Selection Are we using the worst voting procedure?
By Erica Klarreich
November 2, 2002
As Election Day approaches, voters must
be feeling a sense of d╚jř vu. With recent reports of malfunctioning
voter machines and uncounted votes during primaries in Florida,
Maryland, and elsewhere, reformers are once again clamoring for
extensive changes. But while attention is focused on these familiar
irregularities, a much more serious problem is being neglected: the
fundamental flaws of the voting procedure itself, say various
researchers who study voting.
Nearly all political elections in the
United States are plurality votes, in which each voter selects a
single candidate, and the candidate with the most votes wins. Yet
voting theorists argue that plurality voting is one of the worst of
all possible choices. "It's a terrible system," says Alexander
Tabarrok, an economist at George Mason University in Fairfax, Va.,
and director of research for the Independent Institute in Oakland,
Calif. "Almost anything looks good compared to it."
systems abound. One alternative is the instant runoff, a procedure
used in Australia and Ireland that eliminates candidates one at a
time from rankings provided by each voter. Another is the Borda
count, a point system devised by the 18th-century French
mathematician Jean Charles Borda, which is now used to rank college
football and basketball teams. A third is approval voting, used by
several scientific societies, in which participants may cast votes
for as many of the candidates as they choose.
procedures, the plurality system looks only at a voter's top choice.
By ignoring how voters might rank the other candidates, it opens the
floodgates to unsettling, paradoxical results.
In races with two
strong candidates, plurality voting is vulnerable to the third-party
spoilerˇa weaker candidate who splits some of the vote with one of
the major candidates. For instance, in the hotly contested 2000 U.S.
presidential race, Republican George W. Bush won the state of
Floridaˇand, consequently, the presidencyˇby just a few hundred
votes over Al Gore, the Democratic candidate. Green Party candidate
Ralph Nader won 95,000 votes in Florida, and polls suggest that for
most Nader voters, Gore was their second choice. Thus, if the race
had been a head-to-head contest between Bush and Gore, Florida
voters probably would have chosen Gore by a substantial margin.
Should Nader have withdrawn from the race, as many angry Democrats
asserted? Certainly not, says mathematician Donald Saari of the
University of California, Irvine. "We live in a democracy, and
anyone should be able to run for any office," he says. "The problem
was the bad design of the election."
Mathematics can shed light on
questions about how well different voting procedures capture the
will of the voters, Saari says. In ongoing work, he has been using
tools from chaos theory to identify just which scenarios of voter
preferences will give rise to disturbing election outcomes. "With
the muscle power of mathematics, we can address these questions and
finally get some results," he says.
elections with only two candidates, plurality voting works just
fine, since the winner is guaranteed to have been the top choice of
more than half the voters. But as soon as three or more candidates
are on the ballot, the system can run into trouble.
In races with a
large slate of candidates, plurality voting dilutes voter
preferences, creating the possibility of electing a leader whom the
vast majority of voters despise. In the French election last April,
with 16 candidates on the ballot, extreme right-wing candidate
Jean-Marie le Penˇwidely accused of racism and anti-Semitismˇmanaged
to place second with just 17 percent of the vote. He then advanced
to a runoff against the top candidate, incumbent President Jacques
Chirac. Political analysts scrambled to explain le Pen's success,
putting it down to voter disenchantment and a surge in right-wing
fervor across Europe. But the real reason, voting theorists say, is
that the plurality vote distorted the preferences of the voters.
"The fact that le Pen was in the runoff had nothing to do with what
the people wanted," Saari says. The runoff election, in which Chirac
trounced le Pen with 82 percent of the vote, suggests that while le
Pen was at the top of a few voters' lists, he was near the bottom of
many more. "There is no question that under almost any other system,
le Pen would not have made it to the runoff," says Steven Brams, a
political scientist at New York University.
If it weren't for the
plurality system, Abraham Lincoln might never have become president,
Tabarrok says. In the four-candidate 1860 election, Lincoln was a
polarizing figure, popular with many Northerners but abhorred by
many Southerners. Stephen Douglas, Lincoln's closest competitor, was
more broadly popular, and although he didn't get as many first-place
rankings as Lincoln did, he was nearly everyone's second choice,
historians hold. In 1999, Tabarrok and Lee Spector, an economist at
Ball State University in Muncie, Ind., calculated that if almost any
other voting system had been used, history books would refer to
President Douglas, not President Lincoln. "On paper, Lincoln's
victory looks overwhelming, but he actually didn't have broad-based
support," Tabarrok says. With Lincoln now a folk hero, the result of
that election might seem good in retrospect. But that's a separate
matter from whether the voters actually preferred Lincoln on
Election Day, 1860.
