Consequences of Reversing Preferences

Other than standard election disruptions involving shenanigans, strategic voting, and so forth, it is reasonable to expect that elections are free from difficulties. But this is far from being true; even sincere election outcomes admit all sorts of counterintuitive conclusions. For instance, suppose after the winner of an important departmental election was announced, it was discovered that everyone misunderstood the chair’s instructions. When ranking the three candidates, everyone listed their top, middle, and bottom-ranked candidate in the natural order of, respectively, first, second, and third. For reasons only the chair understood, he expected the voters to vote in the opposite way. As such, when tallying the ballots, he treated a first and last listed candidate, respectively, as the voter’s last and first choice.1 Imagine the outcry if after retallying the ballots the chair reported that the election ranking remained unchanged; in particular, the same person won. Skepticism might be the kindest reaction to greet an announcement that the election ranking for a profile — a listing which specifies the number of voters whose preferences are given by each (complete, transitive) ranking of the candidates — is the same for the profile where each voter’s preference ordering is reversed. Surprisingly, this seemingly perverse behavior can sincerely occur with most standard election procedures. It is intriguing that this phenomenon can be explained in terms of simple mathematical symmetries. Of particular interest, the same arguments explain all of the election paradoxes which have perplexed this area for the last two centuries. This issue appears to have been first introduced in [Saari 1995] where a section of this book showed that some procedures allow the same election ranking to occur with a profile and with its reversal. There is no interest in this phenomenon when the common ranking is a complete tie, but when the common ranking is not a tie this effect is called a “reversal bias.” The word “bias” is intended to foreshadow how this anomaly affects election outcomes. Rather than an election ranking, voters more typically care only about who wins, or who is elected for, say, the departmental budget committee. This raises the question whether an election procedure would allow the same winner, or the same two candidates, . . . , or the same k candidates to be top-ranked with a profile and its reversal. Call this situation a “k-winner reversal bias.” Common sense suggests that we should question the reliability of an election procedure if it elects the same committee with a profile and with the profile of reversed preferences; i.e., if the procedure allows a k-winner reversal bias. As one of us (Barney) discovered, an internet discussion group worrying about election methods is particularly concerned about the case k = 1 — which we call the “top-winner reversal bias.” It should be a concern because, as shown here, rather than a rare and obscure phenomenon, we can expect some sort of reversal behavior about 25% of the time with the standard plurality vote.