Explaining … the Condorcet system
I like explaining things so today I’m going to explain the Condorcet system.
If you want to skip ahead and find out how the Condorcet system works, it’s all laid out in the first three paragraphs after “The second solution: the Condorcet system”. It really is that simple. The rest of this essay is just about why it is useful and what problems it solves.
The problem
Say you have an election where you only want one winner; for example, you are trying to elect a Mayor, chair or President. The conventional method is the “First Past the Post” system. Everyone gets one vote and the person with the most votes wins. We all know what a bad system that is: if the field is divided a really bad candidate can come through the middle and win with hardly any support, and — for that reason — what you tend to get is voters voting tactically for what they see as the least bad of the two leading candidates. This kills choice and diversity and is hugely unfair, with many people disenfranchised and many votes wasted.
The classic solution to this problem is the “Alternative Vote” system[i]. Under the Alternative Vote, or “AV”[ii], you rank your candidates in order of preference until you are indifferent. The last place candidate is then eliminated and their votes transferred to their voters’ second preference. Then the second last place candidate is eliminated and the same thing happens until eventually you have a winner.
The AV system has some advantages over conventional First Past the Post. As you get to rank your candidates you are freer to give less popular candidates your vote in the first instance, safe in the knowledge that it will be transferred to your preferred mainstream candidate later on.[iii] This reduces tactical voting. And the system requires candidates to work harder to be elected, requiring them to have a broader base of support, and reducing the number of safe seats.
But the AV system isn’t perfect. Firstly, it doesn’t totally eliminate tactical voting. This excellent video demonstrates some of the reasons why:
Secondly, what AV basically does is it makes sure that votes for minor parties don’t hurt major parties. So if you’re using it for something like electing MPs to Parliament you could argue that AV gives even less proportional and thus less fair results, because it makes it even harder for minor parties to scrape a seat.[iv]
But the major problem with AV is that it is really harsh on a candidate who is everybody’s second choice but nobody’s first choice. Such a candidate might often be the best candidate for the job. Such a candidate stands a high chance of being the candidate that most people would be happy with.[v] But under AV, which as its first step only considers first preferences, such a candidate would be eliminated in the first round — the fact that they were the overall highest choice of the greatest number of people forever lost behind the fact they didn’t get enough first preferences to ever come into consideration.
To me the clearest example of AV’s failings is the London Mayoral election. In four consecutive elections between 2000 and 2012 the city was deeply divided between two candidates (Livingstone for Labour and Norris or Johnson for the Conservatives) who were loved by half the city and despised by the other half. Each election was deeply polarising and left half the capital miserable at the outcome. Meanwhile there were a whole host of excellent third and fourth party candidates (Susan Kramer, Frank Dobson, Simon Hughes, Sian Berry, Siobhan Benita) who it was virtually universally agreed would make excellent mayors and who would have been acceptable to almost all the voters. None of these candidates got a look in because of AV.
The first solution: the Borda count.
The Borda count is a voting system that tries to solve this problem. Most people have heard of it under its alternative name: Eurovision Song Contest rules.
Under the simplest form of the Borda count you rank your candidates in order of preference until you are indifferent. Then, if there are ten candidates, your first choice candidate gets ten points, your second choice gets nine points, your third eight points, and so forth. The candidate with the most points wins.
There are some variations as well: the Modified Borda Count (MBC) gives out more points the more preferences the voter fills in, to encourage people to fill in more of the ballot paper. Range Voting allows you to literally decide how many points to give each candidate.
Whatever the format, Borda solves a lot of the problems with AV. The Borda count tends to be won by the candidate who is most acceptable overall to the most people. However, there are two major problems with the Borda count, one of them is practical and one is philosophical.
The practical problem is that it is very easy to game the system. While the Borda count eliminates the sort of conventional tactical voting you get with the First Past the Post and AV systems, it introduces a whole new, and potentially much worse, set of tactical choices.
Say you are a vaguely left leaning voter in a UK based Labour Conservative marginal seat. You might give your first preference to the Green Candidate, there might be some independent or Left candidate that catches your eye, you might want to give the Lib Dems a high preference, you might have Nats and you might like them. And then you put down your vote (maybe a 3 or a 4, let’s say 4) for Labour. You put the Tories low but maybe not at the very bottom; you still prefer them to UKIP, the BNP, and some of the nastier independents. Maybe out of 10 candidates the Tory gets a 7th place. That’s how you might want to vote, but from the point of view of tactical voting that vote will be absolutely rubbish.
