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Expert View

Alice comes to Brexitland; and figures out what to do

by Peter Emerson | 11.09.2018

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

Many years ago, the 25 children in Duoland used to take majority votes. So when Alice asked, “Dinner kids; shall we have Brussels sprouts?” they had a referendum. Thirteen of them, 52%, said “no”.  So “the will of the kids” was “no Brussels sprouts”, and that was that.

Then Boris asked, “Mushy peas everyone?” So they took another vote. But (a different) 52% said “no”.  So “the will of the kids” was “no Brussels sprouts and no mushy peas either”. And that was another that.

But they still didn’t know what they wanted. What’s more, it was nearly time for dinner; and if a majority says “no” to everything, you’ll never get anything. 

So they had an election instead, and the winner would make the decision. Eleven kids got 2 votes each; but Jacob got 3 votes. So he was the winner – that, after all, is first-past-the-post. (In fact, all 22 thought Jacob was the worst possible choice; ah well, no voting system is perfect, especially in Duoland).

And Jacob said, “Parsnips.”  “Ugghh!” they all screamed. Thus, no deal, and no veg for dinner.

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Solace in Wonderland

So the kids go to Wonderland to try something new. “Any ideas everyone?” asks the Queen of Hearts, “We have Brussels sprouts, peas, and parsnips; any other ideas?”  

Ken says, “Swedes.” So there are four options, and the kids vote in order of preference.  

Nigel huffs; he likes only pumpkins and abstains. Insisting on parsnips, Jacob casts only a first preference, so his parsnips get 1 point. And, because he says nothing about the other veggies, he as it were gives them each 0 points.

Boris and David are also a bit grumpy. They like peas a lot and parsnips a little, so they cast just two preferences. But they’re not giving any points to the swedes let alone those Brussels sprouts. Thus their peas get 2 points each and their parsnips 1 point each.  

Everyone else casts all four preferences. With these votes, a first preference gets 4 points, a second gets 3, a third gets 2, and a fourth gets 1 point. If Boris, David and Jacob had ranked all the veggies instead of being grumpy, their top choices would have got 4 points too.

Here’s the full table of how everybody voted.


We now add up all the points to see which vegetable gets the most points. And the winner is sprouts, the first or second preference of 21 kids. It is the near perfect compromise.


Beyond the looking glass

In the ideal world, especially when a topic is complex and/or controversial, everything should be on the table. During and after the debate, an independent body should choose the options, and ballots should be preferential. Then, in any analysis, every preference cast should be taken into account, and the option with the most points should be the answer.  

This voting procedure, invented by Jean-Charles de Borda in 1770, is called a Modified Borda Count. Multi-option voting was first used in China in 1197, and preference voting was advocated in Catalonia/Spain two years later by Ramón Llull.

Why the Mad Hatters of England are still living in Duoland is odd, not least because this points system of voting was also suggested by Charles Dodgson, better known by his pen name, Lewis Carroll.

Edited by Hugo Dixon

Tags: , Categories: UK Politics

13 Responses to “Alice comes to Brexitland; and figures out what to do”

  • All good fun but as a British citizen in Europe I am not a vegetable but a human being …….altbough some days I feel like a chip to be bargained with. Does this make me a potato?

  • It is interesting.
    It is complicated.
    You have to think.
    Is that too much to ask electors?
    Might it give a “wrong” answer?
    Should we try it?
    Could we hold a parallel ballot – one on the old basis and one on the new basis, and then have a think.
    Something needs to be done.
    It would be nice if it was sorted within my lifetime.

  • A neat parable, but it exposes the false assumption on which relative preferential voting is based. That doesn’t just mean Borda counting, by the way: relative voting systems include the existing largest-minority ‘first past the post’ system, as well as the more fashionable alternatives of single-cross proportional representation (PR) and ‘single transferable vote’ (STV) preferential methods.

    Let’s dig a bit deeper into the parable. Relative methods rely on there being a definite order of preference, they ignore the size of the difference, and they invariably only count from one end. Here are three new voters; let’s see what they say:
    Abigail likes parsnips and swedes equally, doesn’t care either way about sprouts, and mildly dislikes peas. She can’t give two options her first preference, so she has to toss a coin between parsnips and swedes.
    Bill really loves parsnips, and has a strong and equal dislike for the other three options. He could put just one preference for parsnips, but he knows that means they’ll only get one point, so he picks a random order to place the others in so that parsnips get all four.
    Caitlyn doesn’t mind what they have, as long as it’s not peas (she has an allergy). She doesn’t have any way to make that ‘negative vote’ count, so she has to pick a random order to put the others in so that peas can be last.
    All three of those voters hand in their papers. All three say parsnips, then swedes, then sprouts, then peas. Is that really an accurate reflection of their preferences?

