  We study the luminosity gap, dm12, between the first and second ranked
galaxies in a sample of 59 massive galaxy clusters, using data from the Hale
Telescope, HST, Chandra, and Spitzer. We find that the dm12 distribution,
p(dm12), is a declining function of dm12, to which we fitted a straight line:
p(dm12) propto -(0.13+/-0.02)dm12. The fraction of clusters with "large"
luminosity gaps is p(dm12>=1)=0.37+/-0.08, which represents a 3sigma excess
over that obtained from Monte Carlo simulations of a Schechter function that
matches the mean cluster galaxy luminosity function. We also identify four
clusters with "extreme" luminosity gaps, dm12>=2, giving a fraction of
p(dm12>=2)=0.07+0.05-0.03. More generally, large luminosity gap clusters are
relatively homogeneous, with elliptical/disky brightest cluster galaxies
(BCGs), cuspy gas density profiles (i.e. strong cool cores), high
concentrations, and low substructure fractions. In contrast, small luminosity
gap clusters are heterogeneous, spanning the full range of
boxy/elliptical/disky BCG morphologies, the full range of cool core strengths
and dark matter concentrations, and have large substructure fractions. Taken
together, these results imply that the amplitude of the luminosity gap is a
function of both the formation epoch, and the recent infall history of the
cluster. "BCG dominance" is therefore a phase that a cluster may evolve
through, and is not an evolutionary "cul-de-sac". We also compare our results
with semi-analytic model predictions based on the Millennium Simulation. None
of the models are able to reproduce all of the observational results,
underlining the inability of current models to match the empirical properties
of BCGs. We identify the strength of AGN feedback and the efficiency with which
cluster galaxies are replenished after they merge with the BCG in each model as
possible causes of these discrepancies. [Abridged]
