Ivars Peterson's MathTrek
October 20, 2003
Suppose there are three candidates, Heather, Angela, and Kathy, and the election procedure calls for ranking the candidates in order of preference. The voters understand this to mean listing their top preference first, their middle preference second, and their bottom preference third.
Three voters put Heather first, Angela second, and Kathy third. Three voters have Kathy first, Angela second, and Heather third. Four voters have Heather first, Kathy second, and Angela third. Four voters have Angela first, Kathy second, and Heather third. When the votes are tallied, Heather is the top choice for seven voters, Angela is top for four voters, and Kathy is top for three voters. Heather wins.
However, because of a misunderstanding, the officials tallying the ballots actually treat a candidate listed first as the bottom preference and a candidate listed third as the top preference. The surprise is that this "reverse" tally gives the same order of finish (Heather, Angela, and Kathy) as the original tally instead of the expected reverse ranking (Kathy first, Angela second, and Heather third). Heather wins again. Moreover, it's clear you can't get the results of one tally simply by reversing the order of finish in the other tally.
Common sense suggests questioning the reliability of any election procedure if it produces the same result when preferences are reversed.
"Surprisingly, this seemingly perverse behavior can sincerely occur with most standard election procedures," Donald G. Saari of the University of California, Irvine, and Steven Barney of the University of Wisconsin Oshkosh write in the current Mathematical Intelligencer. An election like the one described above actually occurred in an academic department to which Saari once belonged.
Such a counterintuitive outcomein which a candidate can come out on top for one set of voter preferences and for its reverseare a troubling consequence of using election procedures that call simply for a plurality vote and, in effect, bias the results.
"It should be a concern because, . . . rather than a rare and obscure phenomenon, we can expect some sort of reversal behavior about 25 percent of the time with the standard plurality vote," Saari and Barney note. These results come out of considerations of mathematical symmetry related to voting systems.
Interestingly, for three-candidate elections, only the system known as the Borda count never exhibits a reversal bias. In such an election, voters assign 2 and 1 points, respectively, to their top- and second-ranked candidates. The candidate with the highest point total wins.
All other methods that involve some sort of rankingincluding approval voting, where electors can vote for as many candidates as they wish, and the candidate with the most votes winscan admit counterintuitive outcomes.
So, shenanigans aren't always to blame for unexpected results. Sometimes, it's just the choice of election procedure.
Copyright © Ivars Peterson
Klarreich, E. 2002. Election selection. Science News 162(Nov. 2):280-282. Available at http://www.sciencenews.org/20021102/bob8.asp.
Peterson, I. 1998. How to fix an election. MAA Online (Nov. 2).
Saari, D.G. 1997. The symmetry and complexity of elections. Discrete Mathematics Project, University of Colorado at Boulder. Available at http://www.colorado.edu/education/DMP/voting_b.html.
______. 1995. Basic Geometry of Voting. New York: Springer-Verlag.
Saari, D.G., and S. Barney. 2003. Consequences of reversing preferences. Mathematical Intelligencer 25(No. 4):17-31.
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A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the Mathematical Association of America (MAA) book Mathematical Treks: From Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.