Does Brexit demand non-binary decision-making?

Guest post by Peter Emerson, director of the de Borda Institute in Belfast.

Problems which are not binary cannot best be resolved with binary votes. This truism was discovered in 105 CE by Pliny the Younger. The scene was a Roman court of law, and the accused was charged with murder. There were three options: A, acquittal, B banishment and C capital punishment, and, as with Brexit, no majority for anything. The young Pliny realised that if the question was “execution, yes-or-no?” all the A and B supporters would vote ‘no’; but if you asked “innocent, yes-or-no?” the B and C supporters would vote ‘no’, and so on. In a nutshell, with that particular jury and their particular preferences, the answer would depend entirely on the question.

Brexit and Voting Theory

In February 2016, four months before the Brexit referendum, the de Borda Institute issued a press release to suggest a three-option ballot: ‘in the EU’, ‘the EEA’ or ‘WTO’? We also warned that if the question was to be ‘yes-or-no?’ – (‘remain-or-leave?’) – the answer would be ‘no’ (‘leave’).

But we are where we are; and there are lots of options, with again no majority for any of them. So let’s look at the theory, just to see how illogical it is to use binary voting when the problem is not binary.

Imagine a simple example: a parliament of just 9 MPs in a debate on three options, A, B and C, and let’s assume they have the following (1st, 2nd, 3rd) preferences: 4 like ABC; 2 opt for BCA; and the 3 C supporters are split, 2 want CBA and 1 wants CAB.

Preferences No of voters
4 2 2 1
1st A B C C
2nd B C B A
3rd C A A B

A cursory glance suggests that A with the most 1st preferences but just as many 3rd preferences is very divisive. Opinions on C are a bit mixed. So maybe B, the 1st or 2nd preference of all but one MP, best represents the collective will.

Binary Voting

The British parliament, like the Dáil, uses majority voting: the question is either “option ‘x’, yes-or-no?” or maybe “option ‘x’ versus option ‘y’?”. So what happens in the above scenario? If the question is the former category, a majority of 5 doesn’t want A, of 7 doesn’t like B, and of 6 doesn’t give C a 1st preference. So the agreement fails; a ‘no deal’ fails; a second referendum fails; every idea fails. Such is the present scenario: there’s no majority in the House of Commons for anything.

So let us try “option ‘x’ versus option ‘y’?” questions instead, and with three options, we’ll have to have two votes in order to get a ‘fair’ outcome. Assuming everyone votes in accordance with their preferences as shown in the table, if the first round:

  • is A v B, which A wins 5 to 4, the second round will be A v C, and C will win, again 5 to 4;

but if the first round:

  • is B v C, which B wins 6 to 3, for a second round of B v A, then A will win 5 to 4.

and if the first round:

  • is C v A, which C wins 5 to 4, for a second round of C v B, B will win 6 to 3.

In summary, if A is more popular than B which is more popular than C which is more popular than A which… which goes round and round for ever – a ‘cycle’ or ‘the ‘the paradox of [binary] voting’ – the answer could be anything! The agreement succeeds; a ‘no deal’ succeeds; a second referendum succeeds; every idea could succeed. So the whole process is open to manipulation. In a nutshell, majority voting is no good.

Multi-option voting

Let us try multi-option voting instead, as has now been suggested for a second referendum (but not yet for parliament). There are quite a few methodologies: choose the option with the most 1st preferences, (plurality voting), eliminate the one with the least (the alternative vote, AV), choose the one with the highest average preference (Borda count, BC), and so on. OK, let’s have a look.

