  We develop discrete $W^2_p$-norm error estimates for the Oliker-Prussner
method applied to the Monge-Amp\`ere equation. This is obtained by extending
discrete Alexandroff estimates and showing that the contact set of a nodal
function contains information on its second order difference. In addition, we
show that the size of the complement of the contact set is controlled by the
consistency of the method. Combining both observations, we show that the error
estimate $\|u - u_h\|_{W^2_p} \leq C h^{1/p}$ if $p > d$ and $\|u -
u_h\|_{W^2_p} \leq C h^{1/d} \big(\ln\left(\frac 1 h \right)\big)^{1/d} $ if $p
\leq d$. Here the constant $C$ depends on $\|{u}\|_{C^{3,1}(\bar\Omega)}$, the
dimension $d$, and the constant $p$. Numerical examples are given in two space
dimensions and confirm that the estimate is sharp in several cases.
