Election Selection

Are we using the worst voting procedure?

Erica Klarreich

As Election Day approaches, voters must be feeling a sense of déjà vu. With recent reports of malfunctioning voter machines and uncounted votes during primaries in Florida, Maryland, and elsewhere, reformers are once again clamoring for extensive changes. But while attention is focused on these familiar irregularities, a much more serious problem is being neglected: the fundamental flaws of the voting procedure itself, say various researchers who study voting.

Nearly all political elections in the United States are plurality votes, in which each voter selects a single candidate, and the candidate with the most votes wins. Yet voting theorists argue that plurality voting is one of the worst of all possible choices. "It's a terrible system," says Alexander Tabarrok, an economist at George Mason University in Fairfax, Va., and director of research for the Independent Institute in Oakland, Calif. "Almost anything looks good compared to it."

Other voting systems abound. One alternative is the instant runoff, a procedure used in Australia and Ireland that eliminates candidates one at a time from rankings provided by each voter. Another is the Borda count, a point system devised by the 18th-century French mathematician Jean Charles Borda, which is now used to rank college football and basketball teams. A third is approval voting, used by several scientific societies, in which participants may cast votes for as many of the candidates as they choose.

Unlike these procedures, the plurality system looks only at a voter's top choice. By ignoring how voters might rank the other candidates, it opens the floodgates to unsettling, paradoxical results.

In races with two strong candidates, plurality voting is vulnerable to the third-party spoiler—a weaker candidate who splits some of the vote with one of the major candidates. For instance, in the hotly contested 2000 U.S. presidential race, Republican George W. Bush won the state of Florida—and, consequently, the presidency—by just a few hundred votes over Al Gore, the Democratic candidate. Green Party candidate Ralph Nader won 95,000 votes in Florida, and polls suggest that for most Nader voters, Gore was their second choice. Thus, if the race had been a head-to-head contest between Bush and Gore, Florida voters probably would have chosen Gore by a substantial margin. Should Nader have withdrawn from the race, as many angry Democrats asserted? Certainly not, says mathematician Donald Saari of the University of California, Irvine. "We live in a democracy, and anyone should be able to run for any office," he says. "The problem was the bad design of the election."

Mathematics can shed light on questions about how well different voting procedures capture the will of the voters, Saari says. In ongoing work, he has been using tools from chaos theory to identify just which scenarios of voter preferences will give rise to disturbing election outcomes. "With the muscle power of mathematics, we can address these questions and finally get some results," he says.

Singular plurality

In elections with only two candidates, plurality voting works just fine, since the winner is guaranteed to have been the top choice of more than half the voters. But as soon as three or more candidates are on the ballot, the system can run into trouble.

In races with a large slate of candidates, plurality voting dilutes voter preferences, creating the possibility of electing a leader whom the vast majority of voters despise. In the French election last April, with 16 candidates on the ballot, extreme right-wing candidate Jean-Marie le Pen—widely accused of racism and anti-Semitism—managed to place second with just 17 percent of the vote. He then advanced to a runoff against the top candidate, incumbent President Jacques Chirac. Political analysts scrambled to explain le Pen's success, putting it down to voter disenchantment and a surge in right-wing fervor across Europe. But the real reason, voting theorists say, is that the plurality vote distorted the preferences of the voters.

"The fact that le Pen was in the runoff had nothing to do with what the people wanted," Saari says. The runoff election, in which Chirac trounced le Pen with 82 percent of the vote, suggests that while le Pen was at the top of a few voters' lists, he was near the bottom of many more. "There is no question that under almost any other system, le Pen would not have made it to the runoff," says Steven Brams, a political scientist at New York University.

If it weren't for the plurality system, Abraham Lincoln might never have become president, Tabarrok says. In the four-candidate 1860 election, Lincoln was a polarizing figure, popular with many Northerners but abhorred by many Southerners. Stephen Douglas, Lincoln's closest competitor, was more broadly popular, and although he didn't get as many first-place rankings as Lincoln did, he was nearly everyone's second choice, historians hold. In 1999, Tabarrok and Lee Spector, an economist at Ball State University in Muncie, Ind., calculated that if almost any other voting system had been used, history books would refer to President Douglas, not President Lincoln. "On paper, Lincoln's victory looks overwhelming, but he actually didn't have broad-based support," Tabarrok says. With Lincoln now a folk hero, the result of that election might seem good in retrospect. But that's a separate matter from whether the voters actually preferred Lincoln on Election Day, 1860.

History is full of similar situations, Tabarrok says. "One thing we've discovered is how radically the outcome of an election can change by even a small change in the voting system," he says.

In some elections, in fact, any one of the candidates can be the winner, depending on what voting system is being used (see "And the winner is?" below). Saari has calculated that in three-candidate elections, depending on the voting system, more than two-thirds of all possible configurations of voters' preferences will yield different outcomes.

No one's perfect

Is there a best voting procedure? In 1952, Kenneth Arrow, a professor emeritus of economics at Stanford University in Palo Alto, Calif., proved that no voting system is completely free from counterintuitive outcomes. Arrow looked at voting systems that satisfy two harmless-sounding properties. First, if everyone prefers candidate A to candidate B, then A should be ranked higher than B. Second, voters' opinions about candidate C shouldn't affect whether A beats B—after all, if you prefer coffee to tea, finding out that hot chocolate is available shouldn't suddenly make you prefer tea to coffee. These sound like reasonable restrictions, yet Arrow proved that the only voting system that always satisfies them is a dictatorship, where a single person's preferences determine the outcome.

The paradoxical behavior Arrow studied crops up all the time. Saari points to the 2000 Bush-Gore-Nader race in Florida. "It's a beautiful example of Arrow's theorem at work," Saari says.

