A Limit Theorem on the Minmax Set

It is well known that a Condorcet winner may not exist over a multidimensional space. We are concerned in this paper with an extension of the Condorcet's rule: the minmax set. This set, always non-empty, coincides with the set of majority winners whenever they exist. Unfortunately, it may be very large in finite society. We establish that it shrinks to a single point when the population increases smoothly enough under suitable assumptions of single peakedness and intermediate preferences.