AbstractIn clinical trials with two treatment arms, Efron's biased coin design, Efron (1971), sequentially assigns a patient to the underrepresented arm with probabilityp> ½. Under this design the proportion of patients in any arm converges to ½, and the convergence rate isn-1, as opposed ton-½under some other popular designs. The generalization of Efron's design toK≥ 2 arms and an unequal target allocation ratio (q1, . . .,qK) can be found in some papers, most of which determine the allocation probabilitiesps in a heuristic way. Nonetheless, it has been noted that by using inappropriateps, the proportion of patients in theKarms 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 raten-1.