While Arrow's theorem shows that no system is flawless, many capture voter preferences more effectively than plurality voting does. For instance, the paradoxical outcome of the Florida race might have been avoided under the instant runoff, which is advocated by the Center for Voting and Democracy in Takoma Park, Md. In that system, voters rank the candidates, then the candidate with the fewest first-place votes is dropped. That candidate is erased from the voters' preference lists, and ballots of voters who had placed him first are converted into votes for their second choice. From the remaining candidates, once again the one with the fewest first-place votes is dropped. When only two candidates remain, the one with more top votes wins. Since voters communicate their entire ranking when they vote, there's no need to hold repeated elections. In Florida, Nader would probably have been eliminated in an instant runoff, most of his votes converted into votes for Gore.

An instant runoff also reduces the dangers inherent in an election with many candidates. In the French election, most of the voters who selected one of the weaker candidates probably preferred Chirac or the then–prime minister, Socialist Lionel Jospin, to le Pen. Then in an instant runoff, as candidates were eliminated, their votes would have gone to Chirac and Jospin.

Instant-runoff voting could make campaigns both more civil and more issue oriented, suggests Terry Bouricius, New England regional director for the Center for Voting and Democracy. "To win, you have to be highly ranked by a majority of voters, and you also have to appeal to a bunch of voters strongly enough to get their first-place votes," he says. "So, you have to distinguish yourself from the other candidates but also build coalitions."

Chaos in the polling place

Whatever its potential benefits, instant-runoff voting is prone to one of voting theory's most bewildering paradoxes. If a candidate is in the lead during an election season, making a great speech that attracts even more supporters to his cause shouldn't make him lose. But in the instant-runoff system, it can. Suppose, for example, that 35 percent of voters prefer A first, B second, and C third; 33 percent prefer B first, C second, and A third; and 32 percent prefer C first, A second, and B third. In an instant runoff, C will be eliminated, leaving A and B to face each other. A scoops up C's first-place votes, winning a resounding 67 percent to 33 percent victory over B. But suppose A makes such an inspiring speech that some voters who liked B best move A into first place, so now 37 percent rank the candidates as A-B-C, 31 percent as B-C-A, and 32 percent as C-A-B. Now, A faces C in the runoff, not B. The votes that ranked B first become votes for C, and C beats A, 63 percent to 37 percent.

In an article to be published early next year in the Journal of Economic Theory, Saari has catalogued scenarios that give rise to this type of paradox. It can occur in any voting procedure with more than one round, he has found, but never in one-round procedures.

Saari's result draws on a seemingly unrelated field of mathematics: chaos theory, which studies physical systems, such as the weather, in which tiny changes in the starting conditions can have drastic repercussions. Chaos researchers look for points at which the systems' parameters stabilize momentarily and then change direction, since only near those points can a small change produce dramatic effects. Saari realized that in voting theory, only when an election is nearly tied does a small change in voter preferences swing the election in a new direction. By looking at arrangements of ties, Saari has classified the possible paradoxical outcomes for a wide range of procedures.

Saari argues that the way to identify the best voting procedure is to consider which scenarios should result in ties. If three voters have what researchers call cyclic preferences—one prefers A-B-C, one B-C-A, one C-A-B—there should be a tie, he says. Likewise, if two voters have exactly opposite preferences—one prefers A-B-C, say, and the other, C-B-A—their votes should cancel. The only common voting procedure that would give a tie to both of these cases is the Borda count, which gives two points to a voter's top choice and one point to his second choice in a three-candidate election.

Like the instant runoff, the Borda count gives weight to a voter's entire preference ranking. If the Borda count had been used, second-place votes would probably have tipped the 2000 race in Gore's favor, Saari and Brams say. And in France, it's highly unlikely that le Pen would have come in second, Saari says.

Saari has shown that the Borda count is much less prone to the kinds of paradoxes that Arrow studied than most other systems are. Using ideas from chaos theory, Saari has found, for instance, that plurality voting in a six-candidate election gives rise to 10⁵⁰ times as many paradoxical situations as the Borda count does.

Approve or disapprove

Not all the researchers are fans of the Borda count, however. Brams objects that it forces voters to rank all the candidates, even when there are some about whom they have no strong opinion, potentially leading to outcomes that don't really reflect voter preferences.

Brams prefers approval voting, in which people vote for as many candidates as they like. Approval voting, Brams says, gives voters more sovereignty by enabling them to express the intensity of their preferences: a voter who strongly favors one candidate can vote for just that candidate, while a voter who can't stand one candidate can vote for everyone else. A voter with more-moderate views can vote for any number of candidates between these two extremes.

It's hard to predict the outcome of an approval vote since voters' choices depend on where they draw the line between approval and disapproval. But Brams argues that approval voting would significantly alter voter behavior in many elections. In the 2000 presidential race, for instance, approval voting would have enabled Nader supporters to vote for him and also for one of the two stronger contenders.

While the instant runoff, Borda count, and approval voting each has drawbacks, most voting theorists would be happy to replace plurality voting with any one of them. "All methods that allow voters to express their views fully rather than to single out one candidate convey a much more nuanced message to the political machine," says Hannu Nurmi, a political scientist at the University of Turku in Finland.

The fact that U.S. elections have always been plurality votes is no reason to resist change, Tabarrok says. "We chose our voting systems before voting theory existed," he says. "I don't think any voting theorist would choose plurality rule today."

The real lesson to draw from recent election anomalies, voting theorists say, is that citizens should think carefully not just about how well the election machinery counts up the votes but also about how they want the votes to count.

And the winner is?

Different voting methods can produce very different results

In some elections, any candidate can win, depending on which voting system is used, says Donald Saari of the University of California, Irvine. Consider 15 people deciding what beverage to serve at a party. Six prefer milk first, wine second, and beer third; five prefer beer first, wine second, and milk third; and four prefer wine first, beer second, and milk third. In a plurality vote, milk is the clear winner. But if the group decides instead to hold a runoff election between the two top contenders—milk and beer—then beer wins, since nine people prefer it over milk. And if the group awards two points to a drink each time a voter ranks it first and one point each time a voter ranks it second, suddenly wine is the winner. Although this is a concocted example, it's not an anomaly, Saari insists.

