In the 1995 women's figure-skating world championship, Chen Lu, Nicole Bobek, and Surya Bonaly were in first, second, and third place after they finished skating. Then 14-year-old Michelle Kwan surged into fourth with a strong showing in the free skate. In a bizarre twist attributed to an unusual new judging system, Kwan's strong performance caused Bonaly and Bobek to switch places. Bonaly got the silver, Bobek the bronze, and Nobel Prize-winning economist Kenneth Arrow got even more vindication for his work on the drawbacks of rank-order voting.

In the 1950s, Arrow had argued that there is no "good" election method. His research focused on methods in which each voter lists some or all of the candidates in a strict linear order (first choice, second choice, etc.). Arrow proved that any such method must display at least one of the following unreasonable characteristics: (1) a "dictator" always decides who wins, regardless of how everybody else votes; (2) outcomes sometimes conflict with even unanimous electoral preferences; or (3) a seemingly irrelevant candidate changes the relative standing of two others.

The 1995 championship illustrated the third problem. Previously, judges scored skaters from 0 to 6, and the one with the highest average won. But that year, scores were used only to produce rank orders. Final standings were determined by a complex algorithm based largely on the number of times each skater was ranked first, second, third, and so forth. When some judges gave Kwan a higher score than Bobek, some of Bobek's rankings slipped. So even though her numeric scores didn't change, she was demoted to bronze.

Arrow's theorem could have predicted this oddity. But when he concluded that all election methods are inherently flawed, he had neglected an important fact: election methods do not have to be based on rank ordering.

Honeybees hold "elections" each year to choose a new location for their hive; bad decisions could lead to the colony's annihilation. Over 50 million years, natural selection produced a system in which scout bees "score" each candidate site with dances describing the site's direction and distance. The more intense the dance, the greater the chance that other scouts will investigate the site. When a site attracts a sufficiently large majority of followers, it wins.

Sparta, the longest-lasting substantially democratic government in history, voted in a similar way from about 700 b.c.e. until at least 220 b.c.e. Spartans elected Gerontes and Ephors (council members who had the power to dethrone kings) by means of a shouting system. The candidate with the loudest support won.

Both the bees' system and the Spartans' are examples of range voting: each voter scores each candidate within a given range (say, 0 to 99); the one with the highest total wins. As John Harsanyi (also a Nobelist) observed when Arrow's research was published, range voting accomplishes what Arrow deemed impossible. But since his point went against 1950s economic gospel, it was ignored.

The current U.S. election method, in which voters name one--and only one--candidate, is called plurality voting. In elections with more than two candidates, plurality voting is one of the worst methods. Candidates with overlapping constituencies often split votes, causing a less popular candidate to win. For example, Ralph Nader's third-party campaign helped Bush defeat Gore in 2000. With Nader and Bob Barr organizing to be on as many states' ballots as possible in 2008, such danger looms again.

In the following hypothetical situation with 100 voters, 73 percent agree that Hitler is the worst choice. (For simplicity, we limited possible rank orders to four.) But with plurality voting, he wins:

 # Voters Voted for True preference (best> good> bad> worst) 24 Castro Castro> Obama> McCain> Hitler 26 McCain McCain> Obama> Castro> Hitler 23 Obama Obama> Castro> McCain> Hitler 27 Hitler Hitler> McCain> Obama> Castro

Groups such as the FairVote Center for Voting and Democracy in Takoma Park, MD, advocate the Australian practice of instant-runoff voting (IRV). With IRV, voters rank the candidates ex­plicitly. The candidate ranked first by the fewest voters is eliminated, and that candidate's votes transfer to the voters' second choices. This operation repeats until just one candidate remains. In the example, Castro wins after IRV eliminates Obama (whose votes transfer to Castro), then McCain (whose votes transfer to Castro, since Obama has already been eliminated), and finally Hitler.

But IRV can also produce illogical outcomes. Here, even though 53 percent of voters--a clear majority--prefer McCain over Castro, Castro still wins. (Fifty-three percent of voters also rank McCain ahead of Obama, and 73 percent rank him ahead of Hitler.) For mathematical reasons that are difficult to explain, IRV artificially favors extremist candidates. And IRV shares another problem with plurality voting: each can motivate voters to lie.

With plurality voting, if the Obama supporters strategically voted for Castro, Castro would win--an outcome they would prefer to a Hitler victory. With either IRV or plurality voting, McCain voters might be better off pretending their favorite was Obama, thus heading off a Hitler or Castro win. But with range voting, each voter can express how much (or little) he or she likes each candidate, and it is never rational to give your favorite less than the top score. (Assigning two or more candidates the same score is permitted.) Applying 0-to-99 range voting to the previous example, the votes might be:

 # Voters range votes 24 Castro-99 Obama-90 McCain-30 Hitler-0 26 McCain-99 Obama-60 Castro-30 Hitler-0 23 Obama-99 Castro-90 McCain-30 Hitler-0 27 Hitler-99 McCain-0 Obama-0 Castro-0

Obama would win with 5,997, defeating Castro (5,226), McCain (3,984), and Hitler (2,673).

Warren Smith performed computer simulations using Bayesian regret analysis to compare election methods, measuring the quality of election outcomes by summing the utility--or satisfaction--of the voters. These simulations indicate that switching from plurality voting to range voting would improve election outcomes as much as switching from dictatorship to democracy would. Range voting also outperforms all common alternative systems on average--no matter how many honest, strategic, and uninformed voters cast their votes, and no matter how many candidates run. (See William Poundstone's Gaming the Vote for a good summary of this analysis.)

Range voting using a 0 to 9 scale can be done on existing computerized or lever voting machines, punch cards, or paper ballots. Voters would simply select a score for each candidate. In fact, French researchers found that voters make fewer errors on range ballots than they do on plurality ballots.

Changing the way the U.S. president is chosen may seem daunting, but plurality voting is a shaky foundation on which to rest the fate of the country. It doesn't let voters express how strongly they feel, which is a big drawback when some choices are much worse than others. Plagued by the prospect of spoilers and illogical outcomes, it perpetuates a two-party duopoly, minimizing democratic choice.

IRV can't fix the problems of plurality voting. But computer simulations, Spartans, and trillions of honeybee elections offer convincing evidence that range voting can.

Alan T. Sherman, PhD '87, teaches computer science and is part of the National Center for the Study of Elections at the University of Maryland, Baltimore County. Warren D. Smith '84 cofounded the Center for Range Voting (rangevoting.org). Richard T. Carback III is a PhD candidate at the University of Maryland, Baltimore County.

Copyright Technology Review 2008.