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

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

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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).

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.

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.