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Letters:

All the alternative voting systems in this article have a common devil, complexity. How can anyone but a self-indulgent intellectual suggest that the answer to our problems is to further complicate a system that is already stressing the abilities of the full complement of voters? Please leave Instant Runoff and Borda counts where they belong—in dusty intellectual journals and as fascinating conversation at campus mixers. I'd suggest that it is both cheaper and wiser to encourage people who want to make obtuse political statements to "grow up" and make the compromises necessary to see that their ultimate preferences are reflected rather than frustrated by their actions. In short, take a little responsibility.

V. Kurt Bellman
Director of Elections
County of Berks, Pa.

Our plurality voting has one overwhelming advantage: It is simple enough that the majority of people can understand it. As is often said, our system was "designed by geniuses to be run by idiots."

Jerry Malone
Pueblo, Colo.

Voter-behavior and candidate-nomination strategies would be radically different under various systems of vote counting. Therefore, we can never know who would have won the 2000 election under a superior system because we can never know how voters would have voted in a different context. We can't even know who would have been on the ballot. Good voting systems should elicit good candidates and then reward honest voting. Voters should be motivated to vote for their true favorites rather than feeling pressured by a bad voting system into voting for the lesser of two evils.

Jeff Fisher
Vancouver, Wash.

I vote that the purpose of elections is for each person to move government toward his goals, not to elect Tweedledum instead of Tweedledee. Your article misses this point entirely. Rather than "spoil" the election, third parties force the two main parties (in U.S. politics) to pay some attention to the people rather than just each other, lest they lose too many votes to the third party closest to their ideology.

Bob Toxen
Duluth, Ga.

The article remarks that nothing can escape from Arrow's theorem and that everything has drawbacks, but it does not mention any drawbacks of approval voting.

I would like to point out two drawbacks and then a possible remedy, which I call bucket voting. I agree with Brams' objection to the Borda count: like the instant runoff, it forces voters to declare preferences they may not have. But approval voting has the flip side of that drawback: it denies voters the ability to declare any preferences they do have, among the candidates acceptable to them.

In the Bush-Gore-Nader example, very few of those who would have wanted to vote for both Gore and Nader under the approval scheme (because they strongly opposed Bush) would also have lacked any clear preference between Gore and Nader. Approval voting favors candidates who are well known and inoffensive: many voters recognize their names and have mildly positive opinions about them.

Politicians are already notorious for straddling, staging photo-ops, and so on. I believe the blandness bias in approval voting would encourage politicians to be even more reluctant to take a controversial stand and defend it than they already are.

Metaphorically, bucket voting considers votes to be small stones that can be put into various buckets. As a voter, I would enter the voting booth with a bucket of votes and would find an array of empty same-size pails (small buckets), one for each candidate.

I can distribute my votes among the pails however I like, but I cannot put more votes into a pail than it can hold.

Each bucket-voting scheme is specified by a pair of numbers: (bucket size, pail size). By choosing extreme values for these numbers, we can easily subsume approval and plurality voting under bucket voting. Other extreme values, along with a restriction on numbers of votes in different pails, yield the Borda count.

A pair of numbers like (6,3) yields a new scheme that seems better than any of the schemes discussed in the article. If I like 3 candidates and have definite preferences among them, then I can allocate my votes [3; 2; 1] to them.

A voter with tepid opinions on many candidates may prefer an allocation of the form [1; 1; 1; 1; 1; 1]. And so on. For moderate pairs like (6,3), the bucket metaphor can be supported by a simple drag-and-drop interface that is convenient for voters who are not voting theorists or computer geeks. I can see at a glance how I have allocated my 6 votes, correct any departure from my intentions, and then press a button to record my votes.

Problems with paper ballots and old voting machines are leading us toward putting some computing power in voting booths anyway.

Any pail size greater than 1 lets me express a few preferences without forcing me to do so. Keeping the pail size well below the bucket size provides some protection against a zealous minority whose members all want to empty their buckets into the pail of the same extremist candidate. With the pair (6,3), I can express extreme enthusiasm by filling one pail and leaving the others empty, but only if I discard 6-3 of my votes.

Barry K. Rosen
Stormville, N.Y.

The article is very interesting. I have a couple of other suggestions, however:

Recognize the fact that it costs many dollars per vote to win an election. (How much did Gray Davis spend per vote in the recent California election?) Very often the candidate who spends the most gets elected. So, make the rule that the candidate in whose name most money is contributed to the general fund is the winner. At some point—say thirty days before the election—ballots go on sale. Ballots would be printed by machines much like those that now sell state lottery tickets in many states—in fact it may be possible to just use those machines to print ballots with a bit of reprogramming. You would buy a ballot for president, another one for senator, another for congressman, one for mayor, and even one for dogcatcher if that is an elected office.

You would then mark or punch your ballot and fold and seal it. It would become a prepaid mailer to the election office. Ballots would be opened and counted on election day. You could spend as much as you wish for a ballot, with that amount indicated on the ballot.

The money collected would go to the general fund for the political entity the candidates are running to represent. (Federal, State, County, City, etc.) A candidate could raise funds to advertise for more "votes", as they do now, or he could spend those funds to vote for himself. The votes would be secret. No one would know who is ahead until polls close (last mail delivery). You could be required to prove citizenship when buying ballots.

Electronic voting machines or the Internet (charge it to my credit card) could be used, just so a method is used to keep the current tally secret until election time. Any of the methods in your article (plurality, instant runoff, or Borda count) could be used to determine the winner

The Borda Count method in your article would be better if a vote for first place were to count as one, a second place vote as one-half, a third place vote as one-third, etc. Now add up the counts and divide by the total number of voters. The results in your example would be: milk = .600, beer = .600, wine = .633. To make this more familiar to our baseball fans we could say milk and beer had a "batting average" of 600 and wine 633.