    Let’s see what happens if we use an absolute method: we allow each voter to rate each option between +5 and -5, corresponding to absolute support and absolute opposition respectively, and the winner is the option with the best average rating.
    Abigail can now put her parsnips and swedes in equal first place. She’s not absolutely mad for them, so she gives them both +4. Sprouts, not too fussed, 0 points. Mild dislike for peas, -2.
    Bill can give his beloved parsnips a full 5 points! And he can give the others all -4, without having to figure out which he dislikes least.
    Caitlyn can give peas -5 to signify her absolute opposition. She’s under no obligation to rate the others, so she can either give the others 0 or leave them blank.
    Much easier, far less wild guessing, and the results actually reflect the voters’ true preferences!

    That’s called Range voting. It tends to get ignored by electoral reform groups, as they generally have their own pet favourite systems and aren’t interested in checking whether it’s actually the best. But despite that, it’s probably the most commonly used voting system in the world – every time you check a product or service’s aggregated review score with a retailer or magazine, that’s Range voting in a shiny star-shaped wrapper.

  • Well, not having a mathematical bent I find all this confusing, but that is my fault. What is certain is that the present system of ” first past the post ” in UK elections does result very often in a minority percentage of the electorate achieving a majority in Parliament and we are supposed to believe that the ensuing government represents ” the will of the people ” For example Cameron was put into office by 38% of eligible voters; idem for the Leave vote in the EU referendum.

    Something is clearly wrong with this system but neither of the major political parties seems to find it in their interests to change it and no wonder there is no appetite for compromise when one of the political parties get into office. We really have something to learn from the way these things are organized in Germany.

  • What’s wrong with a qualified majority vote in case of a referendum as important as the one this country went through two years ago? The first past the post idiocy split the country in a near ungovernable wreck of a democracy and the less than integer handling of the aftershocks by politicians (the Corbyn faction of Labour absolutely not excepted) did its bit to make this country the laughing stock of the world.

  • 3 To suggest that the Modified Borda Count, MBC, is in the same category as FPTP, as Alex Scott does, is disingenuous. The MBC is non-majoritarian. But Range Voting is majoritarian, and, by the way, so is FPTP.
    To suggest Ms i or Mr j like or dislike x and y equally… but come on, they’re in politics, they have to make a decision. As in a restaurant, you could say you like vegetarian nut roast and beef stroganoff equally, but you can’t have both; as in politics, you have to make a decision. And we’re talking about collective decisions. Each individual can contribute to what then becomes the will of the people. Politics should be about choosing what is best for society as a whole; it’s about decision-making; it’s not about, or it should not be about, thinking, who like ‘this’ or ‘that’ equally.
    Also — and this is its biggest disadvantage — range voting could encourage the bigoted to be even more so.
    4 And to David Quinn, the British have been very good on voting systems. A Tory MP, Mr Hare, ‘invented’ PR-STV in 1857 — but was criticised by his fellow Tories for such a ‘hare-brained’ idea. (Actually, PR-STV had already been invented by a Dane, if not indeed by another Brit, Thomas Hill, in 1821.) And the British imposed PR-STV on the Irish, North and South, in the 1920 Settlement. What’s more, the Brits were behind the German MMP system — half FPTP and half PR-list — in 1949.
    Alas, what was good for the foreign goose was not considered good enough for the domestic gander.
    I might add that it was another Brit, Duncan Black, who was responsible for the re-birth of Social Choice Science in 1942; and yet another, Professor Sir Michael Dummett, who criticised that which afflicts so many — supporters of FPTP and range voting, etc., etc. — society is mesmerised, it seems, by what he called “the mystique of the majority.”

  • Dear British people,

    Is the discussion on Brexit exclusively British? It surely also affects us at the other side of the Norh Sea…. Would you allow me to comment on it? If only because we like you lot over there….? Here’s my observations.

    Brexit started my thinking about how minorities can turn minorities into majorities. I was of course aware of the Putins, Erdohan’s and Trumps crooking the system. But in the UK? No way! No way? I still am flabbergasted.

    As a start : your “first past the post” election system is a bit medieval. But OK: your build in checks and balances have long been able to cope with that. Until the refeendum on Brexit. Thanks David.

    So we (sorry: I mean you, but in the end I also mean we) are in a completely new situation. Where is this going to end? Violence? Unthinkable in the UK! (?)
    What can we do to avoid that?

    And that’s where other types of voting come into the spotlights. Cleary the old types of voting (yes/no) didn’t bring us “the real will of the people”. Nor will a new yes/no referendum on Brexit.

    But what’s the alternative? Peter Emerson is advocating his ideas (in a great parable, even understood by stupid Dutchies). Alex Scott is attacking Peter’s proposals and advocating other proposals .

    Now, I am not the expert on what is the best idea. But please: any new idea is better than your current approach/system. So pleaese sort it out! Instead of the current yes/no road to disaster.

  • Providing multiple options and having everyone express different amounts of support for each is a great idea, and much better than majoritarian binary votes.

    The problem with Borda count is that it forces voters to choose between options even if they have no preference between them, and inaccurately gives these very weak preferences the same weight as a very strong preference between two other options.

    Even in Borda’s day, when he convinced the Academy of Sciences to use his system, the voters quickly found how to manipulate it by putting their most dangerous rival at the bottom of their lists, and the least likely candidate in second place. When many people follow this strategy, it has the perverse effect of electing the worst possible candidate!

