On combining implementable social choice rules

We study if (and when) the intersections and unions of social choice rules that are implementable with respect to a certain equilibrium concept are themselves implementable with respect to that equilibrium concept. Our results for dominant strategy equilibrium are mostly of negative nature; similarly, the intersection of Nash implementable SCCs need not be Nash implementable. On the other hand, we find that the union of any set of Nash implementable social choice rules is Nash implementable (for societies of at least three constituents). This last observation allows us to formulate the notion of the largest Nash implementable subcorrespondence of a social choice rule.