When voters’ preferences in a three-candidate voting situation reflect any significant degree of proximity to perfect single-peakedness, perfect single-troughedness, or perfect polarization, the probability that a PMRW exists is quite high. When voters’ preferences are at all close to reflecting a situation in which a unifying candidate exists, the probability that a PMRW exists is very high. It is very important to note that the associated underlying models that lead to single-peaked, single-troughed, or polarized preferences do not actually have to be the basis of the mechanism by which the voters’ preference rankings on candidates are actually formed. It is only required that the preferences in a given voting situation could have been obtained by one of these models. As a result, Condorcet’s Paradox should rarely be observed in any real elections on a small number of candidates with large electorates, as long as voters’ preferences reflect any significant degree of group coherence or consistency.
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