There’s more than one way to count a three-way People’s Vote

by Peter Emerson | 06.08.2018

Peter Emerson is the Director of The de Borda Institute an international NGO devoted to the use of preferential voting in decision-making.

With the idea of a People’s Vote on Brexit gaining traction with the public, many have started asking what that vote would look like. Several people, including former education secretary Justine Greening and constitutional expert Vernon Bogdanor, have suggested a multiple-option ballot – essentially a three-way choice between Theresa May’s deal, leaving with no deal and staying in the EU.

But there are lots of ways of analysing a multi-option ballot to find the most popular outcome.

You could have a single preference plurality vote, where voters choose a single option and the one with the most votes wins. But that can often be inaccurate, especially when none of the options gets a majority and the winner may be the option with just the largest minority.

Or you could have two votes, carried out in a two-round system (TRS). Bogdanor proposes a variation of this. In TRS, everyone picks their favourite option in the first round. If nothing gets a majority, you then have a second round with everyone voting between the two leading options from the first round. But this methodology can be capricious – for example in the 2002 French presidential election when the extreme right-wing Jean-Marie Le Pen got into the second round because the left-wing vote was split in the first round.

Or you could try an alternative vote (AV), as Greening suggests, where voters make a first and second preference. But this too can be unreliable, for example in a polarised situation, when a good compromise option might get a very low first-preference vote and thus be eliminated, so leaving its huge second-preference vote uncounted.

OK, so what about a points system? Let everybody cast their preferences. In the count, we change the preferences into points: in a three-option poll, a first preference gets three points, a second gets two, and so on. And we simply add up all the points. This is called a Modified Borda Count (MBC).

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To see how each voting method can throw up different results, consider this example of nine voters giving their first, second and third preferences on three options: A, B and C.

  • Four choose A-B-C
  • Three choose C-B-A
  • Two choose B-C-A

An initial glance suggests that B best represents the collective will, because it’s the first or second preference of everybody. But what happens in practice when you apply the different voting systems?

  • In a plurality vote which counts only the first preferences, A wins with 4 votes (C gets 3 and B gets 2).
  • In TRS, there’s a second round between A and C. If the voters’ preferences stay the same as in the first round, C wins with 5 to A’s 4.
  • In AV, B has the smallest first preference score so it is eliminated. Its two votes go to C, so C wins again on 5 to 4.
  • In an MBC points system, however, B wins. The scores are B on 20, with A and C both on 17. The winner is now what it should be.

Alas, because so few people analyse different decision-making voting systems, many in the media and academia would regard all three outcomes as “totally democratic”. As in the example above, however, the result often depends upon the counting methodology.

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Josef Stalin used to say: “It’s not the people who vote that count; it’s the people who count the votes.” He, of course, was just referring to outright cheating. But politicians often choose voting methodologies, first and foremost, because it suits them. For them, accuracy is sometimes of secondary importance.

It’s still unclear where Brexit will end up, and therefore whether three options would be needed in a People’s Vote. For example, May might not get a deal at all. Or MPs might decide that no-deal Brexit is so catastrophic they don’t put it on the ballot paper.

But if a multiple-option ballot is necessary, then the voting method will be very important. As we see above, plurality voting can sometimes be hopelessly inaccurate, while TRS and AV can be very capricious. But a preferential points system, the MBC, is invariably robust and very accurate.

At best, the MBC identifies the option with the highest average preference – and an average, of course, involves every voter, not just a majority of them. Therefore, unlike any binary vote which, as in Brexit, can be so divisive, the MBC includes (almost) everyone’s choices. Indeed, it can be the very catalyst of consensus, and that is what today we all so desperately need.

Edited by Luke Lythgoe

20 Responses to “There’s more than one way to count a three-way People’s Vote”

  • What about a system where every voter has six points to allocate amongst the three choices (Remain, May’s Deal, No Deal)

    So somebody who is an ardent remainer would give (6, 0, 0). Somebody who just wanted to leave would give (0, 4, 2). And so on.

    This is the method I used to use to decide a location for our works Christmas lunch. Those with strong feeling have a vote that counts more than those who really couldn’t care less.

  • Surely it has to be two choice vote – May or remain. No deal is simply the sinister shadow behind failing May.

  • I agree with the above comment. The various versions of ‘3-way’ choice are all theoretically feasible but ‘politically’ impossible as the projected outcomes based on the various options demonstrate. And, with respect, asking the electorate to allocate points would be an operational disaster and make the UK (even more of) a laughing stock than we already are.

  • The Borda system can elect a very unpopular choice which no-one wants just because it is everyone’s second preference.
    Also if one side tactically refuses to put a second preference or third preference and the other side does then you get for example if c voters only put c
    Then c gets 17 B gets 14 and A gets 14 c wins by being clever!
    Also this is a very poorly thought out system .. why would anyone put a third preference?

