  We present a multilevel Monte Carlo simulation method for analysing
multi-scale physical systems via a hierarchy of coarse-grained representations,
to obtain numerically-exact results, at the most detailed level. We apply the
method to a mixture of size-asymmetric hard spheres, in the grand canonical
ensemble. A three-level version of the method is compared with a
previously-studied two-level version. The extra level interpolates between the
full mixture and a coarse-grained description where only the large particles
are present -- this is achieved by restricting the small particles to regions
close to the large ones. The three-level method improves the performance of the
estimator, at fixed computational cost. We analyse the asymptotic variance of
the estimator, and discuss the mechanisms for the improved performance.
