Designs which are minimax in the presence of model misspecifications have been constructed so as to minimize the maximum, over classes of alternate response models, of the integrated mean squared error of the predicted values. This mean squared error decomposes into a term arising solely from variation, and a bias term arising from the model errors. Here we consider the problem of designing so as to minimize the variance of the predictors, subject to a bound on the maximum (over model misspecifications) bias. We consider as well designing so as to minimize the maximum bias, subject to a bound on the variance. We show that solutions to both problems are given by the minimax designs, with appropriately chosen values of their tuning constants. Conversely, any minimax design solves each problem for an appropriate choice of the bound on the maximum bias or on the variance.