Tap writes:

"Alternative Possibilities for Voting Procedure"

Scientific American

"Has there been any progress in developing fairer ways for people to vote in elections? I recall reading some time back about a system in which people would get one vote per candidate, not transferable between candidates; such a system was said to be fairer overall than one vote per voter." — Vitols, Anaheim Hills, Calif.

Donald G. Saari from the department of mathematics at Northwestern University in Evanston, Ill., gives this overview of voting behavior:

"After two centuries of efforts by mathematicians and political scientists, positive results about 'fair voting procedures' are emerging. This is important because 'fairness' can be a casualty when current methods are used in multiple-candidate elections--such as this year's presidential campaign.

To illustrate, suppose that 200 voters prefer Alice to Candy to Becky (denoted by Alice > Candy > Becky), 195 prefer Becky > Candy > Alice, whereas only 20 prefer Candy > Becky > Alice. The plurality election outcome, where we vote for our top-ranked candidate, is Alice > Becky > Candy with a 200:195:20 tally. While we might worry whether these voters prefer Alice or Becky, Candy's feeble support suggests that she is of no interest to these voters.

"This assertion, however, is false. If we compare candidates in pairs, it becomes arguable that Candy is their favorite. These voters prefer Candy to Alice (215 to 200), Candy to Becky (220 to 195), and Becky to Alice (215 to 200); these rankings suggest that these voters actually prefer Candy > Becky > Alice. Notice how this outcome conflicts with and reverses the plurality ranking. Moreover, it shows that Candy's lack of votes more accurately manifests inadequacies of our commonly used election procedure rather than voter disinterest. The example also shows that, inadvertently, we can choose badly.To explain the flaw of the plurality vote, imagine parental reaction should a local school suddenly decide to rank students strictly by the number of A's they receive. The unfairness of this method is obvious; it ranks a student with 3 A's and failures in all other classes above the student with 2 A's and the rest B's. Not only is this flawed procedure equivalent to the plurality vote (by using only top-ranked choices), but it also explains the above example. This is because, like the student with many B's, Candy has the enviable status of being either top- or second-ranked by all voters. Alice, on the other hand, is top-ranked by some voters but viewed a failure by over half of them. If common sense forces us to reject this plurality system when ranking students, why should we embrace this flawed approach when making critical decisions or choosing leaders?

Our widely used plurality method should be replaced — but with what? While some improvement is attained with a runoff election, it has problems. In the example, for instance, Becky wins the runoff (between Alice and Becky), so this procedure avoids selecting Alice. Candy, who appears to be the voters' favorite, is dropped at the first stage, however. Let us consider another method, in which voters have one nontransferable vote per candidate--they vote either yes or no for each candidate. Rather than being fair, this procedure admits almost random conclusions.

In our sample three-person race, for instance, an election could easily produce any ranking of the candidates. Just by varying choices about which voters also vote for their second-ranked candidate, we can obtain either of the two earlier outcomes, or one in which Becky wins and Alice and Candy are tied, or one in which all candidates are tied, or... The source of this chaotic situation is easy to understand by changing the grading analogy to one in which students are ranked according to the number of A's, or the number of A's and B's, or the number of A's, B's and C's, or... In these rules, there is no distinction among grades (so, an A can be judged equivalent to a C or D), and the choice of what grades are included in the count can vary from course to course. Clearly, we must anticipate indecisive chaotic outcomes.

As this one example illustrates, determining 'fair' voting procedures is a complex issue. The problem is that whereas philosophical arguments promoting certain procedures may sound convincing, subtle higher-dimensional mathematical aspects can reveal a host of unexpected complications. Can the system be manipulated? What happens if new candidates enter (such as Pat Buchanan in the primaries or Ross Perot during the fall campaign) or drop out of contention (such as Steven Forbes during the primaries)? Which method most accurately captures the voters' true intentions? Surprisingly, mathematics is proving that only one method satisfies most of these criteria: the Borda Count, named after its founder, the 18-century French mathematician Jean-Charles de Borda.

Although the supporting mathematics is complex, the Borda Count admits an intuitive description. The idea mimicks how we rank students when in each course a student is assigned four points for an A, three for a B, and down to zero for a F. Similarly, for three candidates, the Borda Count assigns two and one points, respectively, to a voter's top- and second-ranked candidate. The Borda Count makes a difference as shown by the example whereby it yields the acceptable Candy > Becky > Alice election ranking, with a 435:410:400 tally.

To see how the Borda Count captures the 'one man, one vote' philosophy, notice that a voter having the preferences Alice > Becky > Candy would vote for Alice in both the [Alice vs. Becky] and the [Alice vs. Candy] pairwise elections and for Becky in the [Becky vs. Candy] election. In these elections, then, he gives two points to Alice and one to Becky--point totals that agree with the Borda Count tally of his ballot. In other words, the Borda Count is the natural extension of what we do in a two-person race; it is an efficient way to compute each voter's 'one man, one vote' actions over pairs of candidates. Indeed, both in pragmatic and theoretical senses, it can be (and has been) proved that only the Borda Count appears to achieve the elusive 'fairness' goal for elections.