History is full of similar situations, Tabarrok
says. "One thing we've discovered is how radically the outcome of an
election can change by even a small change in the voting system," he
In some elections, in fact, any one of the candidates can be
the winner, depending on what voting system is being used (see "And
the winner is?" below). Saari has calculated that in three-candidate
elections, depending on the voting system, more than two-thirds of
all possible configurations of voters' preferences will yield
No one's perfect
Is there a best voting
procedure? In 1952, Kenneth Arrow, a professor emeritus of economics
at Stanford University in Palo Alto, Calif., proved that no voting
system is completely free from counterintuitive outcomes. Arrow
looked at voting systems that satisfy two harmless- sounding
properties. First, if everyone prefers candidate A to candidate B,
then A should be ranked higher than B. Second, voters' opinions
about candidate C shouldn't affect whether A beats Bˇafter all, if
you prefer coffee to tea, finding out that hot chocolate is
available shouldn't suddenly make you prefer tea to coffee. These
sound like reasonable restrictions, yet Arrow proved that the only
voting system that always satisfies them is a dictatorship, where a
single person's preferences determine the outcome.
behavior Arrow studied crops up all the time. Saari points to the
2000 Bush-Gore-Nader race in Florida. "It's a beautiful example of
Arrow's theorem at work," Saari says.
While Arrow's theorem shows
that no system is flawless, many capture voter preferences more
effectively than plurality voting does. For instance, the
paradoxical outcome of the Florida race might have been avoided
under the instant runoff, which is advocated by the Center for
Voting and Democracy in Takoma Park, Md. In that system, voters rank
the candidates, then the candidate with the fewest first-place votes
is dropped. That candidate is erased from the voters' preference
lists, and ballots of voters who had placed him first are converted
into votes for their second choice. From the remaining candidates,
once again the one with the fewest first-place votes is dropped.
When only two candidates remain, the one with more top votes wins.
Since voters communicate their entire ranking when they vote,
there's no need to hold repeated elections. In Florida, Nader would
probably have been eliminated in an instant runoff, most of his
votes converted into votes for Gore.
An instant runoff also reduces
the dangers inherent in an election with many candidates. In the
French election, most of the voters who selected one of the weaker
candidates probably preferred Chirac or the thenprime minister,
Socialist Lionel Jospin, to le Pen. Then in an instant runoff, as
candidates were eliminated, their votes would have gone to Chirac
Instant-runoff voting could make campaigns both more
civil and more issue oriented, suggests Terry Bouricius, New England
regional director for the Center for Voting and Democracy. "To win,
you have to be highly ranked by a majority of voters, and you also
have to appeal to a bunch of voters strongly enough to get their
first-place votes," he says. "So, you have to distinguish yourself
from the other candidates but also build coalitions."
Chaos in the polling place
Whatever its potential benefits, instant-runoff
voting is prone to one of voting theory's most bewildering
paradoxes. If a candidate is in the lead during an election season,
making a great speech that attracts even more supporters to his
cause shouldn't make him lose. But in the instant-runoff system, it
can. Suppose, for example, that 35 percent of voters prefer A first,
B second, and C third; 33 percent prefer B first, C second, and A
third; and 32 percent prefer C first, A second, and B third. In an
instant runoff, C will be eliminated, leaving A and B to face each
other. A scoops up C's first- place votes, winning a resounding 67
percent to 33 percent victory over B. But suppose A makes such an
inspiring speech that some voters who liked B best move A into first
place, so now 37 percent rank the candidates as A-B-C, 31 percent as
B-C-A, and 32 percent as C-A-B. Now, A faces C in the runoff, not B.
The votes that ranked B first become votes for C, and C beats A, 63
percent to 37 percent.
In an article to be published early next
year in the Journal of Economic Theory, Saari has catalogued
scenarios that give rise to this type of paradox. It can occur in
any voting procedure with more than one round, he has found, but
never in one-round procedures.
Saari's result draws on a seemingly
unrelated field of mathematics: chaos theory, which studies physical
systems, such as the weather, in which tiny changes in the starting
conditions can have drastic repercussions. Chaos researchers look
for points at which the systems' parameters stabilize momentarily
and then change direction, since only near those points can a small
change produce dramatic effects. Saari realized that in voting
theory, only when an election is nearly tied does a small change in
voter preferences swing the election in a new direction. By looking
at arrangements of ties, Saari has classified the possible
paradoxical outcomes for a wide range of procedures.
that the way to identify the best voting procedure is to consider
which scenarios should result in ties. If three voters have what
researchers call cyclic preferencesˇone prefers A-B-C, one B-C- A,
one C-A-Bˇthere should be a tie, he says. Likewise, if two voters
have exactly opposite preferencesˇone prefers A-B-C, say, and the
other, C-B-Aˇtheir votes should cancel. The only common voting
procedure that would give a tie to both of these cases is the Borda
count, which gives two points to a voter's top choice and one point
to his second choice in a three-candidate election.
instant runoff, the Borda count gives weight to a voter's entire
preference ranking. If the Borda count had been used, second- place
votes would probably have tipped the 2000 race in Gore's favor,
Saari and Brams say. And in France, it's highly unlikely that le Pen
would have come in second, Saari says.
