[EM] FWD - How to Fix n Election
Donald E Davison
donald at mich.com
Thu Mar 25 04:40:58 PST 1999
--------- Forwarded without comment ----------
Resent-Date: Wed, 24 Mar 1999 10:53:27 -0500
Date: Wed, 24 Mar 1999 07:49:44 -0800
From: "Peter B. Macdonald" <pbmacdonald at netci.com>
Reply-To: pbmacdonald at netci.com
Organization: Private
MIME-Version: 1.0
To: donald at mich.com
Subject: How to Fix an Election
The following appeared on one of my listserve.
Though it might be of interest.
Some of you math types might like to pick it apart.
Regards,
Peter B. Macdonald
http://www.netci.com/taxcap
The other side of Direct Democracy
------------------------------------------------------------
>From Ivars Peterson's Math Trek, Science News Online,
http://www.sciencenews.org/sn_arc98/10_31_98/mathland.htm
October 31, 1998
How to Fix an Election
By Ivars Peterson -- Science News Online
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 ip at sciserv.org.
Ivars Peterson is the mathematics/computers writer and online
editor at Science News.
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T H E C O D E O F H O N O R F O R R E F O R M A C T I V I S T S
Any group of reform activists that are thinking about a petition drive
to place a proposal on the ballot are to present their proposal beforehand
to all other reform activists that they know of. The time for debate and
negative comments is before the petition stage. Once the group makes its
final proposal and enters the petition stage, the debates and negative
comments by all reform activists is to cease.
At this time each activist is to make an honest evaluation. If the
initiative will improve government then each activist is to find it in his
heart to support the initiative, even if it is not exactly what the
activist would like.
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| Q U O T A T I O N |
| "Democracy is a beautiful thing, |
| except that part about letting just any old yokel vote." |
| - Age 10 |
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