Condorcet’s Brexit trainwreck

Brexit as a Condorcet paradox.

Let’s stand a little back from the Brexit trainwreck – the kind you get when Dr. Evil hacks the signalling at Clapham Junction  in rush hour. I have no choice, since as an expatriate I, and a million like me, get no vote.




The options are:
A – Exit with no deal
B – the May deal
C – A softer Brexit (“BINO”) on Norway or Jersey lines; undefined, but probably with staying in most of the EU single market, few restrictions on movement from EU countries, and no say in the rulemaking
D – Remain.

The estimable Simon Wren-Lewis estimates the current factional breakdown of the House of Commons (n=630) over Brexit:

Brexiters – No Deal                                               100
May loyalists – No freedom of movement       200
People’s Vote [second referendum]                  150
Corbyn loyalists                                                       30
Soft Brexit                                                               150

This leads to the following first-choice vote predictions:
     A: 100 for, 530 against
     B: 200 for, 430 against (actual vote was 202 to 432)
     C: <180 for, >450 against
     D: <150 for, >480 against

There is a large majority against anything at all. A neater real-life example of the Condorcet paradox you couldn’t get.

Kenneth Arrow wrapped Condorcet in bomb-proof mathematics to show (hope I get this right) that there is no possible voting scheme (in the most general sense of an algorithm that aggregates ranked preferences) obeying very plausible ground rules that offers a guarantee against circular majorities.

One difficulty with the Arrow proof is that it doesn’t suggest how likely the paradox is. If it’s an asteroid strike risk, we can ignore it for practical purposes. As the number of participants and the number of choices both grow, it’s plausible that anomalies get smoothed out and the system gets better behaved.

It would for instance be hard to get a voting paradox over US health care. The policy space can be represented as a continuum between a full tax-funded NHS at one end and an unregulated health and health insurance market at the other. Voters at the poles will have second and third choices reflecting the continuum of options. Democratic voting lands you somewhere in the middle, as is indeed the case. But you can’t be dogmatic about this. I venture an argument here  that when policies are in a means-ends relationship, circular majorities are in fact pretty likely.

Brexit is different to either in that the choices are discontinuous and heavily constrained by institutional factors: the Article 50 process, the EU’s negotiating position, the Good Friday Agreement about Northern Ireland. Not to mention outright lunacy on the part of the hard Brexiters.

The way you assemble a majority, in the Commons or the country, from the pieces of the trainwreck is sequential ranked voting. But then it’s crucial who goes first. Imagine a second referendum with the structure:

Q1 = approve policy A, Yes/No
       If the majority is against Q1, then
       Q2 = approve policy B, Yes/No
              If the majority is against Q2, then
              Q3 = approve policy C, Yes/No
                    If the majority is against Q3, then
                    Q4 = approve policy D, Yes/No
To guarantee a result, you can omit the last question and just say that D is the default.

As I’m a Remainer, I would like the list to start with the crazy hard Brexit and go A→B→C→D. That gives Remain the best chance. But Brexiters would like the reverse order: D→C→B→A. This favours their preferred option. I can’t see a good formal argument for either scheme. What I can point to is the near-universal example of rules of procedure in deliberative assemblies: you vote first on the most radical amendment, then the next most radical. This scheme has a conservative bias towards the status quo, which looks like common sense. We know the status quo works minimally in that it has allowed the assembly to meet and deliberate.

Author: James Wimberley

James Wimberley (b. 1946, an Englishman raised in the Channel Islands. three adult children) is a former career international bureaucrat with the Council of Europe in Strasbourg. His main achievements there were the Lisbon Convention on recognition of qualifications and the Kosovo law on school education. He retired in 2006 to a little white house in Andalucia, His first wife Patricia Morris died in 2009 after a long illness. He remarried in 2011. to the former Brazilian TV actress Lu Mendonça. The cat overlords are now three. I suppose I've been invited to join real scholars on the list because my skills, acquired in a decade of technical assistance work in eastern Europe, include being able to ask faux-naïf questions like the exotic Persians and Chinese of eighteenth-century philosophical fiction. So I'm quite comfortable in the role of country-cousin blogger with a European perspective. The other specialised skill I learnt was making toasts with a moral in the course of drunken Caucasian banquets. I'm open to expenses-paid offers to retell Noah the great Armenian and Columbus, the orange, and university reform in Georgia. James Wimberley's occasional publications on the web

17 thoughts on “Condorcet’s Brexit trainwreck”

  1. So what would be considered the most radical proposal in this case? I’m assuming it’s the one that makes the most sweeping changes (somehow measured) in the status quo, but is the “status quo” at this point “in the EU” or “heading for a hard Brexit?”

