We use data from the SDSS to investigate the evolution of the large-scale galaxy bias as a function of luminosity for red galaxies. We carefully consider correlation functions of galaxies selected from both photometric and spectroscopic data, and cross-correlations between them, to obtain multiple measurements of the large-scale bias. We find, for our most robust analyses, a strong increase in bias with luminosity for the most luminous galaxies, an intermediate regime where bias does not evolve strongly over a range of two magnitudes in galaxy luminosity, and no evidence for an upturn in bias for fainter red galaxies. Previous work has found an increase in bias to low luminosities that has been widely interpreted as being caused by a strong preference for red dwarf galaxies to be satellites in the most massive haloes. We can recover such an upturn in bias to faint luminosities if we push our measurements to small scales, and include galaxy clustering measurements along the line of sight, where we expect non-linear effects to be the strongest. The results that we expect to be most robust suggest that the low-luminosity population of red galaxies is not dominated by satellite galaxies occupying the most massive haloes.