Paradoxes of Voting

Abstract

Five voting paradoxes are examined under procedures which determine social choice from voters' preference rankings. The most extreme forms of each paradox are identified, and their potential practical significance is assessed using randomly generated voter preference profiles.The first paradox arises when the winner under sequential-elimination simple-majority voting is less preferred by every voter than some other alternative. The fifth paradox occurs when one alternative has a simple majority over every other alternative and one or more of the simple-majority losers beats the winner on the basis of every point-total method that assigns more points to a first-place vote than to a second-place vote, more points to a second-place vote than to a third-place vote, and so forth.The other three paradoxes are solely concerned with point-total procedures. They include cases in which the standard point-total winner becomes a loser when original losers are removed, and in which different truncated point-total procedures (which count only first-place votes, or only first-place and second-place votes, and so forth) yield different winners.The computer simulation data suggest that the more extreme forms of the paradoxes are exceedingly unlikely to arise in practice.