I have long advocated the Instant Runoff method. And it becomes more practical with intelligent voting machines that are coming into use to guide the not-so-bright voters to vote properly. (Even the Florida voters who can't understand a butterfly ballot should be able to use them.) Its main advantage would be to show the elected official where a lot of his/her constituents stand. I, for one, have been voting for a major party for forty years. I would have hardly ever put them first in an Instant Runoff election. Most often one of the minor parties would get my first, and, in a few cases, my second vote, even though the major party of my choice would ultimately get my vote.

Bob Koster
Santa Ana, Calif.

As a former third-party candidate and politically concerned citizen, I appreciate your coverage of potential improvements to our voting system. But I was taken aback by the objection of Steven Brams that the Borda Vote "forces voters to rank all the candidates, even when there are some about whom they have no strong opinion." There is no reason a voter in a 6-way Borda election could not select a first and second choice, then express indifference by allowing the next three to share third place, and reserve last place for a particularly objectionable candidate (awarding 5, 4, 2, 2, 2, and 0 points, respectively). In general, by allowing ties in rank, Brams' objection to Borda votes can be eliminated, without resorting to the far more confusing approval voting system.

Peter C. Everett
Sherborn, Mass.

I'm going to assign your article to my class next term. It is a very well done essay and I expect the students will appreciate the clarity of the exposition. In case the issue comes up again, could I bring your attention to this introduction to social choice theory that I wrote in 1998? This is a brief book (one of the so-called Sage "little green books") intended for upper level undergraduates, not quite so technically difficult as the things your article refers to in the math literature: Paul E. Johnson. Social Choice: Theory and Research. (Thousand Oaks: Sage, 1998). I would also refer readers to a symposium in the Journal of Economic Perspectives, in which the introductory essay downplays some of the worst "gloom and doom" statements of articles like Klarreich's. "When voter preferences are sufficiently similar, a variety of voting systems lead to similar choices, and these choices have desirable properties. The difficulties in aggregating preferences arise in the case of a population with a lack of consensus; this is the situation where the choice of electoral system can make the greatest difference and where apparently minor differences can directly influence the outcome."(p. 4) Those authors tested a variety of voting procedures and found "surprisingly little difference between winners"(p. 18). When there are differences, there is not usually a powerful argument pointing decisively to one system. When society is not in some sort of consensus, voting procedures cannot manufacture one.

Paul E. Johnson
University of Kansas
Lawrence, Kan.

Erica Klarreich's article on how differing voting systems evaluate the intentions of voters was timely. I believe a modified Borda count, in which voters are allotted a full point for their highest-ranked candidate, a half-point for the next-highest, quarter-point for the next, and so on for as many candidates as they wish to rank, would answer most of the objections she cited.

John Huie
Athens, Ga.

Regarding the article, as US voters know (or are at least reminded every four years) the US presidential election is not truly a plurality vote, at least on a national basis. Have the voting theorists looked at how the Electoral College in US presidential elections affects the results?

Richard Wilson
Sunnyvale, Calif.

Thank goodness the election process was not designed by people like Alexander Tabarrok, who feels pure mathematics should outweigh common sense. Plurality voting separates people on both sides of the fence from those sitting on it, so that decision makers can be elected. Can anyone truly believe that this nation or a business could be run successfully by people who can't make a decision or want merely to make a protest statement?

Nowadays, there is enough pre-election data available to determine the front-runners, so why should these indecisive people be allowed to vote again in a run off election. Why should the train be returned to the station to pick up these wafflers; let them take the next train.

Doug Snyder

To Quote "Boss" Tweed: "As long as I count the votes, what are you going to do about it?"

Ben F. Arnold
Springville, Calif.

The author appears to be attempting to find ways for Gore to have won the 2000 presidential election. That is history, and we really do need to get on with life and resolve our current problems with voting.

I do not believe, however, that using a Runoff or Borda Count procedure as described in the article has that much of an advantage over a Plurality. It is interesting that, using your example of the milk, beer, and wine, if a value is assigned to each place holder (i.e. three for first place, two for second place and one for third place) instead of giving credit only for a first and second place, beer comes out the winner two out of three times. Even the Olympics knows you need at least a third place winner.

A recent contest in our state is a fine example. With four candidates for governor, and three of them viable in my opinion, the choice was difficult. Had I been able to rank the candidates as first, second, third and fourth choices, the process would have been simple. Arrow's theorem wouldn't have helped, because the candidate I really wanted to vote for, like Nader, probably didn't have a chance of winning as a first choice, so I ended up voting for my third choice. If each of the candidates was accorded a value based on their placement, then my candidate of choice would probably have won because follow-up polls indicated that a reasonable number would have ranked him as first choice, and a fairly large number would have ranked him as second. Unfortunately, two of the other candidates (one of whom was ultimately the winner) would have probably gotten more first-choice votes, so my candidate of choice would have been eliminated with Arrow's theorem. In our complex society, two choices simply are not sufficient.

A fairly simple algorithm could easily be constructed for allowing ranking of candidates. It would not be unreasonable to have a cutoff point, but it needs to be greater than two candidates. Four seems to be a very good number. Surely our brilliant mathematicians can come up with a voting system that services our needs a lot better than those proposed in the article.

Dixie Luoma
New Brighton, Minn.

Interesting article, especially the links to chaos theory. However, let's not confuse the mathematical aspects of voting with the motives of the people who vote. The obvious motive is to choose (elect) something, but another motive may be to send a message to the people who are believed to set the ballot. For example, it was widely noted before the 2000 presidential election that a vote for Nader (a 'no chance' candidate with appeal to potential Gore voters) might redound to Bush's benefit. Consequently a vote for Nader could be interpreted as a vote for Nader, but also as a message to the Democrat party to select, in future, a different Democrat.

Similarly, the recent gubernatorial election here in California offered, as the two major party candidates, choices that lacked appeal for many potential voters. One result was a low turnout; another result may have been 'message' votes cast for third party candidates.