More information can be found in the book Basic Geometry of Voting by D. G. Saari. Springer-Verlag, New York, 1995.

"The mathematically more sophisticated reader, who would like to explore the connections of voting with dynamical chaos and statistical paradoxes, might want to read the article 'A Chaotic Exploration of Aggregation Paradoxes,' by D. G. Saari in SIAM Review, Vol. 37, No. 1; March 1995."

Lawrence Ford, the chair of the mathematics department at Idaho State University, notes that the goal of fairness is more elusive than it may appear:
"There has been lots of activity over the past 30 to 40 years on mathematical applications in political science. One problem that has been studied extensively seems similar to the problem you describe: Given a set of candidates (at least 3) and a number of voters, select the 'fairest' winner.

"Many solutions have been proposed, but all were shown to exhibit flaws in some cases. Then, in the early 1960s, Kenneth Arrow proved his famous Impossibility Theorem, which essentially states that no system can exist without these flaws, that is, no perfect voting system exists. So 'fairer' is a subjective measure; any voting system can be made to look 'unfair' under the right set of circumstances.

Voting systems such as the one you describe are probably very susceptible to 'strategic voting,' wherein participants can influence the outcome by insincere voting. A similar system that is not susceptible to strategic voting and that is gaining a good reputation as a relatively fair system is 'approval voting.' In it, a voter votes for all candidates he or she approves of. The candidate with the most votes wins. In it, the candidate with the broadest approval base wins (centrists, not extremists, get elected — which is often not the case in a one person/one vote election). One big flaw here is that most voters are fairly positive of their favorites and fairly positive of those they hate, but wishy-washy in the middle. If they choose randomly for or against approval in that middle range, the whole election can become random."

Sam Merrill in the department of mathematics and computer science at Wilkes University in Wilkes-Barre, Penn., provides more details:

The method of voting in multicandidate elections referred to in the question is, I believe, approval voting. Under this procedure, each voter can vote for one, two, three or any number of candidates. In effect, the voter considers each candidate one by one and casts a vote for those whom she (or he) approves but not for those whom she does not approve. The winner is the one approved by the most voters.

Approval voting is now used by several national and international professional societies in their election of officers. These include the Institute of Electrical and Electronics Engineers, the American Statistical Association, the Mathematical Association of America and the Institute for Management Science. Some of the approval-voting elections in these societies have led to winners who would not have won under single-vote plurality (the more common method used in the U.S.) or have enhanced the standing in the voting outcome of candidates who had broad support throughout the profession.

Generally speaking, however, the candidate who has won under approval voting did well among bullet voters (those who voted for only one candidate), as well as among those who voted for more than one candidate. Further details can be found in Steven Brams and Peter Fishburn, 'Approval Voting in Scientific and Engineering Societies, in Group Decisions and Negotiations, Vol. 1, No. 1, pages 41-55; April 1992, and Steven Brams and Jack Nagel, 'Approval Voting in Practice,' in Public Choice, Vol. 71, Nos. 1/2, pages 1-17; August 1991.

Although bills have been introduced in New Hampshire, New York and North Dakota to institute approval voting in public elections, I am not aware of any recent progress on that front.

In a recent study [Steven J. Brams and Samuel Merrill III, 'Would Ross Perot Have Won the 1992 Presidential Election under Approval Voting?' in PS: Political Science and Politics, Vol. 27, No. 1, pages 39-44; March, 1994], we projected the likely outcome had approval voting been used in the three-cornered 1992 presidential election. Based on polling data from the American National Election Study, which asks voters to score each candidate on a scale from 0 to 100, we projected approval-voting totals using three different assumptions. The results consistently indicate that Bill Clinton would have received approvals from about 55 percent, George Bush from about 50 percent, and Ross Perot from about 40 percent. Hence, Clinton would have attained majority support — enhancing his mandate--but his margin over Bush would have been unchanged.

Perot would have doubled the 19 percent he received under plurality because some Bush and especially Clinton supporters also approved Perot but opted to cast their vote for one of the main contenders. Under approval voting, Perot's total would therefore have more accurately reflected his true support. Yet his chances of actually winning might have been reduced, because voters deciding to cast a vote for Perot would not have had to switch away from Clinton or Bush but could support both. In light of the high disapproval rate that Perot faced, he would have been hard-pressed to climb past Clinton's 55 percent approval.

"It has been argued that approval voting benefits centrist candidates, who tend to have a broader appeal, while not denying voters the opportunity to express support for more extreme candidates, and that it should reduce infighting between like-minded candidates because they need to share support from the same voters. Approval voting could be implemented on existing voting machines, and it could be done without a Constitutional amendment."