    1. There again, you can argue either way. I think that “being a member of the EU since 1973” has greater weight as a starting point than “winning a badly designed consultative referendum with interference from Russian spooks and American rightists”, but tastes differ.

    1. Ranked choice (in the simple, common version, at least) eliminates the option with the fewest first-choice votes first, then distributes the second-choice votes of those ballots among the surviving alternatives. Repeat until someone has a majority. So it has an implicit order. And if one option is the first choice of very few but the second choice of almost everyone, ranked choice can lead to counterintuitive results.

      1. Under RCV, it seems to me not that there is an implicit order but rather each person gets to create their own order rather . And there is nothing counterintuitive (to me anyway) if tastes are so divergent that no majority can agree on the best option to look instead for a majority to agree on what is the least bad option: which is the situation of your example.

      2. So, as should be obvious by this point, I am not an expert on voting. Does the following alternative to RCV have a name, and if so, is it any better?

        1) RCV has several stages. In the first stage, people rank all alternatives from most preferred (1) to least preferred (N). At each stage, if none of the alternatives still in contention commands more than 50% of the vote, the alternative with the fewest votes is dropped. The votes of those who are in favor of this least supported alternative are redistributed to all the other alternatives based on the next preferred choice of each individual. Continue until an alternative exceeds 50% of the vote.

        2) Instead of dropping the least supported alternative at each stage,
        * drop each choice in turn and redistribute the votes
        * see if any of the remaining choices exceeds 50%.

        3) If more than one choice exceeds 50% (possible since we are calculating multiple paths) select the one that exceeds it the most.

        So, consider 4 possibilities:
        STAGE 1: A(ranked 1st by 40%) , B(by 10%), C(by 20%) & D(by 30%).

        None pass the 50% threshold, so try dropping each in turn and redistribute the votes:
        STAGE 2:
        Drop A: B(10% + 30%), C(20% + 5%), D(30% + 5%)
        Drop B: A(40%+9%), C(20%+0%), D(30%+1%)
        Drop C: A(40%+5%), B(10%+10%), D(30%+5%)
        Drop D: A(40%+5%), B(10%+15%), C(20%+10%)
        In parentheses above, the first percentage is the first round vote for that choice, the 2nd %age, following the plus sign, is what gets redistributed to that choice from the one that is dropped.

        In Stage 2, A almost makes it when B is dropped, but not quite. Moving to stage 3, we have to track each sequence. That is, see what happens when having already dropped A, we drop B, C or D; having already dropped B, we drop A, C or D, etc.

        STAGE 3:
        Having dropped A [recall the results above: B(40%), C(25%), D(35%)], calculate what happen when
        B is dropped: C(25% + 24%), D(35%+16%)
        C is dropped: B(40% + 9%), D(35%+16%)
        D is dropped: B(40%+10%), C(25%+25%)

        Having dropped B [A(49%), C(20%), D(31%)]
        drop A: C(20% + 29%), D(31%+20%)
        drop C: A(49%+10%), D(21%+10%)
        drop D: A(49%+5%), C(20%+26%)

        Perform similar calculations for “Having dropped C” & “Having dropped D”.
        Notice that none of the totals are as high as A when first B and then C are dropped (59%), except those for A when first C and then B are dropped. A is the winner.

        I have not read, don’t pretend to understand, Arrow’s proof of his theorem, so this may well run afoul of that: except that people have not been asked to vote on alternatives in paired horseraces. Rather they have been asked to rank alternatives and at each stage, all possible orders of dropping one is considered. The first stage in which any choice exceeds 50% is the last stage of calculation and the choice with the most support in that stage is the winner (in case more than one choice exceeds 50% in that stage)

    2. Oh. Looks plausible. But does RCV rule out the Condorcet paradox? Is Arrow’s proof is right, it can’t offer an absolute guarantee – but maybe it can reduce the risk to an acceptable level.

      RCV in the UK would be against Tradition. Perhaps the risk of national disaster will be enough to allow the needed innovation, in a referendum or in the Commons.