If you’re voting tactically the only real thing that matters in a Borda count is the points difference between the two strongest candidates. If you want the Labour candidate to beat the Tory candidate you must put Labour 1st and Tory 10th. And conversely if you want the Tory to win you must give them your 1st preference and Labour the 10th. Anything else just results in a really weak vote which doesn’t have much impact on the election. Take the vote I outlined above. It would take four votes like that to counteract the effect of just one vote which had Tory 1st and Labour 10th.
Now these kinds of tactical considerations quickly make Borda elections very problematic. Any system which strongly encourages you to give extreme and fringe candidates far higher preferences than you actually want to give them is asking for trouble. And given how often tactical voting leads to the wrong result in relatively stable and well understood systems such as First Past the Post, imagine how much worse the consequences would be under Borda where no one really understands the rules and where small changes in vote orders can result in big changes in outcomes. Now it is true that some of the more modified versions of the Borda count are less susceptible to this kind of tactical manipulation than others, but it is still a problem they all suffer from to some extent.
That’s the first problem. The second problem is spelled out in an argument between two[vi] really really interesting 18th century French aristocrats, scientists and rebels: Jean-Charles, chevalier de Borda[vii], inventor of Eurovision Song Contest rules…
… and his great critic and rival Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet.[viii]
Condorcet felt that an election system should always give you the same winner regardless of who else was standing. If candidate A would beat candidate B if they were in a head-to-head contest, then a fair system is a system in which candidate A always beats candidate B. The system should never be such that if a certain other candidate (C) joins in the race then actually B then beats A. He called this rule his Condorcet criterion: an election should always be won by the person who would win regardless of what combination of other people are running.
None of the systems we have discussed thus far meet the Condorcet criterion.[ix] Condorcet set out to invent an election system that did. In so doing he also invented a system which was immune to the kind of vote gaming and tactical voting that makes the Borda count so impractical.
The second solution: the Condorcet system
None of the maths in Condorcet is that hard so you can count the votes by hand. It is however fairly time consuming so it is more normal to use a computerised counting system if you can. You cast your vote in a Condorcet election the same way you cast your vote in AV or Borda: you rank your candidates in order of preference until you are indifferent.
But after you have voted what Condorcet does is it then divides the election up into lots of mini elections, each with two candidates. So say you have an election with five candidates: A, B, C, D, and E. Condorcet immediately sets up all the possible one-on-one elections, or duels (A vs B, A vs C, A vs D, A vs E, B vs C, B vs D etc… all 5C2 = 10 different combinations). It then, in turn, decides who wins each duel by comparing the votes cast for each. So when considering the A vs B duel you go through the votes and each time a voter has given A a higher preference on their ballot than B you give A one point in that duel. Each time you get a ballot where B has a higher preference than A then B gets one point. At the end you count up whether A or B has won more points within the duel and then announce that either A or B has won that duel.
The overall winner is then the candidate that wins all their duels, in other words the candidate that comes out on top in every single mini election in which they participate.
“That’s all very well”, said Borda[x]in a series of essays, counter-essays and letters in the early 1790s, “but what if there is no such winner? What about rock/paper/scissors type scenarios where you have candidate A beating candidate B and candidate B beating candidate C but at the same time you have candidate C beats candidate A?”
“That’s not very likely is it?” said Condorcet. “We’re talking about personal preferences here. The whole of politics, be it about ideas or personalities, is based around personal likes and dislikes. These likes and dislikes are hugely unlikely to be circular. It’s not very likely that the same group of rational people[xi] are going to, in the same breath, tell us that they prefer rock to scissors, that they prefer scissors to paper, but that they also prefer paper to rock. Ask someone what they would like for dinner and give them three options: how often are they going to say ‘well that’s got me totally stumped because I prefer chicken to fish, and fish to lamb, but I also prefer lamb to chicken’? Besides we’re in the middle of the French Revolution and I’m busy inventing the terms left wing and right wing. If voters and candidates fall on a left-right axis, or indeed any axis at all, then those kids of paradoxes don’t come up.”
“Yeah but people are weird, it could happen,” said Borda. “I like my way better, it’s clearer”.
“Yes but it’s not fair,” said Condorcet. “If there is my kind of winner, let’s call them the Condorcet winner, then they should win the election. And if for whatever reason there isn’t one, because of one of those weird rock/paper/scissors type setups then we’ll simply call that a tie and come up with some tiebrake rules”
And so that, basically, is the Condorcet system. There are various different kinds of Condorcet system, and they all boil down to a difference in how they deal with those sort of rock/paper/scissors ties. But usually they don’t come up: 99% of the time there’s only one Condorcet winner, and they are elected.
Tiebreak rules.
Most rules for breaking ties in the Condorcet system rely on the same basic ideas and concepts.
Firstly, in order to work out what is going on in an election that is tied under Condorcet rules it’s helpful to draw out a network diagram in which each of your candidates is a point (a “node”) and then you can draw arrows between all the candidates showing which candidate beat which other candidate in their duels.
