Today, guest scientist Andrew Boyd votes. The
University of Houston presents this series about
the machines that make our civilization run, and
the people whose ingenuity created them.
The 2000 presidential
election between George W. Bush and Al Gore brought
to our attention the problems involved with
counting votes. Dangling chads notwithstanding,
political scientists have long been aware that
there are much deeper problems with voting systems;
problems so fundamental they leave us scratching
our heads and asking what's going on.
Trouble first surfaced during the Enlightenment, as
Jean-Charles de Borda and the Marquis de Condorcet
debated the merits of different voting schemes. But
it was not until 1951 that Nobel laureate Kenneth
Arrow fully laid bare a problem that Borda and
Condorcet had been struggling with.
Borda advocated letting people rank each candidate
with a number, adding the points, and choosing the
candidate with the best total score. We could view
the method of voting we use today as a special case
of Borda's method -- where our favorite candidate
re-ceives one point and everyone else receives
none.
Condorcet, on the other hand, advocated a vote
between every pair of candidates. The candidate
that wins in every comparison is elected. The
practical problem with Condorcet's method is that
it may fail to produce a winner. We see this all
the time in athletic competitions. The Astros beat
the Reds, the Reds beat the Cubs, and the Cubs beat
the Astros. Who's the winner? In voting, this is
known as Condorcet's Paradox.
But there's a hidden problem in Borda's method of
numerical ranking, too. Imagine Smith and Jones are
running for office, we cast our votes, and Smith
wins. Now suppose a new candidate enters the
election and we vote again. Even if we all feel the
same way about Smith and Jones, we may find Jones
now wins. This is a very real problem in U.S.
elections, and the democratic and republican
parties constantly worry about candidates from
third parties claiming votes.
The fact that candidates entering or leaving the
race can change the order of the remaining
candidates is very alarming. If I prefer chocolate
ice cream to vanilla, and someone offers me
strawberry ice cream, why should I now prefer
vanilla to chocolate? Yet this is exactly what can
happen with Borda's method. We might ask if there
is a voting system -- any system at all -- that
doesn't threaten to flip-flop the two candidates,
when a third can-didate enters the race.
Remarkably, Arrow proved that for any system
meeting the most basic standards of common sense,
the answer is No.
The implications for voting are stunning. But the
impact of Arrow's work on economics and social
choice goes far deeper. If we can't combine
individual preferences in any reasonable way, can
we even talk about society's preferences? If we
can't talk about society's preferences, how can we
develop economic or social policies and claim they
represent what society prefers?
Arrow did more than prove a result that now bears
his name. Like many of the best results in science,
engineering, and mathe-matics, Arrow's theorem
distills a known problem into its most basic
pieces, and, in doing so, helps us see the world in
a surprising new way.
I'm Andy Boyd, at the University of Houston, where
we're interested in the way inventive minds work.
(Theme music)
Dr. Andrew Boyd is Chief Scientist and Senior Vice
President at PROS, a pricing and revenue optimization
software firm. Dr. Boyd received his A.B. with Honors
at Oberlin College with majors in Mathematics and
Economics in 1981, and his Ph.D. in Operations
Research from MIT in 1987. Prior to joining PROS, he
enjoyed a successful ten year career as a university
professor.
Arrow presented five postulates that any "sensible"
or "fair" voting system should satisfy. He then
mathematically proved that these postulates were
mutually contradictory -- no voting system could
satisfy all five.
For brevity, we've focused on the most famous
postulate, the independence of irrelevant
alternatives, which loosely states that when
candidate A is preferred
to B, then
A should still be
preferred to B if other
candidates enter or leave the election. As basic as
this may seem, the other postulates are even more
so, and are therefore simply referred to as "basic
standards of sensibility."
Arrow's postulates and links to a proof can be
found at:
http://encyclopedia.thefreedictionary.com/Arrow's+theorem
This episode has been substantially revised as Episode 2427.
The Engines of Our Ingenuity is
Copyright © 1988-2004 by John H.
Lienhard.
