This is part of a one-week discussion on a second Brexit referendum, with ten contributors. The other contributions are available here.

Let us peep into the Pandora’s Box labelled “Brexit referendum”. What we see is not a simple route map to a clear destination, but a jumble of conflicting ideas. The essential problem is easy to state, if hard to solve. It is that referendums conventionally pose a binary choice—but in this case, there are three distinct outcomes.

First, the United Kingdom could leave the European Union without a deal. Second, there could be some compromise arrangement, such as the Withdrawal Agreement negotiated between Theresa May’s government and the other 27 EU member states. Third, the UK could remain a member of the EU after all. (In order not to make the problem even more complex, you can consider other possible arrangements, such as “Norway plus”, under the general heading of “compromise”.)

The challenge is to find a democratic winner among these three options. This means either offering voters all three choices, and deciding how to measure their preferences, or dropping one of the three options and offering a binary choice between the other two. Let’s start with the three-option route—or rather, routes. All three could be offered on the same ballot paper. We would then have what’s known as an Alternative Vote referendum (the very system we rejected eight years ago for electing members to the House of Commons). Voters would be invited to state both their first and second preferences. If one of the options receives more than 50% of first preferences, the contest is over. But if none reaches the 50% mark, the least popular option is eliminated. Its supporters’ second preferences then come into play. For example, suppose the first-preference result is: A 40%, B 33%, C 27%. Option C is eliminated, and the second preferences of C’s supporters are counted. Suppose that 27% is divvied up as: A 9%, B 18%. Then the final result is: A 40+9 (49%); B 33+18 (51%). B is the winner. In this example, the clear “first-past-the-post” victor, option A, is not the winner. How would that be viewed by A’s supporters? Would the announcement of B’s victory heal the nation’s divisions? Just asking.

(For lovers of electoral theory, the Condorcet process could even make C the winner. The Marquis de Condorcet, an 18th century French philosopher, proposed that the way to choose among three options would be to count the votes as three binary choices: A versus B, A versus C, and B versus C. In our example, suppose C is the overwhelming second choice of both A’s and B’s first-choice supporters. It could well be that more people prefer C to either A and B. But under the Alternative Vote system it is eliminated after the first count.)

Yet another method is a two-stage referendum. One idea being considered in Downing Street is to ask voters in stage one to vote Remain or Leave. If Remain wins, that’s the end of the matter: the UK stays in the EU. But if Leave wins, then we move to stage two, say, a fortnight later. Voters would then choose between the government’s deal and no deal.

A neat solution? Not necessarily. Imagine the following conversation between a parent and their child:

Parent: “Shall we go out today or stay at home?”

Child: “What will we do if we go out?”

Parent: “We might go walking with Aunt Theresa, or read ‘The Canterbury Tales’ with Uncle Jacob. If you want to go out, I’ll lock the front door behind us, then toss a coin to decide where we go.”

Child: “That’s not fair. I like Aunt Theresa but can’t stand Uncle Jacob. How can I decide whether or not to go out if I’m not sure what ‘going out’ actually means?”

By the same token, voters who plump for Leave in stage one will not know what Leave actually entails. There are likely to be people who want a compromise deal with the EU, but fear crashing out without a deal. Should they vote Leave and risk a catastrophic decision at stage two—or vote Remain in order to avert disaster, even though it’s not their first choice? And what about those who want a “pure” exit, but see no point in a compromise that requires Britain to obey EU rules but with no say in them?

This is all wonderful stuff for games theorists, but it’s not obvious that a great national decision should depend on the extent to which millions of voters apply game theory to the way they vote.

Hence, there is no perfect way of asking voters to choose among three options. But if we are to have a binary referendum, one of the three options—Remain, Compromise or No Deal—must be left off the ballot paper. Which?

* “Remain”, on the grounds that the UK has voted to leave, and the issue now is simply to decide what kind of leave—even though polls find consistently that “Remain” is currently by far the most popular of the three options?

* “Compromise”, on the grounds that it is the least popular option among voters and cannot muster a majority in Parliament, even though most government ministers and business leaders think it is the least bad way of implementing Brexit?

* “No Deal”, because most ministers and business leaders, along with the Bank of England and a large majority of economists, warn of the extreme perils of this outcome, even though it is arguably the only way the keep the 2016 referendum promise to “take back control”?

There is no objectively “right” answer. Perhaps the best way forward is for Members of Parliament to use their judgment. A large majority oppose a No-Deal Brexit, fearing it would cripple Britain’s economy, society and public finances. They could feel they have a duty to prevent this—and that means voting for a “Remain v Compromise” referendum.

And all of this leads to a clear, if paradoxical, conclusion. In charting the way ahead, we cannot escape the need for political leadership and careful deliberation. For direct democracy to succeed, we need representative democracy to succeed first.

______________

Peter Kellner is the former president of YouGov, a polling firm, in Britain.