Unequivocal majority and Maskin-monotonicity

The unequivocal majority of a social choice rule is a number of agents such that whenever at least this many agents agree on the top alternative, then this alternative (and only this) is chosen. The smaller the unequivocal majority is, the closer it is to the standard (and accepted) majority concept. The question is how small can the unequivocal majority be and still permit the Nash-implementability of the social choice rule; i.e., its Maskin-monotonicity. We show that the smallest unequivocal majority compatible with Maskin-monotonicity is n - left perpendicular n-1/m right perpendicular, where n >= 3 is the number of agents and m >= 3 is the number of alternatives. This value is equal to the minimal number required for a majority to ensure the non-existence of cycles in pairwise comparisons. Our result has a twofold implication: (1) there is no Condorcet consistent social choice rule satisfying Maskin-monotonicity and (2) a social choice rule satisfies k-Condorcet consistency and Maskin-monotonicity if and only if k >= n - n - left perpendicular n-1/m right perpendicular.