The 2-to-2 Games Theorem of [16, 10, 11, 17] implies that it is NP-hard to distinguish between Unique Games instances with assignment satisfying at least (12 − ε) fraction of the constraints vs. no assignment satisfying more than ε fraction of the constraints, for every constant ε > 0. We show that the reduction can be transformed in a non-trivial way to give a stronger guarantee in the completeness case: For at least (12 − ε) fraction of the vertices on one side, all the constraints associated with them in the Unique Games instance can be satisfied. We use this guarantee to convert the known UG-hardness results to NP-hardness. We show: 1. Tight inapproximability of approximating independent sets in degree d graphs within a factor of Ω logd2 d , where d is a constant. 2. NP-hardness of approximate the Maximum Acyclic Subgraph problem within a factor of 23 + ε, improving the previous ratio of 1415 + ε by Austrin et al. . 3. For any predicate P−1(1) ⊆ [q]k supporting a balanced pairwise independent distribution, given a P-CSP instance with value at least 12 − ε, it is NP-hard to satisfy more than |P−1(1)| + ε fraction of constraints. qk.