  1. In plurality voting (like the UK’s first-past-the-post electoral system), the option with the most 1st preferences, in this case A with 4, is the winner.  This methodology considers only the voters’ 1st preferences.
  2. With the two-round system, TRS (as in French elections), if no one option gets an absolute majority in the first round, the two leading options – in the above example, A on 4 and C on 3 – go into a second round majority vote, which C wins by 5 to 4.
  3. With AV (which Australia uses)[1], the least popular option B is eliminated and its 2 votes go to B’s voters’ 2nd preference, option C, and so C wins this count as well; again it’s C’s 5 to A’s 4. Both AV and TRS consider only some of the voters’ 2nd and subsequent preferences. In the above example, B’s six 2nd preferences are not even counted!
  4. Or there is a BC (as in part of Slovenia’s electoral system) – in our example, a 1st preference gets 3 points, a 2nd gets 2 and a 3rd gets 1 – to give scores of A 18, B 19 and C 17, so the winner is now B.  Like the Condorcet rule, the BC takes into consideration every preference cast by every voter. It is accurate. Indeed, “when there are more than two [options]” the Borda and Condorcet rules are “the two best interpretations of ‘majority rule’”  (Iain McLean, Oxford Concise Dictionary of Politics, 2003, p 139).

The Way Forward

It is worth repeating that problems which are not binary cannot best be resolved with binary votes.  If the UK is going to get out of this Brexit mess, it could indeed have an ‘indicative vote’ in parliament, to see if there is any idea which has majority support.  To take a series of majority votes, however, would be “daft” – to quote Lord Desai’s description of that methodology when it was used on five options of Lords reform[2]… and every vote was lost!

The appropriate voting procedure would be for the House of Commons to have a free and preferential vote under the rules laid down for the modified Borda count, MBC[3], in which case, something is bound to come out on top!

Another possibility is a second referendum, but let us first admit that the 2016 vote was not good. Not only was a lot of the campaigning very poor, to put it at its mildest, but the methodology itself was “blunt” – to use Professor Vernon Bogdanor’s phrase. Any two-option ballot, now, would be unfair to at least some of the electorate. The wiser course would be to follow the New Zealand example: ask an independent commission to determine how many and which options should be on the ballot paper – they held a five-option referendum in 1992, admittedly under a TRS system; this Institute would recommend at least four options, and probably a maximum of six – and then hold a preferential referendum under the rules of the MBC.

Another recent idea which this Institute proposed last September but which is only now gaining some traction is for a citizens’ assembly, much on the lines of Ireland’s recent exercise. Inter alia, the latter recommended that consideration be given to multi-option referendums. Unfortunately, the citizens took this decision by a majority vote… which is a bit like coming to a peace agreement by first having a punch-up.  Then, in what this Institute thinks was the ‘experts panel’s’ last-ditch stand to defend majority voting, the latter asked the Citizens’ Assembly to vote on how such a multi-option referendum should be counted, by another majority vote. As if there are only two ways multi-option votes can be counted; this is nonsense!

Sadly, there are many politicians and political scientists who are as it were mesmerised by what the late Professor Sir Michael Dummett called “the mystique of the majority”  (his Principles of Electoral Reform, 1997, OUP, p. 71).


One of the reasons why the world is in a mess is because of its use of adversarial voting procedures. There’s Brexit. There’s Catalonia and an illegal binary referendum, which Republika Srpska wants to emulate: it could start a war. There was Scotland, which Donetsk did emulate, and it did start a war. And there is the overblown ego of Trump, in charge of another binary mess.

So what can Ireland do? Our electoral system is pretty good. But our majority vote decision-making in Dáil Éireann is terrible! Is it not time we used (electronic) preferential voting? And then we could suggest the House of Commons does the same.

[1] AV is the same as Ireland’s STV except that it is not PR, so it is held in single-seat constituencies.

[2] He proposed (but did not name) a BC (Hansard, 22.1.2003).

[3] In an MBC on n options, a voter may cast m preferences, so n ≥ m ≥ 1, points are awarded to (1st , 2nd … last) preferences cast according to the rule

(m, m-1 … 1)

and not

(n, n-1 … 1) or (n-1, n-2 … 0).

The above (m, m-1 … 1) rule was first advocated by Jean-Charles de Borda in 1770.


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