Saari has shown that the
Borda count is much less prone to the kinds of paradoxes that Arrow
studied than most other systems are. Using ideas from chaos theory,
Saari has found, for instance, that plurality voting in a
six-candidate election gives rise to 1050 times as many paradoxical
situations as the Borda count does.
Approve or disapprove
the researchers are fans of the Borda count, however. Brams objects
that it forces voters to rank all the candidates, even when there
are some about whom they have no strong opinion, potentially leading
to outcomes that don't really reflect voter preferences.
prefers approval voting, in which people vote for as many candidates
as they like. Approval voting, Brams says, gives voters more
sovereignty by enabling them to express the intensity of their
preferences: a voter who strongly favors one candidate can vote for
just that candidate, while a voter who can't stand one candidate can
vote for everyone else. A voter with more-moderate views can vote
for any number of candidates between these two extremes.
to predict the outcome of an approval vote since voters' choices
depend on where they draw the line between approval and disapproval.
But Brams argues that approval voting would significantly alter
voter behavior in many elections. In the 2000 presidential race, for
instance, approval voting would have enabled Nader supporters to
vote for him and also for one of the two stronger contenders.
the instant runoff, Borda count, and approval voting each has
drawbacks, most voting theorists would be happy to replace plurality
voting with any one of them. "All methods that allow voters to
express their views fully rather than to single out one candidate
convey a much more nuanced message to the political machine," says
Hannu Nurmi, a political scientist at the University of Turku in
The fact that U.S. elections have always been plurality
votes is no reason to resist change, Tabarrok says. "We chose our
voting systems before voting theory existed," he says. "I don't
think any voting theorist would choose plurality rule today."
The real lesson to draw from recent election anomalies,
voting theorists say, is that citizens should think carefully not just about
how well the election machinery counts up the votes but
also about how they want the votes to count.
And the winner
is? Different voting methods can produce very
In some elections, any
candidate can win, depending on which voting system is used, says
Donald Saari of the University of California, Irvine. Consider 15
people deciding what beverage to serve at a party. Six prefer milk
first, wine second, and beer third; five prefer beer first, wine
second, and milk third; and four prefer wine first, beer second, and
milk third. In a plurality vote, milk is the clear winner. But if
the group decides instead to hold a runoff election between the two
top contendersˇmilk and beerˇthen beer wins, since nine people
prefer it over milk. And if the group awards two points to a drink
each time a voter ranks it first and one point each time a voter
ranks it second, suddenly wine is the winner. Although this is a
concocted example, it's not an anomaly, Saari insists.
If you have
a comment on this article that you would like considered for
publication in Science News, please send it to
Brams, S.J., and D.R. Herschbach.
2001. The science of elections. Science 292(May 25):1449. Available
Malkevitch, J. 2002. Voting and
elections. American Mathematical Society (March). Available at http://www.ams.org/new-in-math/cover/voting-introduction.html
Tabarrok, A., and L. Spector.
1999. Would the Borda count have avoided the Civil War? Journal of
Theoretical Politics 11(April):261- 288. Abstract available at
Taylor, A.D. 2002. The manipulability of voting systems. American
Mathematical Monthly 109(April):321-337.
Further Readings :
Fishburn, P., and S. Brams. 1983. Paradoxes of preferential voting.
Mathematical Magazine 56(September):207-214.
Peterson, I. 1998. How to fix an
election. Science News Online (Oct. 31). Available at http://www.sciencenews.org/sn_arc98/10_31_98/mathland.htm
Saari, D. 2001. Decisions and
Elections: Explaining the Unexpected. Cambridge, England: Cambridge
University Press. See http://books.cambridge.org/0521004047.htm
______. 2000. Chaotic Elections!
A Mathematician Looks at Voting. Providence, R.I.: American
Mathematical Society. See http://www.ams.org/bookstore/pspdf/elect.pdf
______. 1999. Chaos, but in
voting and apportionments? Proceedings of the National Academy of
Sciences 96(Sept. 14):10568-10571. Available at http://www.pnas.org/cgi/content/full/96/19/10568
______. 1995. A
chaotic explanation of aggregation paradoxes. SIAM Review
Saari, D.G., and F. Valognes. 1998. Geometry,
voting, and paradoxes. Mathematics Magazine 71(October):243-259.
Voting and Democracy
6930 Carroll Avenue, Suite 610
Park, MD 20912
Web site: http://www.fairvote.org/
Steven J. Brams
New York University
715 Broadway, Fourth Floor
New York, NY 10003-6806
Web site: http://www.nyu.edu/gsas/dept/politics/faculty/brams/brams_home.html
University of Turku
Web site: http://www.utu.fi/yht/valtio-oppi/hallinto/hannu2.html
Donald G. Saari
University of California, Irvine
Web site: http://www.math.uci.edu/faculty/dsaari.html
100 Swan Way
Oakland, CA 94621-1428
Web site: http://www.independent.org/index.html
From Science News, Vol. 162,
No. 18, Nov. 2, 2002, p. 280.