    Borda’s own peers abandoned Borda count after realizing its flaws, and it’s weird that people are still pushing it 200 years later. Much better voting systems have been invented since, which provide the same conceptual advantages of Borda without the huge strategic flaws.

  • Jean-Charles de Borda invented the Modified Borda Count, MBC. In an n-option ballot, voters may cast m preferences — and obviously, n ≥ m ≥ 1 — and points are awarded to (1st, 2nd … last) preferences cast according to the rule (m, m-1 … 1). Unfortunately, it soon became a BC, and points were awarded according to the rule (n, n-1 … 1) or even (n-1, n-2 … 0), which cannot cater for partial voting. And at worst, a BC can morph into a plurality vote. Not so an MBC.

    But Hmm has an amazing interpretation of history! The person in l’Académie des Sciences — by now renamed l’Institut Français — who in 1800 chose to abandon the BC and opt instead for majority voting was the new President of the Institute, a certain Napoléon Bonaparte. He then held three national referendums, in 1800, 1802 and 1804, so to become Consul, Consul for Life and then Emperor… and the rest, as they say, is history.

    Save to say that the two best measures of collective opinion are still the MBC and the Condorcet rule. And just as the sports team which scores the most goals is often the one which wins the most games — i.e., the team with the best goal difference is normally (but not always) the League Champion — so too in any preferential ballot, the MBC social choice is often the same as the Condorcet winner. And as quite a few social choice scientists have recommended, to be really really sure that the outcome of any ballot is indeed accurate, for any one voters’ profile, the two counts can be conducted, and if the MBC winner is the same as the Condorcet winner, you’re probably spot on! More of all this on http://www.deborda.org of course.

    Napoléon, by the way, was a bit of an amateur: he got only 99%. The first dictator to dictate properly — that is, to get 100% — was an Irishman. Sorry about that. But thus, in 1818, Bernardo O’Higgins became Chile’s El Supremo.

  • Very interesting string of comments above.
    I especially like the simple but obvious idea of having all options on the ballot paper.
    Concerning the methods described, despite all their plus points, it seems to me that fair and wise decisions would not be reached if those taking part in the vote either abstained because of complexity, or input erroneous entries due to misunderstanding.
    In an ideal world, a system would be both simple and yet be able to reflect the range of feelings held by voters.
    What would happen if we simply asked voters to distribute for example 13 points amongst the options / candidates on the ballot paper? In some combinations, you could rank 2 options equally, include small or large differences between options, and indicate reasonably accurately your relative views on each available option.
    And all you’d need would be a population who could add up to 13, or an automated system that did the maths for you.
    No idea what this is called, but if anyone wants to add the formal label, please be my guest.

  • Albert Weale’s argument (UK in a Changing Europe, 05.09.18) also covers this issue, and also makes a lot of sense, though one has to wonder whether sense itself is able to command much support in the current political climate. In addition, there is a potential problem with the three-way voting by rank offered by Weale and Greening – that of the Condorcet paradox – where ‘remain’ ≥ ‘WTO’ ≥ Chequers ≥ ‘remain’. The ‘will of the people’ may well be ambiguous, making the ways in which the votes are taken, counted and balanced critical. As Weale quite rightly says, “When there is a choice to be made over three alternatives, the order in which those alternatives are voted on matters greatly.”

    In this case, I suggest it is not only the order, but also the result of each two-way vote, which is critical. The logical order of the three-way choice facing Parliament, and thus the country, appears to be somewhat uncontroversial. We have voted to leave, so the next question is Chequers (or what we might term the Article 50 deal) versus no deal.

    The unanswered question is whether or not the winner of the second vote is actually preferred to the remain option (i.e. suspend or rescind Article 50). Some might object that Article 50 cannot, under the present rules of the EU, be either suspended or rescinded. But this is to deny any democratic or other public decision to challenge present laws, and hence change or re-define them. Whatever the outcome, the lawyers will have a field day, one way or another.

    I suggest that the appropriate form of a ‘peoples’ vote’ should be:

    1 Do you accept the outcome of Article 50 – which ever it actually turns out to be (either Article 50 deal or no deal) – Yes or No

    2 Given the result of question 1, do you consider this result to be better than Remain? – Yes or No

    We have been told, repeatedly, that no deal is better than a bad deal, but the current argument appears to be about what a ‘bad deal’ actually means – relative to what?

  • Three options, A, B and C. And 13 points. (Keith Medhurst.) So I could give option A 8 points, for example, and then with my remaining 5 points, I could give B 5 and C 0, or B 4 and C 1, or B 3 and C 2, etc. In all, then, giving A between 13 and 0 points, I think there are over 90 different ways of voting. Simple? And then there’s B and C. Still simple? And tactical voting?

    With the MBC, however, it’s A-B-C, A-C-B; B-C-A, B-A-C; or C-B-A and C-A-B. Six ways of voting. And because its the MBC (and not the BC), tactical temptations are minimal.