  • Agreed.
    But Brextremists love the threat of a No Deal because they misguidedly believe that Europe will blink first.

  • no deal brexit shouldn’t beer offered as an option bearing in mind the disastrous consequences it would bring to the uk.

  • There’s another method that would be even more representative than the MBC, and that is Range Voting.

    A Relative preferential system such as MBC, or any of the others above, can capture the order of preferences but not their degree: it cannot tell the difference between thinking the second choice is nearly as good as the first, and thinking it’s indistinguishable from the worst. Range Voting is an Absolute preferential system: it asks the voter to decide how much they like each option independent of the others, and then bases the results on the average opinion of each.

    The main electoral reform groups invariably ignore Range Voting, as each has their own favourite (which one at least has explicitly refused to question!). But despite being rare in political elections, it is used far more often than any other system – the ubiquitous ‘star’ reviews used by countless online retailers are simply Range Voting with a shiny branded wrapper. As such, it’s probably by far the easiest reform option to ‘sell’ to the people and have it understood.

  • Range Voting sounds good to me. Its seems to have advantages that express the degree of support you feel for an option.

  • In response to Keith Underhill’s point, presumably voters refusing to give a second preference in order to manipulate the vote could simply be discounted as spoilt papers.

    But it is more difficult to be sure that some type of tactical manoeuvre would not emerge which discourages voters from being completely honest, even with MBC or range voting.

    Also, the problem if you have a three way choice is that greater emphasis is given different ways leaving. Voters spend more time thinking of leaving in some shape or form. If twenty ways of leaving were offered, some would chose one of these options who would otherwise have remained.

    What this means is that first preference votes for Remain would probably be less than 50%, once the press campaigns had seduced voters away to the other choices. The lack of a first preference majority for remaining in the EU would then be seized upon by the Brexit press, no matter how the results were subsequently processed.

    The two round system might overcome this problem. Voters have some time to refocus on the critical two choices in the final run-off, giving less room for argument.

  • A prototype of the Modified Borda Count (MBC) was invented in 1199 by Ramón Llull, but Jean-Charles de Borda was the first to do a proper mathematical analysis of this voting procedure, and hence the name.

    The rule is as follows. In an n-option MBC, a voter may cast m preferences, so n ≥ m ≥ 1. Points are awarded to (1st, 2nd … last) preferences cast, according to the formula (m, m-1 … 1), and the winner is the option with the most points.

    • he who casts only one preference gives his favourite just 1 point;
    • she who casts two preferences gives her favourite 2 points {and her second choice 1 point};
    • he who casts n preferences give his favourite n points, {his second choice (n-1) points, his third (n-2) points, etc.}

    In effect, then, the voter is encouraged to submit a full ballot of n preferences, so to recognise the validity of all his neighbours’ aspirations. The second consequence is that the protagonist will be, not only well advised to ask her supporters to cast lots of preferences, but also incentivised to persuade her erstwhile (majoritarian) opponents to give her option their 2nd or 3rd preferences.

    The methodology, then, is well suited to situations such as exist in Britain today, when passions are roused. This Institute first tested it in Belfast in 1986; it was a public meeting in which over 200 took part – pretty well everything from Unionist paramilitary to Sinn Féin, (and this, still eight years before the cease-fire). They met, some for the first time; they proposed various constitutional arrangements, and those which complied with the UN Charter were accepted; they debated; then they cast their preferences, and thus they found their best compromise: NI to have devolution and power-sharing within a tripartite Belfast-Dublin-London agreement – a sort of Belfast Agreement, 12 years ahead of its time. Five years later, we did it all again with electronic voting – not bad for 1991.

    Since then, the MBC has been demonstrated and/or used in many settings, not only in NI but throughout these islands and elsewhere, not least in conflict zones as in the Balkans and the Caucasus, with the first major breakthrough being in Dublin City Council in 2013. Our experience to date shows that, in nearly all scenarios, most voters cast most if not all of their preferences.

    In summary, then:

    + a plurality vote can be hopelessly inaccurate; {its electoral equivalent is the UK’s first-past-the-post, FPTP, (or fake post-truth polling)}.

    + TRS, which is based on a first round plurality vote, can also produce screwed results; {it is used in French elections}.

    + AV is like a knock-out series of plurality votes, with the least popular being eliminated and its votes transferred at each stage, and it can also be capricious; {it is used in Australian elections and, in its PR format, in Ireland}.