Such messages can be sent—whether by abstentions or third party votes—under any of the voting systems discussed in the article, but it must be realized that there is more to a voting system than the math.

Ideally, the voting system should be such as to encourage people to vote in a thoughtful manner. Perhaps the greatest weakness—and strength—of the plurality system is that it encourages a two party political system.

Larry Levin
Juniper Hills, Calif.

I foresee no chance that our system of voting will be changed for the better any time soon—or late, for that matter. Plurality voting is too simple and (unfortunately) common-sensical to be abandoned, whatever its faults. Said the ever-perceptive Alexis de Tocqueville in 1835, “A proposition must be plain, to be adapted to the understanding of a people. A false notion that is clear and precise will always have more power in the world than a true principle which is obscure or involved.”

Jana Hollingsworth
Seattle, Wash.

The instant-runoff example presented in the paragraph "Chaos in the Polling Place" purportedly shows a fickle unpredictable outcome in a three-way race. But that is because only one half of the possible outcomes were given. Outcomes A-C-B, B-A-C and C-B-A were omitted. When these three are also included and assigned votes (not percentages, for simplicity) of 34, 33 and 33 respectively, the results are far from chaotic.

For this scenario A now captures 69 (35+34) while B garners 66 (33+33) and C receives 65 (32+33) and is eliminated as before. In converting the second choice votes from those that chose C first, A receives 32 while B gets 33. Now A wins as before but with 101 to 99 for B—definitely not a landslide of 67% to 33%.

Presently voters favoring one of the two major parties dare not vote for a third party candidate for fear of benefiting their rival party. But the instant-runoff has the potential to break this major-party-only stranglehold because any vote for a third party that is not successful for that candidate will be converted to their major-party choice. Even better, apathetic non-voters will have good reason to return to the polls.

This voting method could be submitted directly to the electorate through the initiative process in about 30 states. This may be necessary as partisan state legislators are unlikely to proffer acts that would diminish the power of their parties.

Russell Corzine
Onia, Ariz.

I thought your article on voting math was pretty simplistic. It ignored the fact that we have primary elections in the US that allow us to settle out a field of candidates. This winnowing is done by members of parties who have an interest in politics, not just someone who happens to show up. This would seem to go a long way to answering concerns about "plurality" being too narrow a system.

Also, it is claimed that during the 2000 Presidential election "...for most Nader voters, Gore was the second choice". This sidesteps that most Nader voters also said they would not have voted or encouraged others to vote if he dropped out.

Lastly, the schemes mentioned are favorites of the voting theorists but all share the flaw of being too complex to explain easily. Our system many not be perfect but it is clear and has done a good job for us.

Ken Schleede
Webster, N.Y.

I work as an election officer in Kansas. I was recently given the article in your magazine. As I respect the information in your article it does not raise the biggest issue we as election administrators deal with every election. You must first get people to get out and vote before any of this is even an issue. If you are only getting 50% of the voters to vote you are not getting the majority vote no matter what you do. With funds being a bigger problem, the theories discussed seem to be very improbable.

Jolene Campbell
Manhattan, Kans.

Milk as an evil: Regarding the article, a commonly heard remark by voters is "I voted for the lesser of two evils." Thus I would suggest a variation on the "Instant-runoff" voting algorithm. Instead of casting out the contender with the fewest top-choice votes, I would cast out the one with the most bottom-choice votes, that is, remove the most objectionable choice. In the example given in the sidebar, milk draws the most third-choice votes (9), and is thus cast out. When re-tallying with milk eliminated, wine becomes the clear winner.

In this example, the result is the same as the Borda Count method, but it is not always so. I prefer this method because (a) it eliminates the arbitrary weighting (2, 1, and 0 in the example) of the Borda Count, and as I indicated in my opening, it minimizes "evil" in the perception of the voters. It's often more important to avoid an evil than to select the very best among several good choices.

I confess though, if I were voting in this election, wine would be my choice, so maybe this preference colors my judgment of the best voting method.

Alan W. Harris
La Canada, Calif.

In an article, a quote appeared from Donald Saari of the University of California, Irvine: "We live in a democracy, and anyone should be able to run for any office." Could the author or editors of this fine publication please get back in contact with Donald Saari and please let this obviously misinformed individual know that we are most assuredly NOT a "Democracy"? We are a Republic, and a democracy was something that our forefathers fought whole-heartedly against. Why is it that the majority of the public seems to forget this?

Frank Moore
Atlanta, Ga.

The article gives some examples of paradoxical behavior of the Instant Runoff voting system. How about a slight twist on the Instant Runoff system... Normally, for each round, the Instant Runoff system eliminates the candidate with the least number of 1st place votes.

I suggest that, for each round, you eliminate the candidate with the most number of last place votes. This slight modification seems to solve every example of paradoxical behavior that the article brings up.

William Robinson

The article states, "George W. Bush won the state of Florida ... by just a few hundred votes over Al Gore...." Not so. George Bush won exactly five of the only nine votes that, in the final analysis, counted.

Dale Hurliman
Beavertown, Pa.

Thanks for a timely and informative article about a subject I have studied for some years. I think it would have been better to have clearly pointed out that while a non-plurality vote would likely have resulted in a different outcome in Florida, without examining the detailed votes of all the other states, the final result must remain unknown. There's also the little matter of our electoral college....

Billy Harvey

I used to play politics in a town that had two political sides. The way to win city council elections was to talk many of your own people out of running. There was no primary and whoever had the most candidates lost.

Dan Tillmanns
Ridgecrest, Calif.

Erika wrote a very good article but failed to mention our unique method of selecting presidents. No mention of our Electoral College and the reasons behind this institution. No comparisons between the states that have a winner take all versus states that don't. People select electors and not necessarily presidents. Some States require electors to vote for their chosen candidates and some don't.

Perhaps it is beyond the scope of the article, but I would have enjoyed a more philosophical argument over how we conduct presidential elections.

