We revisit the longstanding question of whether first brightest cluster galaxies are statistically drawn from the same distribution as other cluster galaxies or are “ special ” , using the new non-parametric , empirically based , model presented in Vale & Ostriker ( 47 ) for associating galaxy luminosity with halo/subhalo masses .

We introduce scatter in galaxy luminosity at fixed halo mass into this model , building a conditional luminosity function ( CLF ) by considering two possible models : a simple lognormal and a model based on the distribution of concentration in haloes of a given mass .
We show that this model naturally allows an identification of halo/subhalo systems with groups and clusters of galaxies , giving rise to a clear central/satellite galaxy distinction , obtaining a special distribution for the brightest cluster galaxies ( BCGs ) .

Finally , we use these results to build up the dependence of BCG magnitudes on cluster luminosity , focusing on two statistical indicators , the dispersion in BCG magnitude and the magnitude difference between first and second brightest galaxies .
We compare our results with two simple models for BCGs : a statistical hypothesis that the BCGs are drawn from a universal distribution , and a cannibalism scenario merging two galaxies from this distribution .
The statistical model is known to fail from work as far back as Tremaine & Richstone ( 45 ) .
We show that neither the statistical model nor the simplest possibility of cannibalism provide a good match for observations , while a more realistic cannibalism scenario works better .
Our CLF models both give similar results , in good agreement with observations .
Specifically , we find < m _ { 1 } > between -25 and -25.5 in the K-band , \sigma ( m _ { 1 } ) \sim 0.25 and < \Delta _ { 12 } > between 0.6 and 0.8 , for cluster luminosities in the range of 10 ^ { 12 } to 10 ^ { 13 } h ^ { -2 } { L _ { \odot } } .