Nash implementation of supermajority rules

A committee of n experts from a university department must choose whom to hire from a set of m candidates. Their honest judgments about the best candidate must be aggregated to determine the socially optimal candidates. However, experts’ judgments are not verifiable. Furthermore, the judgment of each expert does not necessarily determine his preferences over candidates. To solve this problem, a mechanism that implements the socially optimal aggregation rule must be designed. We show that the smallest quota q compatible with the existence of a q-supermajoritarian and Nash implementable aggregation rule is $$q=n-\left\lfloor \frac{n-1}{m}\right\rfloor$$ q = n - n - 1 m . Moreover, for such a rule to exist, there must be at least $$m\left\lfloor \frac{n-1}{m}\right\rfloor +1$$ m n - 1 m + 1 impartial experts with respect to each pair of candidates.