The development of response-adaptive randomization (RAR) has taken many different paths over the past few decades. Some RAR schemes optimize certain criteria, but may be complicated and often rely on asymptotic arguments, which may not be suitable in trials with small sample sizes. Some Bayesian RAR schemes are very intuitive and easy to implement, but may not always be tailored toward the study goals. To bridge the gap between these methods, we proposed a novel framework in which easy-to-implement Bayesian RAR schemes can be derived to target the study goals. We identified a specific study goal for the popular Bayesian RAR scheme that assigns more patients to better performing arms, and showed that it fits in the new framework given that goal. For different study goals, we illustrated the new framework in the setting where multiple treatment arms are compared to a concurrent control arm. Through simulation, we demonstrated that the RAR schemes developed under the new framework outperform a popular method.