  We study the relation between the maximal violation of Svetlichny's
inequality and the mixedness of quantum states and obtain the optimal state
(i.e., maximally nonlocal mixed states, or MNMS, for each value of linear
entropy) to beat the Clauser-Horne-Shimony-Holt and the Svetlichny games. For
the two-qubit and three-qubit MNMS, we showed that these states are also the
most tolerant state against white noise, and thus serve as valuable quantum
resources for such games. In particular, the quantum prediction of the MNMS
decreases as the linear entropy increases, and then ceases to be nonlocal when
the linear entropy reaches the critical points ${2}/{3}$ and ${9}/{14}$ for the
two- and three-qubit cases, respectively. The MNMS are related to classical
errors in experimental preparation of maximally entangled states.