Free and fair elections? Not in this country when we have the two parties who control the ballot box and refuse to debate the other candidates.

Study the Minnesota election for governor during their last election. A third candidate who was included in the debates and went from single digits in the polls before the election to winning the election. Voter turnout in Minnesota also increased because of his inclusion.

I believe the French election outcome would have been different if le Pen was included in the debates. The Florida mess most likely never would have happened if all candidates debated.

The bottom line is I believe that one method is a good as the other, but what needs to be addressed is how the media distorts elections. Specifically how censorship by omission denies voters information to make an informed choice. To me that is the real problem with our elections.

David L. Lee
Normal, Ill.

What would happen if one were to combine the Borda count with approval voting? Voters could pick as many candidates as they liked to rank, or perhaps it would be better if candidates could pick a set number of candidates to rank. That way, no one would be forced to rank a candidate that they did not feel strongly about, while at the same time giving support to all the candidates they favor.

Meredith Killian

Thank you for the interesting article about those 3 voting systems. Most agree that Instant Runoff is at least a little better than Plurality, but there are important ways in which Borda is worse than Plurality. For instance, Borda is the only proposed method that I know of that can fail to elect a candidate who is the voted favorite of a majority. That sort of Borda behavior leads to horrendous strategy problems & dilemmas, such as co-operate/defect dilemmas where even Plurality wouldn't have them.

Borda is great if all anyone wants to do is vote sincerely in order to do his/her part to maximize social utility—if no one is interested in getting the best result for themselves that they can. Experience shows that our electorates aren't of that type. Even Mr. de Borda himself said, "My method is intended for honest men." Not suitable for public political elections.

Not only can someone win in Instant Runoff because some voters decide to rank him/her lower, but it gets better than that: It's been shown by Brams that it's possible for some voters to make a winning candidate lose by moving him from last place to 1st place.

A scarier version of that is one in which a candidate is going to lose, but then his corruption is discovered, and some voters move him from 1st place to last place, thereby causing him to win.

Here's another criterion that Instant Runoff (IRV) fails, but which Borda, Approval, & Plurality pass: Participation: Adding to the count 1 or more ballots that vote X over Y shouldn't change the winner from X to Y.

But maybe IRV's worst problem is that, like Plurality, it's always looking only at 1st choices. Would you make an important & irreversible decision based only on a fraction of the available information? One result of that is that, when IRV eliminates your 2nd choice because your traveling vote hasn't yet reached it, IRV isn't counting your preference for 2nd choice over last choice. IRVists like IRV because they can express all their pairwise preferences; but IRV may or may not count them. At least Approval reliably counts every preference that you consider important enough to be one of those that you vote. With Approval, the voter is the one who decides which of his pairwise preferences will be counted. IRV would decide that for you, often to the detriment of your outcome.

With Approval, no one ever has strategic incentive to bury his/her favorite by voting someone else over him/her. That can't be said for Instant Runoff or Borda.

Need for favorite-burial easily arises in IRV, when your favorite is big enough to eliminate your needed compromise, without being big enough to win.

If people want a rank-method, there are much better ones than Instant-Runoff or Borda, for public political elections.

Mike Ossipoff

In the article, your statement that "...Republican George W. Bush won the state of Florida—and, consequently, the presidency..." is by no means universally accepted as fact. Indeed, the circumstances surrounding that debacle strongly suggest that George W. was effectively appointed by the Supreme Court.

That aside, of course there are voting systems in which the will of the electorate could be more fairly expressed—so what?

The examples you cite merely bear witness to the fact that true will of the electorate is the very last thing that voting in the real world is about.

Rich Falzone
Houston, Tex.

I really enjoyed Erica Klarreich's article, though I believe it could use two slight additions. Another "spoiler" of note in the Florida election was the highly inaccurate voter purge list used by the governor's office. This list effectively disenfranchised thousands of minority voters in Florida, whose votes could have easily outweighed Bush's lead. For more information on this topic, you can contact the NAACP, which has settled a civil rights suit against the state of Florida over this purge list. Also pertinent is the fact that the Green party strongly advocates the use of the instant runoff voting system described in the article.

Sam Douglass

Erica Klarreich raises some interesting issues in her article, but what’s missing from her analysis (and that of her sources) is what happens after the election, namely forming a government. This is where instant runoff, the Borda count, approval voting, and other proportional representation systems potentially can distort the will of the voters more than a plurality system.

As Terry Bouricius is quoted in Klarreich’s article, these systems require coalition building, and that’s where the problems begin. In the last Austrian election, for example, the Social Democrats won just over 33% of the vote, the Freedom Party and the People’s Party each just under 27%. As a result of the coalition-building that followed the election, the third- place People’s Party ended up leading a government with the Freedom Party, completely excluding the Social Democratic Party which had actually beaten them both. In post-war Germany, the small Free Democratic Party, which has never won more than 13% of the vote and rarely more than 10%, has been in government more than either major party, and has usually controlled more cabinet ministries than its vote share would imply. In both cases, small parties—and their supporters!—have had their power magnified, not diminished, by the election laws.

In short, while plurality systems may distort results before the election and during vote counting, instant runoff and other system may distort the results after the election when government coalitions are formed.

Which is the fairer system? There is still no easy answer. Considering the effects of elections laws on how governments are formed may only make answering that question harder, but it needs to be examined.

Sean Blinn
Bedminster, N.J.

"The real lesson to draw from recent election anomalies, voting theorists say, is that citizens should think carefully not just about how well the election machinery counts up the votes but also about how they want the votes to count." I believe the Florida Election controversy presents a strong case for the plurality voting system. In our two-party system, third parties often represent a dissonance within one of the two major parties. For many would be Gore voters, Nader offered them a means to express their displeasure with the priorities of the Democratic Party. The fact that Gore lost the election makes their message that much louder.

