The randomness, efficiency (power and variability), and desirable allocation proportions are important components for evaluating a response-adaptive design in clinical trials and conflicted demands in applications. The aim of this paper is to provide designs dealing with these dilemmas. We first give a general framework for efficient response-adaptive randomization procedures that attain the Cramér-Rao lower bounds of the allocation variances for any desired allocation proportions. The general framework is flexible for us to define new families of efficient designs with good properties for both two and multiple-treatment clinical trials. We also prove that, among all response-adaptive randomization procedures with the same limit allocation proportions, the selection biases and entropies as measures of the randomness of the designs have their optimal values. Basing on the theory on efficiency and randomness, we propose a new family of doubly adaptive biased coin designs for multi-treatment clinical trials that can target any allocation proportion and are asymptotically best in terms both the randomness and efficiency so that their randomness is asymptotic optimal and asymptotic allocation variance attains the Cramér-Rao lower bound. Theoretical properties, including the strong consistency, the asymptotic normality, and the functional central limit theorem for both the sample allocation proportions and the estimators of the distribution parameters, are developed by using the technique of Gaussian approximation and Gaussian comparing theorems.

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