A referendum on an electoral system called the single transferable vote will take place in British Columbia on May 17. Before making up their minds, the citizens of this province might want to know what mathematicians think about various voting methods.
The current procedure asks voters to choose one out of several candidates, whereas the STV model requires the ranking of candidates in the voter's order of preference. Which system serves democracy better?
The beginnings of voting are lost in history. Written sources attest to the existence of various election methods in antiquity and through the Middle Ages. Confusion in choosing the right system was common. In 1130, for instance, the ambiguity of the procedure led to the election of two popes, an event that created a rift within the Catholic Church.
In 1770, French mathematician Jean-Charles Borda proposed a new rule, an ancestor of STV. He asked that voters rank the candidates and that points be accordingly assigned. In a three-candidate election, for example, the top-ranked on a ballot received three points, the second two and the third one. The candidate with the most points won.
Various point rules can be used: six for first place, five for second and zero for third; or 10 for first, two for second and one for third. Our current system, in fact, gives one point for first place and zero for the rest. But what system is better?
Mathematicians found the answer to this question, and they have bad news and good: The bad news: Borda's 3-2-1 count is not ideal; it can still lead to distorted results. The good news: Within the point method, the Borda count is the best. Moreover, our current 1-0-0 count is by far the worst; it gives the least amount of information about what voters want and often yields results that speak against the will of the majority.
This becomes clear from examples. In 1970, James Buckley, the centre-right candidate in New York, won a seat in the U.S. Senate even though more than 60 per cent of the votes went to the two centre-left candidates. A less obvious but even more disturbing case is the more recent Bush-Gore race for the U.S. presidency. If those voting for Ralph Nader could have made Al Gore their second choice (a reasonable assumption), Mr. Gore would have won comfortably, in accord with the popular vote.
But there are other systems. The runoff method, for instance, uses the 1-0-0 point rule in combination with several rounds of voting. Only more than 50-per-cent support makes the winner. Otherwise, the last candidate is dropped and the vote is repeated. An alternative is to exclude all but the first two candidates and vote a second time. The first version is too tedious to be efficient in a national election, whereas the second can lead to distorted results. In the 2000 election for the Romanian presidency, for instance, this method led to a runoff between a left-wing extremist and a right-wing one; the centre vote had been split among several candidates.
The detractors of the Borda count argue that the method is too complicated and would, therefore, become more of a drawback than a benefit. But the uninformed voter does not fare better in the current system, either. Practice has shown that, when the advantages of the Borda count are clearly explained, the electorate is more eager to adopt it. This method has been successfully tested in Slovenian parliamentary elections, for voting best player in Major League Baseball, for choosing the winner in the Eurovision song contest, and for making administrative decisions in universities.
STV combines the Borda count with proportional representation without the inconvenience of bringing non-candidates into the legislature. This method is based on an idea proposed in 1880 by Australian mathematician J. B. Gregory, whose original system was introduced in 1907 in Tasmania, where it is still in use.
As far as mathematics is concerned, no voting procedure is perfect because each method can lead to unrealistic results. But STV is far better than the current system, which often distorts the will of the majority. So, by endorsing the proposed reform, the citizens of British Columbia will make our elections fairer.
Florin Diacu is a professor of mathematics at the University of Victoria.