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No. 1921:
Arrow's Paradox

by Andy Boyd

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."

Multi-candidate ballot

This episode has been substantially revised as Episode 2427