This is not say that I wouldn't support a new voting system. To the contrary, I think it would eventually allow for more parties to have a louder voice within our system. I just wanted to express that in plurality elections, the mathematical deficiencies of the system allow for unique forms of expression.

Wilkes Joiner

The article reminds me of Winston Churchill's famous quote, "Democracy is the worst form of government in the entire world...except for all the rest." The deficiencies of plurality voting remind me of Joseph Stalin's, "It is not the people who vote that count, but rather the people who count the vote."

Mark W. Randolph

I really, really, really wish you had considered writing about Condorcet voting in your recent article about voting techniques. It requires the same ranking-of-candidates that IRV does, but doesn't have the same mathematical flaws that you detail.

Condorcet voting analyzes the one-on-one matchups of all the candidates by looking at the rankings. Here is a good example that I came up with. Say that you have an election with a right-wing extremist, a left-wing extremist, and a centrist:

25 million people vote for the centrist

35 million people vote for the right wing, then the centrist

40 million people vote for the left wing, then the centrist

Doesn't it seem the centrist should win? In Condorcet voting, the centrist wins. 65 million people prefer him to the right wing, 60 million prefer him to the left wing. In IRV, the centrist is eliminated first, and the left-wing candidate wins with less than half of the votes compared to the number of people voting.

There are other simple scenarios that just as easily demonstrate Condorcet's superiority over Approval and Borda.

Curt Siffert
Portland, Ore.

All of the alternatives discussed in your article about "voting theory" fail to consider another choice. If you include a choice "none of the above," this does not become just another parameter in the equation. It changes the equation. Should such a choice be in the plurality, "all of the above" would have to go! It would also give those of us who don't like the choices presented a reason to vote. As it now stands, we must make our choice, not based upon the "best person" but based upon the "least damaging person."

I have voted in every election since 1953. Most of the time my vote has gone to "the lesser of two (or more) evils." I would love to have the choice of "none of the above."

David A. Mathewes
Canton, N.C.

The article discusses the interesting paradoxes possible with voting procedures as currently practiced in most elections. However, contrary to the schemes discussed in the article, how about a "perfect" voting procedure that does not have counterintuitive outcomes: an approval rating system, such as what judges use in the Olympics? Imagine a system in which a voter rates each candidate with a real number anywhere between -100 to 100, with a negative value indicating a voter's dislike for a candidate. No matter how many candidates enter a race, a ranking is obtained by ordering the averaged rating of the candidates. Interesting advantages abound, such as the negative value allows the possibility that a winning candidate could have an approval rating less than zero, indicating a particularly troubled electorate.

Mark A. Lamb
Glendora, Calif.

For the sake of argument, let us consider the mentioned Florida election. I think it likely that with the substantial philosophical differences between President Bush and Vice President Gore that we could safely say that in a run-off type of election that Ralph Nader would have been everyone's second choice other than those who voted directly for candidate Nader. This would have presented the absurd case where almost all of the voters preferred Ralph Nader if their own first choices weren't to win.

From the article, I infer that the author thought that Gore should somehow come out the winner even though counting first and second choices would reveal a substantial margin for Nader. The gist of this is that the case in Florida demonstrates that still further rules would have to be incorporated and this is simply a further means where corruption and bias can seep into the system, which is difficult at best to maintain.

What is being suggested is that the candidate least objectionable to the greatest number of people should be the winner instead of the plurality winner—that person who is the choice of the greatest number of people. Could our founding fathers have been so stupid?

Thomas Kunich

Arrow's Impossibility Theorem is certainly one "quiver" in shooting down the concept of an ideal democratic process of voting. Another is an extrapolation of Heisenberg's Uncertainty Principle applied to measuring the wills of a large populace.

Heisenberg's uncertainty principle limits how accurate one can measure states in the microscopic. The result is that we must live with the fuzzy model of electron shells as opposed to the orbital model.

But Heisenberg's principle applies equally in the macroscopic scale. Counting enormously large quantities is just another example of measuring. Voting is a measure of the citizens' preferences, after all. When there are 50 people in a room, it's simple to count hands and determine who chose coffee over tea. However, when there are 50,000,000 people, counting will run into difficulties as imposed by the principle. We saw exactly those effects in the Florida voting debacle, and the subsequent recounts.

The solution isn't to impose recounts when two or more candidates are nearly tied: the principle will still impose fuzziness in the counting. The solution might be to reformulate the measurement, by conducting a run-off election among those who are within the margin of error of each other. This solution doesn't sidestep the principle, but is basically a different measurement. If one can't get the most accurate measurement one way, the solution isn't to pull that tape measure out again and again. The solution may be to measure a different quantity instead. Thus, using alternative voting methods, as suggested in the article, may be one way to deal with another fundamental problem with voting.

Eric Dew

This research ignores the fact that many (most?) votes are not FOR, they are AGAINST! How would "Plurality" be affected if voters were allowed to cast their vote against a candidate? Certainly "Runoff" would immediately eliminate possible votes for other candidates.

In the recent California election, being able to vote against the two major candidates for Governor would probably have let one of the minor candidates win.

John Kopf

The article did not mention incentives to distort the revelation of voter preferences in an election. It ignored non-IRV preferential voting methods. And it misevaluated some problems.

Approval systems cannot express voters' beliefs unless they are used in a deceitful fashion: treating acceptable candidates as unacceptable, and so creating a situation like that of plurality voting. Borda count systems can report non-count information so as to reveal voter preferences, but can be manipulated through the mere addition of even a voteless candidate. Plurality voting creates an incentive to cheat in a generally shortsighted attempt to keep a more undesirable but still acceptable candidate out of office, by voting for the "lesser of two evils"; this has had cascading effects on the quality of elections in many respects, and has helped transform the word "debate" so as to cover non-debate, joint press conferences.

There are other forms of preferential voting. Proxy or contingent proxy voting, akin to the Electoral College, can provide negotiated improvements for a loser's supporters, whether or not the proxy's vote precedes the general election. Limited, IRV-like preferential voting, where only, say, one two-candidate preference is revealed by each voter, would likely suffice in American elections.