Secondly, from this diagram it should be possible to find the small group, usually a circle, who beat all the other candidates from outside that group. These are basically your rock, paper and scissors candidates — within the group there might not be a clear winner but it is clear that the winner should come from within that group. This group is known as a “Smith Set”.
Thirdly you can write down a number next to each arrow in your network diagram showing the margin of victory in that duel (in other words the number of votes that duel was won by). This is called “defeat strength”.
Here are some of the most common tiebreak methods. These methods might seem confusing but it’s worth bearing in mind that a) Condorcet was right and you rarely have to resort to these methods and b) these methods produce the same result in all but the most unusual of circumstances, so it rarely matters which one you pick. “Schulze” and “Ranked pair” are probably the two most commonly used types.
- Smith/IRV is a hybrid method where you use the Condorcet system to identify your “Smith set” and then — if you need to — you have an AV election from within the people in the Smith set. You do this with the same votes from your Condorcet election, simply by knocking out all the candidates not in the Smith set, redistributing their votes under AV rules, and then going from there as you would in a normal AV election.
- The Schulze method, otherwise known as “Schwartz Sequential Dropping” (SSD)[xii], works by taking your network diagram (see pic) and then crossing out the weakest (in terms of “defeat strength”) link from that diagram. You alternate doing this, and dropping entirely people who are no longer able to win, until you come up with a simplified diagram which only contains one winner.
- Ranked pairs, otherwise known as “Tideman”, is effectively doing Schulze backwards. You start with your network diagram but you keep it blank (no arrows on it). Then you rank your duels by margin of defeat. Then you start drawing on your arrows to mark the duel victories, drawing on the strongest victory (highest margin of defeat) first. But you skip over any arrows that would result in you generating a circular preference, with the result that by the end you have drawn out a simplified version of the network diagram where there is only one winner.
- The Kemeny–Young method lists out every possible sequence of outcomes, and then next to each sequence lists out which duel results you would need to get that outcome. It then assigns a score next to that outcome based on how many votes are votes in favour of those duel results[xiii]. The outcome with the highest score then wins.
- Minimax, otherwise known as “Simpson”[xiv], chooses the person in the Smith set whose biggest defeat is the smallest, and names them the winner.
- Copeland works a bit like a football league table. Candidates get a point for each duel they win and minus one for each point they lose, and the candidate with the most points wins. Copeland itself often produces draws, at which point they use something more like the rules of Conkers to separate the winners: candidates add together the points scores of the people they beat to give a revised score, and the person with the highest revised score wins.
- The Nanson or Baldwin methods did not actually evolve as forms of Condorcet at all, but it does just so happen to give the same result as Condorcet systems: in other words a Condorcet winner will always win under Nanson[xv]. Nanson is hybrid between the Borda and AV methods where the scores are initially computed using the Borda method but then the last placed candidate is knocked out as they would be in an AV election, and then the scores are calculated again, this time totally ignoring any votes cast for candidates that have already been eliminated. Repeat until you have a winner.
- Dodgson’s[xvi] method asks the question “what is the smallest number of ballots we would need to reverse the order of to produce just one single Condorcet winner?” That number is found,[xvii] those ballots are reversed, and the winner is then calculated using those altered ballots.
[i] If you think about it, the Alternative Vote is just a version of the Single Transferable Vote with the number of winners reduced to one. I wrote about STV here.
[ii] There’s also a system called the Supplementary Vote or SV which is a form of AV in which the number of preferences you can give is arbitrarily restricted. There’s no good reason for doing this and SV is a stupid system. It is however used in Fiji, Sri Lanka, and for mayoral elections in the UK. No one really knows why; there’s a theory that in the UK it was because Labour party apparatchiks felt it was the system most likely to prevent Ken Livingstone winning as an independent: oops! You also sometimes hear the term Instant Runoff Voting or IRV used to describe either SV or AV. In this article I’ve used AV throughout for simplicity, even when talking about Mayoral elections where the term SV would be more correct.
[iii] Or it would if people understood the system. The number of people voting mainstream party first preference, fringe party second preference in the Mayor of London elections shows how many people vote without realising how the system works.
[iv] You could argue this isn’t really AV’s fault. AV is not a form of proportional representation and if you are using any sort of non-proportional system to elect your parliament then of course your result won’t be proportional. This is why you should only use these non-proportional systems when electing individuals such as Mayors or Presidents.
[v] I wrote a really bad article for Lib Dem Voice about Condorcet once. I wouldn’t read it, but I do have quite a good example involving Marmite to explain this point.
