We apply systematically a framework to settle the NP-hardness of some properties related to pure Nash equilibrium in Scheduling and Network Design Games. The technique is simple: first, we construct a gadget without a desired property and then embed it into a larger game which encodes a NP-hard problem in order to prove the complexity of the desired property in a game. This technique is very efficient in proving NP-hardness of the existence of a Nash equilibrium. In the paper, we illustrate the efficiency of the technique in proving the NP-hardness of the existence of a pure Nash equilibrium in Matrix Scheduling Games and Weighted Network Design Games. Moreover, using the technique, we can settle not only the complexity of the equilibrium existence but also that of the existence of good cost-sharing protocol.