We derive the maximum bias functions of the MM-estimates and the constrained M-estimates or CM-estimates of regression and compare them to the maximum bias functions of the S-estimates and the $τ$-estimates of regression. In these comparisons, the CM-estimates tend to exhibit the most favorable bias-robustness properties. Also, under the Gaussian model, it is shown how one can construct a CM-estimate which has a smaller maximum bias function than a given S-estimate, that is, the resulting CM-estimate dominates the S-estimate in terms of maxbias and, at the same time, is considerably more efficient.