[vi] We now know that this isn’t actually strictly true. Both electoral systems I go on to discuss, and the problems associated with each, were actually already discovered and discussed at length some 500 years earlier by a 13th century explorer and monk from Majorca called Ramon Llull. Llull may also have invented computer science. However as Llull’s work on elections was lost and was only rediscovered in 2001 his work doesn’t really form part of our story.
[vii] Borda was a scientist but also a sailor, back in the age when the line between “Captain in the French Navy” and “pirate” was a thin one. He captained a squadron of six ships around the Caribbean for a while, and fought on the side of the Americans during the American Revolutionary War; his ship lost a bruising battle with the British and he was captured, only to negotiate his way back to freedom and France. There he perfected various navigational methods and instruments and, when the French Revolution occurred, he was one of the leading campaigners for, and later implementers of, decimalisation and metrication. As part of his work on the subject he worked with the legendary mathematicians Lagrange and Laplace on standardising the metre and measuring the circumference of the earth.
[viii] Condorcet was a philosopher, mathematician, political scientist and pamphleteer. He did academically important work on calculus and atheism, made some of the first attempts to introduce mathematics into social sciences, and wrote some of the first French articles to denounce slavery, racism, and sexism. He was a passionate supporter of the French Revolution and became first a deputy (MP) and then later the Speaker of the revolutionary assembly. He was chosen for this role because he was seen as being independent both of the moderate Girondins and the radical Montagnards. However, at the trial of King Louis XVI he came out against the death penalty, and suggested a compromise where Louis would be sent to the galleys. This marked him out in Robespierre’s eyes as a Girondin sympathiser and from that point his days were numbered. Knowing this he went into hiding, and died shortly thereafter: seemingly poisoned — possibly by his own hand.
Condorcet’s wife, Marie-Louise-Sophie de Grouchy, Madame de Condorcet, was possibly even more interesting than he was. Often patronisingly referred to as a “young beauty”, “society hostess” or “Thomas Payne’s translator”, what she really was was one of the key organisers and convenors of the pre-revolutionary Jacobins and subsequently of the Girondins. She is also widely thought to have ghost written Condorcet’s writing on women’s rights, and is often mentioned as one of France’s first feminists. Her boldness and forthright political views appalled a visiting Thomas Jefferson. After Condorcet died Sophie and her daughter Eliza threw themselves into the United Ireland movement. The Irish rebel leader Arthur O’Connor subsequently became her son-in-law and with her help went on to become one of Napoleon’s generals.
[ix] First Past the Post doesn’t: in series four of the Wire you see that Carcetti couldn’t have beaten Royce unless Tony Gray had run as a spoiler candidate. AV doesn’t either: most of the Green candidates and a good number of the Lib Dem candidates of the last 16 years would have won the London Mayoral election if they had been lucky enough to get the historical winner alone in a one-on-one. No version of the Borda system does either: although there aren’t any well-known examples, it is a simple enough job to create one.
[x] Or words to that effect, in 18th century French (I’d love to know the 18th century French for rock/paper/scissors). Actually it was Condorcet himself, not Borda, who first outlined this “Condorcet paradox”, Borda just decided that it was a problem.
[xi] Condorcet’s philosophy is all about the fact that people are rational, and should behave rationally.
[xii] Also known as “Cloneproof SSD” (CSSD), and “Beatpath”.
[xiii] So say you have four candidates. One possible result is A first, B second, C third, D fourth. For such a result you would need A to beat B, C, and D, B to beat C and D, and C to beat D. So you go through all the votes and count up the number of times A beats B, A beats C, A beats D, B beats C, B beats D, or C beats D. You assign one point for each time one of those things happens (ignoring votes where the opposite happens) and then you add up the total and that gives you the number of points for that one possible result. Repeat for all the other possible results.
[xiv] Also known as “Simpson-Kramer”, “Simple Condorcet”, and “Smith/Minimax.”
[xv] It just so happens that mathematicians were able to prove that while a Borda count doesn’t guarantee that the Condorcet winner will always win, it does guarantee that the Condorcet winner will always be in the top half of the candidates. Since being in the top half in each round of AV is enough to guarantee that you won’t be eliminated that round, this means that a Condorcet winner can’t lose a Nanson election.
Technically there’s a slight difference between Nanson and Baldwin. Baldwin does the classic thing of eliminating the candidates from the bottom one at a time, whereas Nanson eliminates the entire bottom 50% of remaining candidates at every round. But the overall effect is the same.
[xvi] Yes, the same guy that wrote Alice in Wonderland.
[xvii] This is mathematically the same thing as a well-known problem in computer science known as “minimising the Kendall tau distance” so the algorithms to identify and select these ballots are already well known. This means you can use Dodgson’s method with ease provided you are using a computer to count the votes.
