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PR is not the panacea to our dated voting system

Ballot boxes are counted after a set of local elections - Credit: AFP via Getty Images

If our letters page is anything to go by, The New European’s readers are no fans of First Past The Post. But what exactly is the best way to ‘make every vote count’? MICK O’HARE considers the options.

December 2019’s general election suggested that voters wanted an opportunity to think again about Brexit. Parties advocating Remain or backing a second referendum scooped 53% of the votes – minds appeared to be changing. But instead the Conservatives won a thumping mandate to “Get Brexit Done” despite only receiving 43.6% of votes cast.

Of course the Tory majority is an especially egregious example of Britain’s electoral process, outlined in the Electoral Reform Society’s (ERS) report into that election, which strips bare the folly of Britain’s first-past-the-post (FPTP) system. And if this example seems chosen to appeal to the ‘liberal, metropolitan-elite, Remoaner’ readership of The New European then consider the flipside. The Brexit Party received 644,255 votes yet not a single MP in parliament. Had they been the Tories, who only needed 38,262 votes to elect each of their 365 MPs, they would have had 17. The Liberal Democrats would have had 96. And pity the Greens. They picked up more than 850,000 votes, yet only have one MP. 

In fact the crazy truth is that under FPTP a party can, theoretically at least, win a majority if only 326 people vote for it (If only one person in 326 constituencies votes for Party A, then that party will have 326 MPs and a Commons majority. In the other 324 constituencies Party B could win each seat by thousands of votes, yet be out of office). It’s demonstrably unfair and despite the Conservatives’ commitment to FPTP, they know it too. The ERS describes it as “disenfranchisement on an industrial scale”. They and other opponents of FPTP believe proportional representation (PR) is the answer, and readers of TNE have taken to discussing different forms on the letters page.

Certainly there’s an acceptance that the number of MPs should be linked to the number of votes a party receives, and that each vote should have significance (in 2019 14.5 million votes were cast that had no effect on the make-up of parliament).

But PR comes in many forms. Should we elect parties or people? Should MPs be elected purely by the number of votes cast for their party (party list proportional representation, as used in the Netherlands) or should there be a constituency link (the additional member system used in Germany)?

Then there’s single transferable vote used in Ireland, which allows voters to rank candidates in order of preference. It keeps a constituency link and means tactical voting is less relevant (apparently 32% of Britons voted tactically rather than for the candidate of their choice in 2019).

There are more versions of PR with various levels of complexity: Borda count (a points-based system), range voting (voters score each candidate from 1-10 – the winner is the one with most points) and two-round systems (used mostly to elect presidents, as in France) among others.

So which one? Crucially, are any truly ‘fair’? And can mathematics help us choose – the objectivity of maths should be free from questionable human notions of ‘fairness’?

Unfortunately maths, being the cruel betrayer of imperfections that it is, can find fault with most PR systems. Perhaps the most obvious is the problem inherent to all: that smaller or extremist parties can call the shots because they can hold the balance of power. Introducing a cut-off of, say, 5% tends not to counteract this, but raising the threshold to 10 or 15% risks discounting minority parties entirely, negating the whole purpose of PR (although the disenfranchised 5% could be allowed to recast their votes).

But beyond the mathematics of coalitions are deeper problems with PR systems. Maths has given us Arrow’s Impossibility Theorem, which in simple terms states that any voting system must abandon at least one criterion of ‘fairness’ in order to work (the percentage cut-off being an example).

Another example of Arrow’s theorem (named after economist and Nobel laureate Kenneth Arrow) shows the flaws in ranking candidates or parties in preferential order.

If there are two parties with extreme views (X and Z) each preferred by 35% of voters, the centrist party, Y, would be the second preference of all these voters but is the first preference of only 30%. Y would win if second-rank votes were taken into account and would arguably be the least-worst party (but also the least favoured of all three).

Yet if the system eliminated the last-placed party (Y) and redistributed the votes among X and Z (as would happen, for example, under single transferable vote) an extreme party would be elected against the wishes of 65% of voters.

The Gibbard-Satterthwaite theorem takes Arrow’s further. It’s complex but essentially points out that almost every voting system in which voters rank candidates or parties can be manipulated, sometimes by a single voter and in which ‘lying’ about one’s preference can strengthen one candidate unfairly or undermine another.

An obvious example is the range voting system. Voters rate candidates from 0-10 so a moderately right wing voter might gave the moderately right wing candidate A 10, the centrist B 7, the moderately left winger C 4, while ranking the extreme right wing candidate 1 and the extreme left candidate 0.

Other voters would, obviously, have other preferences. The candidate with the highest score is elected but the system fails if voters aren’t honest – If the voter above gives candidate A 10 and all the others 0 rather than ranking them with a range of points then it becomes FPTP by other means. Range voting however is beneficial in a three-horse race if every voter gives their favourite candidate 10 and their least favourite 0, meaning the middle candidate’s score will always fall between those two ranges (or at worst, equal one of them) and therefore ‘lying’ has less effect. But how many UK constituencies are simple three-horse races?
Maths has far more to say on voting systems but one other perhaps worth considering is restricting an MP’s voting strength in parliament to how many votes it required to get them elected. Based on December 2019, a Tory MP’s parliamentary vote would be worth 38,262 points while the Green Party’s Caroline Lucas’s would be worth 864,743. It might give weight to what voters actually intended, but it also shows how maths, with the best intentions, can give us something of a headache.

We shouldn’t be surprised. Donald Saari of the University of California showed years ago that you can invent a voting system that produces any result you desire, from any particular voting pattern. And we haven’t even started – and nor will we – on the intricacies of the Banzhaf power index or the Alabama paradox.

So it seems, as far as mathematics is concerned, there is no truly fair system and that’s before we add in other instances of inequitability such as the wealth required to run for office, gerrymandering or a lack of impartial information allowing voters to make an informed choice. If you are searching for electoral perfection bear in mind the Condorcet criterion, named after 18th century mathematician the Marquis de Condorcet which states that the winning candidate in any vote should be the one who would win a two-candidate election against each of the other candidates. Ideal but highly unlikely suggesting that, even without mathematical complications, apotheosis seems unachievable.

Any voting system must offer a balance between many interests and opinions while maintaining stable and effective government, and – preferably – retaining a link between MPs and constituents. It’s easier said than done. As popular-science writer Ian Stewart, emeritus professor of mathematics at Warwick University, wrote: “Mathematicians have studied voting systems for centuries. They have turned up paradoxes and surprises. What they haven’t done is come up with the answer. With good reason: it doesn’t exist.”

PR is not the panacea we might expect. But it’s pretty much accepted that one of its forms is far preferable to FPTP. As with all political conundrums compromise is key and while mathematics can expose hidden flaws in systems that, at first take, seem plausible it can offer no conclusive answer. At least some form of PR would end what Lord Hailsham denigrated four decades ago as “elective dictatorships”. 

But so would simply electing MPs by lottery (known as “sortition” in ancient Athens) – a system far more likely to give a representative slice of the population. Maybe there’s something better than PR out there…

What do you think? Have your say on this and more by emailing

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