How to Fix an Election

Voting sounds like a simple matter. Just pick a candidate, then count the ballots and announce the tally.

When there are three or more candidates (or choices), however, the results may not actually reflect the true preferences of the voters.

Suppose that a group of 15 people must decide which one of three beverages (milk, beer, or wine) to stock in the communal refrigerator. Six people prefer milk to wine to beer; five people prefer beer to wine to milk; and four people prefer wine to beer to milk.

If each person were allowed to vote only for his or her favorite beverage, milk would win, beer would come second, and wine would end up third. A close look at the preferences, however, reveals that nine voters actually prefer beer to milk. Similarly, nine voters prefer wine to milk, and 10 prefer wine to beer. These pairwise comparisons suggest that the voters really prefer wine to beer to milk--a ranking opposite to the plurality outcome.

What if there were a runoff election when the initial round doesn't produce a winner with more than half the votes? In this case, wine would be dropped from the ballot. In a head-to-head contest, beer would defeat milk.

"The voters don't change their opinions at all," notes mathematician Donald G. Saari of Northwestern University in Evanston, Ill. "You just change the voting procedure, and you get a different result."

Saari and Fabrice Valognes of the University of Caen in France describe voting paradoxes and mathematical methods for studying these outcomes in the October Mathematics Magazine.

Such problems with elections bothered a number of mathematicians in 18th-century France. In 1770, Jean-Charles de Borda (1733-1799) wondered whether the use of plurality voting by the Academy of Science distorted the membership's preferences, allowing "inferior" candidates to get elected. He proposed a voting system now called the Borda count, which assigns points to different preferences.

In a three-candidate race, two points would go to the voter's first choice, one point to the second, and zero to the third. The winner would be the candidate with the highest point total.

Applied to the beverage example, wine would win, beer would come second, and milk would be third. That outcome happens to agree with the pairwise rankings.

However, there appears to be no particular reason to choose a 2-1-0 weighting scheme over another set of weights, such as 6, 5, 0; 4, 1, 0; or even 1, 1, 0.

In the 1780s, Marie-Jean-Antoine-Nicolas de Caritat, marquis de Condorcet (1743-1794) argued in favor of an alternative scheme in which the winner is the candidate who beats all other candidates in pairwise elections. In the beverage example, wine would win a majority vote over each of the other beverages. Milk would be the clear loser.

The Condorcet procedure can fail, however. For example, suppose 5 people prefer A to B to C; 5 people prefer B to C to A, and 5 people prefer C to A to B. A natural way to proceed is to run A against B, then run the winner against C. In this case, A would win the first round overwhelmingly, only to lose to C by a landslide. It seems obvious that C should also easily defeat B. Yet B convincingly defeats C.

"Whichever candidate is voted upon last, wins--decisively," Saari and Valognes remark. "In particular, there is no Condorcet winner or loser."

So, which voting method is best?

Saari used mathematical ideas from the study of dynamical systems, sometimes loosely called chaos theory, and algebraic geometry to identify situations in which different voting systems fail. The results indicate that, for more than two candidates, you can always find examples of voting procedures where the election results favor a specified outcome.

"You can get whatever result that you want," Saari says. Yet "nobody changes his or her mind."

It turns out that, despite some problems, the original Borda count is the best voting scheme. "It significantly reduces the number of paradoxes that might arise," Saari says. Moreover, "if something goes wrong in the Borda count, it will go wrong in every other procedure."

The worst scheme is the simple plurality vote. In elections in which voters must select candidates to fill two or more positions, giving the voters the option to choose any number of candidates up to the full allotment (approval voting) messes up the results even more.

That may explain the quirkiness often found in lists of the 100 best U.S. films or the top mathematicians of all time.

"Manipulating elections means taking advantage of voting paradoxes," Saari says. It's useful to be able to identify what can go right and what can go wrong.

In general, "who you elect reflects the procedures you use more than who you want," he adds. "Bad procedures can lead to lousy elections results."

References:

Saari, D.G. 1995. Basic Geometry of Voting. New York: Springer-Verlag.

______. 1995. A chaotic exploration of aggregation paradoxes. SIAM Review 37(March):37.

______. 1992. Millions of election outcomes from a single profile. Social Choice and Welfare 9:277.

Saari, D.G., and F. Valognes. 1998. Geometry, voting, and paradoxes. Mathematics Magazine 71(October):243.

Additional information is available at Donald Saari's Web page at http://www.math.nwu.edu/~d_saari/.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.