  In this paper we continue the analysis of the two-scale method for the
Monge-Amp\`ere equation for dimension $d \geq 2$ introduced in [10]. We prove
continuous dependence of discrete solutions on data that in turn hinges on a
discrete version of the Alexandroff estimate. They are both instrumental to
prove pointwise error estimates for classical solutions with H\"older and
Sobolev regularity. We also derive convergence rates for viscosity solutions
with bounded Hessians which may be piecewise smooth or degenerate.
