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. While the results for dominant strategy equilibrium are mostly of negative nature, the situation is di¤erent in the case of Nash implementation. We …nd that the union of any set of Nash implementable social choice rules is Nash implementable (for societies of at least three constituents), while the intersection of …nitely many such correspondences is “almost”Nash implementable (even in the case of two-person societies). The former observation allows us to formulate the notion of the largest Nash implementable subcorrespondence of a social choice rule.