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In 1980 Duke abruptly left the Klan






A Test Drive of Voting Methods
By William Poundstone















William Poundstone is author of
Gaming the Vote: Why Elections Arent
Fair (and What We Can Do About It)
(Hill and Wang, 2008). E-mail:
william.poundstone@sbcglobal.net Poundstone / Test Drive

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Mathematics is not normally a subject that inspires heated
controversies. A notable exception is the mathematics of voting. What is
the fairest way of voting? is a simple question, and you might hope that it
would have a simple answer. It doesnt. This issue is the biggest, longest-
running dispute in voting theory.
There is little controversy over what voting method is worst. That
would be the plurality vote the system used in almost all American
elections. Any voting theorist will tell you that the plurality vote is
especially vulnerable to vote splitting. When two candidates base of support
overlaps, both candidates are penalized. This is most easily seen in a case
like the 1912 Presidential election, where two Republicans (William Howard
Taft and Teddy Roosevelt) ran against each other, splitting the Republican
vote. The result was the election of a Democrat (Woodrow Wilson) who
probably wouldnt have won otherwise. A more common example of vote
splitting is the familiar political phenomenon of a spoiler. In 2000, Ralph
Nader almost certainly took enough of the liberal vote away from Al Gore in
the crucial state of Florida to hand the election to George W. Bush.
Spoilers arent uncommon. Of the U.S. presidential elections since
1828, at least five were probably decided by spoilers. In effect, weve
elected the wrong president eleven percent of the time. Poundstone / Test Drive

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So whats the solution? Everyone agrees that we need to collect a little
more information from voters. The ballot should be designed so that voters
can express how they feel about the candidates who werent their first-place
choices. It is also necessary to find a way to make use of that additional
information in tallying the ballots.
There are two obvious approaches. One is to ask the voters to rank the
candidates in order of preference. Dozens of ranked-ballot methods have
been devised. The two most historically important are named for Jean-
Charles Borda and the Marquis de Condorcet, rival members of the French
Academy in the eighteenth century. Borda and Condorcet sparred over the
merits of their respective systems, initiating a feud whose skirmishes
continue to the present day. Though Bordas and Condorcets schemes can
use an identical ballot, their methods of tallying ranked votes are completely
different, and sometimes, so are the winners.
Poundstone / Test Drive

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In Bordas system, the rankings are converted into points. In an
election with four candidates, every first-place ranking is worth 3 points,
second-place is 2 points, third-place is 1 point, and fourth (last) place is
worth nothing. The points are added, and the candidate with the most points
wins.
Sounds reasonable? Borda thought so. Condorcet objected that the
winner of an election should be able to beat every other candidate in two-
way votes something not guaranteed by Bordas system. Today,
computers can readily determine the Condorcet winner by using the ballot
rankings.
Another important ranked system is instant runoff voting (IRV), a
nineteenth-century invention that is now used in Australia, Ireland, and other
nations. IRV simulates a series of runoffs in which the least popular
candidate of each round is eliminated. Each time a candidate is eliminated,
ballots ranking that candidate highest are transferred to the highest-ranked of
the remaining candidates. That way, votes for minor candidates are not
wasted and ultimately count toward the voters preferred front-runner.
Though these capsule descriptions of IRV, Condorcet, and Borda may
all sound perfectly reasonable, it is possible to have an election in which the
three systems produce three different winners from the same set of ranked
ballots.
A fundamentally different approach to voting is to ask voters to rate or
score the candidates. In approval voting, the ballot is essentially a report
card with pass-or-fail grades. The candidate who gets the most passing
grades (approval votes) wins. A generalization of approval voting called Poundstone / Test Drive

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range voting has voters rate the candidates on a numerical scale say, 1 to
10. The candidate with the highest average score wins. Approval and range
voting are much more closely related to each other than the ranked systems
are but, yes, its possible for range and approval voting to have different
winners.

Which system is best? There is no easy answer because we are dealing
with the paradoxical mathematics of voting, the human behavior of voters
and candidates, and philosophical questions about what it means for a voting
method to be fair. Over the past twenty years, the ensuing debates have
resulted in frayed nerves, strained personal relationships, and very little
consensus. Donald G. Saari, a Borda proponent, and Steven Brams, an
approval voting advocate, have found it necessary to avoid discussing their
differences in order to remain friends. No such pact of silence restrains the
raging war of words between range voting theorist Warren D. Smith and Poundstone / Test Drive

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Rob Richie, a political activist whose organization promotes IRV. The
ongoing lack of agreement has surely hampered efforts to replace the
plurality vote with something better.

Id like to showcase a novel way of visualizing a few of these abstract
issues. It was devised by Ka-Ping Yee, a recent graduate of Berkeleys
computer science Ph.D. program. Yee became interested in voting in 2004,
when Berkeley adopted IRV for its city elections and the Presidential race
raised concerns about electronic voting. He read up on Kenneth Arrows
impossibility theorem and tried to convince the members of his Berkeley
residence, Kingman Hall, to adopt approval voting. The motion failed, but
the residence later adopted Condorcet voting, another method Yee had
discussed.
Much of the voting literature focuses on what can go wrong with
electoral methods on mathematical paradoxes (often rare) and aberrant
voter behavior (often conjectural). These are unquestionably important
matters to discuss. But Yee took the opposite tack. Suppose that every
voting method works exactly the way its proponents want it to work. Which
method would be fairest then?
To answer this, Yee used computer simulations to generate colorful
maps of multicandidate elections. In these simulations, the publics views
on political issues form a normal distribution. That is the familiar bell-
shaped curve that describes a wide range of natural variations. Dont get too
hung up on this normal distribution business. Yee isnt saying that political
beliefs follow a normal distribution, only that this provides a good test drive Poundstone / Test Drive

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for voting systems. It eliminates the more contrived paradoxes and restricts
attention to a simple case where it is possible to have a good intuitive sense
of what an elections outcome ought to be.
A
B

This diagrams horizontal scale represents the range of possible
convictions on a particular issue. The height of the curve at any point tells
how many voters favor that particular position. The curve is highest
somewhere in the middle. This represents the center of opinion. As you go
further to the left or right, the height of the curve diminishes.
There are two candidates, A and B. B is a little to the right of center,
and A is well to the left of center. Should the candidates otherwise be
equally qualified and appealing, and should the issue represented by the Poundstone / Test Drive

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chart be the only one that matters, you would expect that B would beat A
under any reasonable voting system. B is closer to the political mainstream.
In the diagrams below, Yee postulates two independent political
issues. After all, most campaigns have many issues at stake. You might think
of the two here as small v. large government and social conservative v.
social liberal. A candidate who takes a given position on one issue is free to
take any position at all on the other issue. You therefore need a two-
dimensional square to chart the range of possible