser environment than the most luminous galaxy may therefore have a higher $P_{\mathrm{CEN}}$.

The total magnitudes of the galaxies in the redMaPPer catalogue are given by the $i$-band `cModel_Mag` [$m_{\mathrm{i}}$; @Abazajian2004], while the colours of the galaxies are determined using `modelMag` [@Abazajian2004] in the $u, g, r, i$ and $z$-bands. All magnitudes and colours have been corrected for Galactic extinction using the dust maps of [-@Schlegel1998].

![image](f_1){width="15cm" height="13cm"}

### Richness of the redMaPPer clusters {#sec:richn}

The catalogue contains clusters with richnesses[^5] of $\lambda \geq 20 \, S(z)$, where $S(z)$[^6] is a correction factor that is used to account for the survey depth of the SDSS. We show the effect of the survey depth on the measured completeness of the redMaPPer clusters as a function of their halo masses $(M_{h})$ in Fig. \[fig:redM\_completeness\_fig\]. The $M_{h}$ have been derived using the halo mass-richness relation of [@Rykoff2012], assuming richness[^7] is a proxy for $M_{h}$. The various lines in Fig. \[fig:redM\_completeness\_fig\] are constructed from the values in figure 22 of [@Rykoff2014]. The redMaPPer algorithm used five redshift bins 