Multistage designs allow considerable reductions in the expected sample size of a trial. When stopping for futility or efficacy is allowed at each stage, the expected sample size under different possible true treatment effects ($δ$) is of interest. The $δ$-minimax design is the one for which the maximum expected sample size is minimised amongst all designs that meet the types I and II error constraints. Previous work has compared a two-stage $δ$-minimax design with other optimal two-stage designs. Applying the $δ$-minimax design to designs with more than two stages was not previously considered because of computational issues. In this paper, we identify the $δ$-minimax designs with more than two stages through use of a novel application of simulated annealing. We compare them with other optimal multistage designs and the triangular design. We show that, as for two-stage designs, the $δ$-minimax design has good expected sample size properties across a broad range of treatment effects but generally has a higher maximum sample size. To overcome this drawback, we use the concept of admissible designs to find trials which balance the maximum expected sample size and maximum sample size. We show that such designs have good expected sample size properties and a reasonable maximum sample size and, thus, are very appealing for use in clinical trials.