  In the present paper, personalized treatment means choosing the best
treatment for a patient while taking into account certain relevant personal
covariate values. We study the design of trials whose goal is to find the best
treatment for a given patient with given covariates. We assume that the
subjects in the trial represent a random sample from the population, and
consider the allocation, possibly with randomization, of these subjects to the
different treatment groups in a way that depends on their covariates.
  We derive approximately optimal allocations, aiming to minimize expected
regret, assuming that future patients will arrive from the same population as
the trial subjects. We find that, for the case of two treatments, an
approximately optimal allocation design does not depend on the value of the
covariates but only on the variances of the responses. In contrast, for the
case of three treatments the optimal allocation design does depend on the
covariates as we show for specific scenarios. Another finding is that the
optimal allocation can vary a lot as a function of the sample size, and that
randomized allocations are relevant for relatively small samples, and may not
be needed for very large studies.
