  We aim to efficiently allocate a fixed simulation budget to identify the
top-mc designs for each context among a finite number of contexts. The
performance of each design under a context is measured by an identifiable
statistical characteristic, possibly with the existence of nuisance parameters.
Under a Bayesian framework, we extend the top-two Thompson sampling method
designed for selecting the best design in a single context to the contextual
top-mc selection problems, leading to an efficient sampling policy that
simultaneously allocates simulation samples to both contexts and designs. To
demonstrate the asymptotic optimality of the proposed sampling policy, we
characterize the exponential convergence rate of the posterior distribution for
a wide range of identifiable sampling distribution families. The proposed
sampling policy is proved to be consistent, and asymptotically satisfies a
necessary condition for optimality. In particular, when selecting contextual
best designs (i.e., mc = 1), the proposed sampling policy is proved to be
asymptotically optimal. Numerical experiments demonstrate the good finite
sample performance of the proposed sampling policy.