      Krugman plays down the disaster risk, but he doesn’t regularly talk to British businessmen like Andy Haldane of the Bank of England. The businessmen have stopped being polite about hard Brexit in public; you can imagine what they say in private.

  2. Nice article, thanks. This is good: “Hagfish produce slime the way humans produce opinions—readily, swiftly, defensively, and prodigiously.” It’s funny the things that stick in the mind. My not very consequential hagfish post (though I still like it) was in 2012.

    Brexit delay takes both parties, as Barnier has just reminded Westminster MPs. The EU would surely be ready to extend the deadline for a second referendum or a general election. For Corbyn’s different-coloured unicorn? Not so sure. For May’s (“let’s have another go at squaring the Ulster circle with clever wording”)? I doubt it.

  3. I’m so old, I remember when

    a conservative bias towards the status quo

    was considered the acme of Conservatism.

  4. The Brexit issues are complicated, for sure. The multiple-choice voting adds another layer of complexity, one that may be insoluble. Perhaps the only way around that is to reduce the choices to two.

    Reading your post has led me to read up on the Condorcet paradox, which led me to peruse some sources about Pareto efficiency. Then I read the article on the Clapham Junction Rail Crash. Interesting how a series of preventable but seemingly innocuous small breakdowns led to a catastrophe. The most surprising thing to me was that it was just 30 years ago. I would have guessed that by the 1980’s there would have been much stricter enforcement of safety regulations for things like fatigue and inspections of work.

    Somehow I got sidetracked onto the fascinating information about Hagfish slime. I can’t figure out how I got there from here, but it was a most interesting break.

    But my favorite part of your post was the picture at the top. Yes, that is truly a perfect photo of Brexit.

    1. Update–Now I see that it was an unrelated (but most interesting) comment from Keith that led me there. That, in turn, led me to your reference to your previous post on that subject, plus your delightful analogy to human opinions. I shall memorize that line.

      What a cool blog this is, even when (or perhaps, especially when) it’s not focused on dysfunctional American politics.

    2. The main problem here are mutually incompatible red lines that the various Brexit factions have. I don’t see a way out of this. It comes all down to them figuring out that you can’t simultaneously have your cake and eat it, too.

      Posing it as a voting problem is an interesting thought experiment, but still falls short of the complex underlying game theory issues.

      There are basically three options now (James’s Option C is not really on the table; it might have been two years ago, but too much time has been wasted): Crash out, Article 50 revocation, or taking the current deal. There really is no time left to arrange anything else within the next two months.

      May seems to be trying to run the clock down by forcing Parliament to decide between a no-deal Brexit and her deal, hoping that a majority will not want to be held responsible for the consequences of a no-deal Brexit. But too many of her own party are just fine with crashing out, so she would have to count on opposition support, and Corbyn wants a Brexit, too, and may count on May getting all the blame for a crash out. An Article 50 revocation would normally not be likely (because they Tory leadership is committed to a Brexit, one way or the other, and the government controls the time table in the Commons [1]), except that Bercow (Speaker of the House of Commons) is enough of a wild card that a cross-party majority may just be able to do it despite existing convention if too many people hate both alternatives.

      [1] Technically, Parliament has supremacy, but centuries of procedural precedent mean that the government gets to decide what the Commons vote on. If the government doesn’t like a bill, it can stop it from coming to a vote. The main exception here is a motion of no confidence, but that has already failed. On the other hand, Bercow did allow a normally unamendable business motion to be amended, signalling that he may not feel as bound by convention when push comes to shove. But that may end in a constitutional crisis.

    3. The photo caption for the train says it happened at the Gare Montparnasse in Paris in 1895. You’ll have to take it from there.

  5. I am still figuring out what I think about this algorithmic issue. (I suspect this will continue. And, could barely spell “algorithmic.” Took several tries. Need more caffeine.) Really though, since this is humans we’re talking of, is it a surprise?

    Personally I think they might as well vote again, and maybe a few more times after that. You can’t really get too much on a high horse when you barely won on such a big issue. Doing nothing is not the worst thing.

    Meanwhile, as I’ve said before … not nearly enough blame is being cast at the EU. Seriously. What is up with that? (I watched a few moments of Cumberbatch and turned it off. It is all so one-sided already.) I don’t understand Europeans. When did they become such doormats?

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