In the paradoxical situation posited for IRV, where a candidate loses by attracting more votes, it does not matter much who wins, since the victors in either case have substantial support, and it is more important simply to have a decision. But this paradox only comes about by convincing voters less likely to share a candidate's beliefs, who hold the convincer at a lower preference than a third competitor; this is an ineffective short-term strategy even without the paradox, where winning is the goal, since it generally takes more effort than it would to convince more sympathetic voters.

Of course, there are grave problems with American elections apart from, if not due to, their general use of plurality voting: restrictions on campaign donations conferring significant advantages on plutocrats and on large, standing organizations; severe 3rd-party-and-independent ballot access restrictions still remaining in some states; corrupt infotainment organizations allowed by corrupt judges to make de facto corporate contributions in their role as newsmakers (e.g. deciding on debate participants); and bribery of voters through distorted information: the concealment of taxes effectively exacted from them and then used as if for charity.

David Hudson

After reading the article, I felt I had to comment. The authors state we should not resist change just because all US elections have been plurality votes. Perhaps they have been but if they've worked successfully that way for over 200 years, why change? These voting theorists advocate going to another, in their view, better system. Considering the difficulty some states seem to have just tabulating votes in a plurality system, do they really expect going to a more complex system is going to improve anything? If anything, it just adds more potential for error. A lot of the basis for their argument to switch revolves around the result of the 2000 Presidential Election, seeming to advance the belief that Al Gore was the winner of that election and switching to one of the preferred systems would have prevented the actual result. Regardless of your political views, and who you believe should have won, I don't think anyone can argue that just having a plurality system resulted in the outcome. It was due to having an electoral college in a plurality system. To achieve the result they seem to imply the voters wanted would be as "simple" as eliminating the Electoral College in favor of determination by purely popular vote. And, just because something theoretically works better than the current system, or because the current system was put in place before voting theorists existed, doesn't necessitate abandoning the current system. To use a bit of an analogy, I believe everyone would agree that Einsteinian physics is superior to Newtonian physics but when we send probes to the planets we use Newtonian physics because it's not as complex and works just fine. I believe the same holds true for plurality voting.

John A. Ferko
Minot, N.D.

Your article is really skewed up. This is why our nation has national primaries for Pete's sake! Just because the third, fourth or fifth party candidate is on a nationwide ballot doesn't mean they'd be an automatic second choice of any voter. If you really wanted a true "election selection" then include the top two candidates of the Republicans and Democrats on the final ballot for a better runoff comparison. The second or third place finisher in one of the two major political parties would be a much more viable choice by millions of voters over most of the "third party" candidates on the final ballot.

VP Gore lost in his own home state. That's why the 2000 election night results went against him.

Brian Adams
Springfield, Mo.

I see a least two problems with the proposed modified voting schemes. The main problem in Florida in 2000 was the incompetence of the voters. Problems would be increased by more complicated schemes. Problem two is that the other methods will result in picking the candidate that is disliked the least. Candidates will then strive for the mediocre middle. We will have the choice between Tweedledum and Tweedledee.

Roger Bowers
New Hartford, N.Y.

References:

Brams, S.J., and D.R. Herschbach. 2001. The science of elections. Science 292(May 25):1449. Available at http://www.sciencemag.org/cgi/content/summary/292/5521/1449.

Malkevitch, J. 2002. Voting and elections. American Mathematical Society (March). Available at http://www.ams.org/new-in-math/cover/voting-introduction.html.

Tabarrok, A., and L. Spector. 1999. Would the Borda count have avoided the Civil War? Journal of Theoretical Politics 11(April):261-288. Abstract available at http://www.sagepub.co.uk/frame.html?http://www.sagepub.co.uk/
journals/details/issue/abstract/ab007632.html.

Taylor, A.D. 2002. The manipulability of voting systems. American Mathematical Monthly 109(April):321-337.

Further Readings:

Fishburn, P., and S. Brams. 1983. Paradoxes of preferential voting. Mathematical Magazine 56(September):207-214.

Peterson, I. 1998. How to fix an election. Science News Online (Oct. 31). Available at http://www.sciencenews.org/sn_arc98/10_31_98/mathland.htm.

Saari, D. 2001. Decisions and Elections: Explaining the Unexpected. Cambridge, England: Cambridge University Press. See http://books.cambridge.org/0521004047.htm.

______. 2000. Chaotic Elections! A Mathematician Looks at Voting. Providence, R.I.: American Mathematical Society. See http://www.ams.org/bookstore/pspdf/elect.pdf.

______. 1999. Chaos, but in voting and apportionments? Proceedings of the National Academy of Sciences 96(Sept. 14):10568-10571. Available at http://www.pnas.org/cgi/content/full/96/19/10568.

______. 1995. A chaotic explanation of aggregation paradoxes. SIAM Review 37(March):37-52.

Saari, D.G., and F. Valognes. 1998. Geometry, voting, and paradoxes. Mathematics Magazine 71(October):243-259.

Sources:

Terry Bouricius
Center for Voting and Democracy
6930 Carroll Avenue, Suite 610
Takoma Park, MD 20912
Web site: http://www.fairvote.org/

Steven J. Brams
Department of Politics
New York University
715 Broadway, Fourth Floor
New York, NY 10003-6806
Web site: http://www.nyu.edu/gsas/dept/politics/
faculty/brams/brams_home.html

Hannu Nurmi
Department of Political Science
University of Turku
FIN-20014 Turku
Finland
Web site: http://www.utu.fi/yht/valtio-oppi/hallinto/hannu2.html

Donald G. Saari
Department of Mathematics
University of California, Irvine
Irvine, CA 92697-3875
Web site: http://www.math.uci.edu/faculty/dsaari.html

Alexander Tabarrok
The Independent Institute
100 Swan Way
Oakland, CA 94621-1428
Web site: http://www.independent.org/index.html