Generalized Efron's biased coin design and its theoretical properties

In clinical trials with two treatment arms, Efron's biased coin design, Efron (1971), sequentially assigns a patient to the underrepresented arm with probability p > ½. Under this design the proportion of patients in any arm converges to ½, and the convergence rate is n-1, as opposed to n-½ under some other popular designs. The generalization of Efron's design to K ≥ 2 arms and an unequal target allocation ratio (q1, . . ., qK) can be found in some papers, most of which determine the allocation probabilities ps in a heuristic way. Nonetheless, it has been noted that by using inappropriate ps, the proportion of patients in the K arms never converges to the target ratio. We develop a general theory to answer the question of what allocation probabilities ensure that the realized proportions under a generalized design still converge to the target ratio (q1, . . ., qK) with rate n-1.