    + The Borda Count, BC, is a corrupted form of de Borda’s MBC: either (n, n-1 … 1) or (n-1, n-2 … 0). With the latter formula especially, it cannot cope with partial voting. {The BC is used in part in Slovenian elections.}

    In a nutshell:
    + plurality voting is adversarial;
    + range voting allows the voters to be consensual but incentivises them to be adversarial;
    + TRS is also, fundamentally, adversarial;
    + AV allows for and encourages some inter-party co-operation;
    + in the worst-case scenarios, that is, when tensions are high, the BC is little better than a plurality vote;
    + an MBC not only allows the voters to be consensual, it actually encourages them to be so.

    Another very accurate and robust system is the Condorcet rule and, in many settings, the MBC winner is also the Condorcet winner. A good comparison is a sports league: in most years, the Condorcet winner, which is like the team which wins the most matches, is also the sort of MBC winner, the team with the best goal difference; but there can be exceptions. Of the two, the MBC is non-majoritarian and therefore gets my first preference. It is also very difficult to manipulate, especially if the rules on composites are applied… of which more anon.


  • This is great but I had never heard of De Borda and I doubt whether Justine Greening and other politicians have either, and certainly not David Cameron.
    The People’s Vote people should announce that we have identified a voting system which is agreed as the most valid by UCL / De Borda / other institutions trusted by the public. An idiot’s version should be available, couched in less mathematical language, so that journalists do not need to produce their own dumbed down version, which may be less accurate .

  • It’s not a three way choice as the underlying issue is to leave or not to leave. There should be two questions 1) Do you wish to leave or remain? 2) If the majority answer to 1) is to leave, do you wish to leave with the deal on offer or no deal?

  • I’ve read some of Peter Emerson’s work, and his arguments against majoritarian binary votes are very good and should be read and understood by everyone. Majority domination of the minority is not a desirable goal, and only seems that way as a perverse consequence of our binary plurality voting systems. It results in false dilemmas, increasing divisiveness and political polarization, and even partisan violence.

    I don’t understand why he advocates Borda Count, though. Assigning points to rankings is an ok concept in theory, but it’s still based on the flawed ranked ballot concept, and it fails spectacularly when voters are strategic (electing the WORST candidate, precisely because everyone thought they had no chance of winning and dishonestly ranked them highly). It likewise suffers from the “army of clones” problem, where parties unfairly increase their chances of winning by running many similar candidates, as happens in countries like Nauru that use Borda.

    A much better idea is to just collect the point values directly, using score/range/ratings ballots. All of Emerson’s arguments in favor of the ranked points system apply even more favorably to rated points systems.

    Ranked ballots provide a distorted picture of voter preferences, since they incorrectly assign equal weight to both very strong and very weak preferences, and force distinctions where none exist in the mind of the voter. Rated ballots collect more information, which makes them more likely to find the consensual candidate that best pleases the voters.

    Gibbard proved that ALL reasonable voting systems have incentives to vote strategically, and yes, rated systems are no exception, but even raw Score Voting with strategic voters still produces better winners than ranked systems with strategic voters.

    Additionally, there are several alternative ways to count rated ballots that reduce the incentive to exaggerate: STAR Voting, 3-2-1 Voting, Majority Judgment, etc. These use rated ballots, but count them in a way that discourages strategic exaggeration, encouraging honest voting and finding the candidate that maximizes voter satisfaction.

    STAR and 3-2-1 are much better than Borda, and we should be using systems like these to end conflict and find consensus among the voters.

  • The Modified Borda Count (MBC) is a decision-making procedure and, if done correctly, it does not suffer from the “army of clones” problem because the options to be listed on the ballot paper are subject to impartial arbitration by a team of consensors, who are always tasked to produce a (short) list which is balanced and fair.

    If we’re talking about an election, then the appropriate methodology is the Quota Borda System (QBS) which, like PR-STV, incentivises the various parties to nominate only as many candidates as they think they can get elected. So QBS does not suffer from this problem either.

    Score voting or range voting is not suitable for a conflict zone, where passions are always high; and not suitable for other societies if and when the topic in question is contentious, as was Brexit. Granted, score/range voting ALLOWS voters to be consensual, but they INCENTIVISE them to be the very opposite: winner-takes-all-and-loser-gets-nought, and all that. Whereas, with the MBC, (almost) everybody wins something and (almost) nobody wins nothing.

    One more comment. As I to said in my last piece, the MBC is extremely difficult to manipulate, not least because those who try have first to guesstimate the intentions of the other voters. In a two-option ballot, this is relatively easy. In a six-option poll, where there are 720 different ways of casting a full ballot, it is obviously rather more difficult. So just because nothing is perfect, the work of Gibbard and others should not be used to imply that all systems are equally imperfect. And as I(and others) have said elsewhere, the most accurate methodology is a combined MBC/Condorcet, and if the MBC social choice coincides with the Condorcet social choice, then you can be something like 99% sure that this outcome is an accurate reflection of the will of the people.

