Computational aspects of equilibrium notions for games have been extensively studied, including settings where the goal is to find an equilibrium that possesses some additional properties. Our work extends this direction by considering games in which players are subject to some form of constraint on their strategic choices. We also consider the relationship between Nash equilibria and so-called generalized or social equilibria in this context. Our results demonstrate that the complexity of finding an equilibrium varies significantly between games with slightly different strategic constraints. We also demonstrate that these constraints are useful for modeling problems involving strategic resource allocation and also are of interest from the perspective of behavioral game theory.