In this work, we study Schmidt's winning sets in S-gap shifts. We introduce the non-dense orbits winning (NDOW) property, meaning that all exceptional sets of non-dense orbits are winning. We show that when S is either a nontrivial finite set or piecewise syndetic, the shift has the NDOW property. In contrast, when S is the set of all primes, the system does not have this property. We also prove some dimension results for exceptional sets. This is a joint work with Guohua Zhang.