Engines of Our Ingenuity

No. 2427
ARROW’S PARADOX

by Andrew Boyd

Today, we vote. The University of Houston’s College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

Political scientists have long been aware that there are problems with voting systems; problems so deep and 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 receives 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 with Borda's method of numerical ranking, too. Imagine we can get chocolate or vanilla ice cream for our picnic group. We cast votes, and chocolate wins. Now suppose someone suggests strawberry as an option. We add it to the list and vote again. Even though we all feel the same way about chocolate and vanilla, we may find vanilla now wins. Seems silly. But it’s a very real problem in U.S. elections, and the democratic and republican parties constantly worry about candidates from third parties claiming votes.

One of the most famous examples occurred in the 2000 presidential election where George W. Bush narrowly defeated Al Gore. That same year, Ralph Nader won close to three percent of the popular vote. Political scientists believe that had Nader not been on the ticket, most of his votes would have gone to Gore, changing the outcome of the election.

We might ask if there’s a voting system — any system at all — that doesn't threaten to flip-flop two candidates when a third candidate 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 reasonably combine individual preferences, how can we develop economic or social policies then claim they represent what society prefers? In a real sense, Arrow used mathematics to show that we can’t; that instead, rhetoric, gamesmanship, and back-room deals must necessarily be part of the political process.

I’m Andy Boyd at the University of Houston, where we’re interested in the way inventive minds work.

(Theme music)


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."

A simple, brief description of Arrow's postulates and links to a proof can be found at: http://www.websters-online-dictionary.org/definition/ARROW%2527S+PARADOX.

A more detailed description, including an outline of a proof, can be found at http://en.wikipedia.org/wiki/General_Possibility_Theorem.

Multi-candidate ballot

This is a substantially revised version of Episode 1921



The Engines of Our Ingenuity is Copyright © 1988-2008 by John H. Lienhard.