Let’s say we do have a “People’s Vote” on Brexit. How should the ballot be structured?

Any second referendum is likely to be regarded by significant portions of the population as a political stitch-up.

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Recent polls show that public support for a second referendum, a “People’s Vote”, is growing. But if there is to be another referendum, should “Remain in the EU” be an option? Is it simplest to omit “Remain” from the ballot, and to pose a straight choice between “Deal” and “No deal”?

Either way, the 2016 referendum, existing withdrawal legislation and the Conservative and Labour 2017 manifesto pledges would stand. And with only two options to choose from, the winner would, by definition, command the majority of valid votes.

But omitting “Remain” would be contentious. Remainers could say that voters were denied a valid option, which might be either the first or second preference of a national majority. A spoilt ballot campaign would be likely, endangering the popular sense of legitimacy.

But if “Remain” is an option, how or where on the ballot paper should it appear? This is a deceptively important question. The key to getting what you want from a referendum is to know how to design it: you must structure the ballot and select the formula most likely to turn voters’ individual decisions into your social choice.

Most Remainers’ preferred design would be a categorical choice between three options: “Leave + Deal”, “Leave + No Deal”, and “Remain”, with the winner being whichever wins the most votes – a  plurality, not necessarily the majority. But this clearly splits the “Leave” vote. Even if the two “Leave” options combined were to win a majority, “Remain” would almost certainly win the plurality, and hence the referendum. This is why a “People’s Vote” on this model is so attractive to Remainers.

If it were decided that a majority of valid votes should be required, a two-round system could be used. There are three possibilities here. In the first round, voters might choose between “Leave” and “Remain”. If “Leave” won again, the second round offers “Deal” or “No deal”.

Alternatively, all three options could be offered in the first round, with the two most popular going head-to-head in the second. Or the first round could offer “Leave + Deal” and “Leave + No deal”, with the winner going up against “Remain” in the second.

The problem here is that voters are asked to vote again twice, in effect, being asked to vote in three referendums. It is possible to put both rounds on the same ballot paper, but that does not resolve the problem that the ordering of options (what is offered in which round) influences the outcome.

Alternatively, voters could rank their preferences, the option with the fewest first preferences is eliminated, and votes are redistributed according to second preferences, until one option achieves a majority. This design makes “Hard Brexit” and “No Brexit” more likely, as “Leave + Deal” would probably be eliminated first.

The problem is that this method seems to award the deciding votes to the supporters of the least popular option, whose papers seem to be “counted twice”. The overall winner may also not have won the most first preferences, generating claims that it really came second. Questions over whether it really was the real “will of the people”, having failed to secure a true majority, would be divisive.

The promise that a second referendum would unite the country is probably illusory. The politics of referendum design would reopen today’s divisions, above all the problem of “Remain” appearing on the ballot. Omitting it ignores a possible and popular option; but including it discards two other “people’s votes”, 2016 and 2017 (three if you include the Conservatives’ 2015 manifesto commitment).

Any second referendum is therefore likely to be regarded by significant portions of the population as a political stitch-up. It seems, then, that, if we are to have referendums, they have to be carefully designed in the first place. Retrofitting doesn’t work.

Christopher Fear is a teaching fellow, and James Connelly a professor of political theory, in the Department of Politics, University of Hull.