By Nick Fonda
Local Journalism Initiative
The title would have caught my attention any time, all the more so in election season. To the south of us, the Excited States have been in election frenzy since the Republican candidate announced he was running almost two years ago. (The early announcement has been successful in stalling and deferring almost all of the legal cases facing the former president.) Here in Canada, we might find ourselves in election season at any moment between now and October 2025. Our election season is thankfully much shorter, mandated by law to be from 36 to 50 days.
The intriguing title of the video was Why Democracy is Mathematically Impossible.
The opening line of the video softens the title somewhat by rephrasing it as, “Democracy might be mathematically impossible.” If democracy might be impossible, surely that leaves some room to hope that democracy might also be possible.
Certainly, democracy as its practiced in Canada and in the country next door is in grave need of reform. In both countries, political power irregularly but inevitably swings between only two parties. Both countries use the first-past-the-post (FPP) system and in both countries the party garnering the greatest number of votes is by no means assured of winning the election. Both countries have experienced requests (so far unheeded) on the part of some of the populace for electoral reform.
In 2015, Justin Trudeau was well aware that many of his fellow Canadians would welcome electoral reform. After all, for both the NDP and the Green Party, a change to proportional representation was a key plank to their electoral platforms. Trudeau borrowed the plank and promised that the 2015 election would be the last one contested under FPP. Like many democracies around the world, he said, Canada would adopt some form of proportional representation.
For many—but not all—Canadians, the idea of having our representatives in Parliament correspond proportionally to the popular vote is appealing. It reflects an innate desire for fairness. If Party X gets 40% of the Vote and Party Y gets 20%, then Party X should have 40% of the seats in Parliament and Party Y should have 20%.
Might that be mathematically possible?
The video begins by stating that the methods we use to elect our leaders is irrational, before delving into the history of democracy and the math involved in democratic elections.
The simplest form of a democratic election is to list the names of all the candidates on a ballot and ask the voter to indicate their preferred candidate. This electoral system dates back to antiquity. It has been used to elect members to Parliament in England since the 14th century. There are 44 countries around the world—including 30 former British colonies—that use some form of this system. It came to be known as FPP probably because of the horseracing metaphor often used to describe elections.
Even though it is still used by almost one quarter of the countries in the world, it is a flawed system on a couple of levels. Because we have political parties, and because of disparities in the size of federal ridings (from about 25 000 voters to 120 000 voters), a majority of votes won’t guarantee electoral victory. In the last century, England has had 21 majority governments, but only twice did those majority governments garner a majority of the votes.
Another problem with FPP is the spoiler effect. Parties with similar platforms split the vote, and a less popular party ends up winning. In 2000, Ralph Nadar was a third-party candidate in a race that was only ever between Bush and Gore. In the key state of Florida, Nadar took some 100 000 votes. If about 600 of those votes had gone to Gore, George W. Bush would never have become president.
There are alternatives to FPP.
In 1784, Jean Charles de Bordas of the French Royal Society of Science proposed an electoral system that ranked candidates. The following year, a fellow-nobleman and fellow member of the Society, the Marquis de Condorcet dismissed Bordas’ idea and proposed a somewhat different ranked-ballot system by which voters rank the candidates in their order of preference. Condorcet came to be known as the grandfather of a branch of mathematics called Social Choice Theory. He also gave his name to the Condorcet Paradox. While his ranked-ballot system could work quite well, it could also result in a Mexican standoff in which no clear winner could be determined.
Curiously, Condorcet’s system had been proposed almost five centuries earlier by Ramon Llull, a philosopher, theologian, poet, and prolific writer. For over three centuries, the Catholic Church elected its pope using Llull’s ranked-ballot system. Condorcet was unaware of Llull’s electoral ideas because his Ars Eleccionis (The Art of Elections) went lost and wasn’t discovered until 2001. A philosopher and social economist who was a proponent of education and women’s rights, Condorcet also drafted a constitution for the newly-born French Republic. Sadly, this effort landed him in jail where he died at the age of 50.
When voters rank the candidates, they essentially have a second chance to express their wishes. If a voter’s candidate is eliminated (by placing last), the voter’s second choice is given his vote. A candidate who places second on multiple ballots can see himself emerge as the winner by virtue of picking up the votes of eliminated candidates.
This voting system was used in a municipal election in Minneapolis and resulted in an unexpected side-effect. In 2013, after having been in office by acclamation for several terms, the mayor of Minneapolis announced his retirement. In all, 35 people from the city of 425 000 stepped forward to fill his shoes. With that many candidates, it was reasonable to assume that at least one (if not many more) might be aggressive, combative, and generally unpleasant. Instead, aware that placing second on the ballot might be even better than placing first, the candidates were unfailingly polite, courteous, and friendly. (A voter’s first choice is likely based on policy, but a second choice could well be based on personality.) The mayoral debate ended with the 35 candidates, linked arm in arm on the stage, singing Kumbaya.
Following Condorcet, dozens of mathematicians turned their attention to his paradox and finding a mathematically satisfactory system of electing a democratic government. Among them was Charles Dodgson, little known for teaching Math at Oxford but well known as Lewis Carroll, author of Alice in Wonderland.
While all of the systems proposed were better than FPP, all of them had particular flaws or weaknesses that, under certain conditions, could be problematic.
Then, in 1951, an American mathematician, Kenneth Arrow, published Arrow’s Impossibility Theorem. In it he outlines five conditions necessary for a reasonable and logical voting system. He then proved mathematically that, in an election with three or more candidates, it is impossible to satisfy all five criteria. In part for his theorem, he was co-winner of the Nobel Prize in Economics in 1972.
More recently, and on a more encouraging note, a Scottish economist, Duncan Black, published a paper entitled On the Rationale of Decision Making. Black’s voting system is based on the work of both de Bordas and Condorcet. He places voters and candidates on a line between two opposites: left and right, pro and anti. Black showed that the preference of the median voter reflects the majority decision. In Black’s system, the voter indicates the candidate he approves of, or disapproves of and to what extent. In places where Black’s system has been tried the side effects were all positive: increased voter turnout, minimal negative campaigning, and the elimination of the spoiler effect. Approval rating is used to elect the Secretary General of the United Nations. Kenneth Arrow came to accept the viability of Black’s electoral system.
It there is hope that democracy is mathematically possible, perhaps there is hope that Justin Trudeau will remember his promise of 2015 and introduce electoral reform.
Swinging back and forth between the same two parties is of interest to big-money donors but to no-one else. While Canada is not yet officially in election season, like autumn, it’s in the air. We’re at a curious moment in Canada, with a great many citizens dissatisfied with the current Prime Minister, and almost as many already dissatisfied with the Prime Minister in-waiting. Proportional representation, in any form at all, would be a welcome change from FPP, a form of democracy as flawed as it is antiquated.
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