We describe a template for evaluating response-adaptive randomization procedures that involves a Taylor series expansion of the noncentrality parameter of a test of treatment effects for binary responses. The template involves targeting an optimal allocation, the rate of convergence, and the variability of the randomization procedure. We find that the Hu and Zhang's doubly-adaptive biased coin design and an urn design by Ivanova both maintain power and result in a modest reduction in expected treatment failures for two treatments. For K>2 treatments the problem becomes more subtle. Finding an optimal allocation requires optimization techniques, and the solution is on the boundary. To control for this, we can add an addtional minimum sample size constraint. But the solution is not continuous. We use smoothing techniques to create a smooth allocation function and find that we are able to increase average power over complete randomization.