Implementation via approval mechanisms

We focus on the single-peaked domain and study the class of Generalized Approval Mechanisms (GAMs): First, players simultaneously select subsets of the outcome space and scores are assigned to each alternative; and, then, a given quantile of the induced score distribution is implemented. Our main finding is that essentially for every Nash-implementable welfare optimum - including the Condorcet winner alternative - there exists a GAM that Nash-implements it. Importantly, the GAM that Nash-implements the Condorcet winner alternative is the first simple simultaneous game with this feature in the literature.