Asymptotically best response-adaptive randomization procedures

We derive a lower bound on the asymptotic variance of the allocation proportions from response-adaptive randomization procedures when the allocation proportions are asymptotically normal. A procedure that attains this lower bound is defined to be asymptotically best. We then compare the asymptotic variances of five procedures, for which allocation proportions converge, to the lower bound. We find that a procedure by Zelen and a procedure by Ivanova attain the lower bound and a procedure by Eisele and its extension to K>2K>2 treatments can attain the lower bound but are, in general, not asymptotically best. We discuss the tradeoffs among the benefits of randomization, the benefits of attaining the lower bound, and the benefits of targeting an optimal allocation. We conclude that none of these procedures possesses all of these benefits.