  • I’m no mathematician, but it seems to me that in the case of Brexit, where there is a fundamental two-way split with two options on one side of that divide, anyone casting three preferences with the first assigned to remaining (3 points) would be casting as many points for leaving as for remaining. On the other hand, casting only 2 votes would give a 2-1 preference for remaining.
    I’m not sure how that would pan out en masse, but trying to introduce a new voting system with such inherent subtleties at short notice would be likely to lead to the same complaints about a poll with insufficient information as have bedevilled the original Brexit referendum.

  • Following my previous line of thinking, all that Brexiteers would have to do to achieve their goal would be to vote for the two forms of Brexit. That way they would receive three times as many votes as Remainers could by only voting for their preferred option. Any further choices made by Remainers would simply add the same or more points to the Brexit side.
    In short, MBC only seems to work where choices are not subdivided or, perhaps, where a dummy choice is added. This might be, in the case of Brexit, a ‘Remain just as before’ option but with the addition of a ‘Remain but campaign for a reduction of free movement within the EU’.

  • Two large minorities strongly advocate either a complete break with the EU (No deal Leavers) or staying in the EU (Remainers). It’s probable that the majority, worried about, but also bored, with Brexit, just hope for a half decent outcome, and their votes will depend on the effectiveness of the various campaigns leading up to the vote. One way to get a fair reflection of people’s wishes could be as follows:
    A single ballot paper with three binary choices:
    Leave with the deal negotiated/ Leave with no deal.
    Leave with the deal negotiated/ Remain in the EU.
    Leave with no deal/ Remain in the EU.
    Voters should probably be instructed to make all three choices or have their votes invalidated.
    Inevitably the voter will vote twice for their real preference and once for their best alternative (Any ballot paper that has equal preference for all three options would be invalid and meaningless).
    The voter’s preferred choice (two Xs on the ballot paper for the same option) is then recorded as two votes for that option. The votes for the three options are added up and if one has more than 50% it wins outright.
    Failing that (as seems likely) each second preference vote (the remaining single X on each ballot paper) is recorded as one vote. These votes, with effectively half the value of the first preference votes, are added to each option’s total. The option with the highest number of votes wins. It is quite possible that this will be less than 50%, but it might be as close to a fair expression of the popular will as we can get.
    This is equivalent to each voter having one and a half votes, one for their first preference, half for their second.

  • Something similar, based on a Borda count, is used in Nauru – the initiative of a Desmond Dowdell, the Minister of Justice, by birth an Irishman: a 1st preference gets 1 point, a 2nd gets a half, a 3rd a third, and so on. An excellent system.
    But please don’t make voting compulsory.
    With an MBC, he who abstains has no influence; she who submits a partial ballot has a partial influence; those who fill in a full list of preferences have a full influence. And of course, the very mathematics of an MBC count encourage not only the voter to submit a full ballot, but also the protagonists to campaign for them to do so.

  • Re John’s comments. The correct procedure, as in New Zealand in 1992 when they had a five-option referendum, is to leave the choice of options to an independent commission. They had five options: FPTP, AV, AMS, MMP and PR-STV. Now you could say that’s unfair: 2 majoritarian systems FPTP and AV, and 3 others; or 2 proportional systems MMP and PR-STV, and 3 others; or 2 preferential systems, AV and PR-STV, and 3 others… etc., etc., (and if you’re not careful) ad nauseam. So leave the choice of options to an independent authority and, as a general rule, aim for 4 to 6 options.

  • Very interesting indeed that so much work has gone into these Brexit vote choices when the case for Brexit was already made with almost all of the people commenting on this website clearly supporting overturning a referendum vote. The majority has already spoken in the Referendum.

    If Theresa May’s deal (treaty) that has been voted down 3 times is so bad then why is that even an option on this remainer cooked up extra referendum (people’s vote).

    The commenters on this post, using their apparent intellectual abilities to dream up a word like Brextemists and their wanton need for a vote that kills democracy and since the vote has already been taken is verging on a banana republic overturning votes as and when they like.

    The insanity of these people, the remoaning never stopped, they like sheep follow the EU into oblivion, they have no faith in our great country and would much rather prefer to remain as a vassal state than break away from the EU. I myself was a remainer but now I think anyone who wants to remain part of this club is not being true to their country and their self. We can make bigger steps outside of the EU and if as I suspect it is mainly about money and greed with the remain camp then they will be very happy once we begin making money from the rest of the world in large amounts. Who know how our standing is the world now since Theresa May has made our country look foolish but I believe we still have the best people, the greatest minds and some of the best products in the world. I am ready to take on this brave new world as it was voted for by the majority, remainers need to get onboard or risk becoming a vote overturning, vote rigging communist like shitocracy.