There are many ways to hold a three-way vote, none optimal, says L. Alan Winters of the University of Sussex

The case for a second Brexit referendum to leave the European Union relies on at least one of two premises.

First, that in June 2016 it was insufficiently clear what “leave” would entail, but that this is now clearer. And second, that trying to combine direct and parliamentary forms of democracy has so confounded Britain’s usual political decision-making processes that the resulting impasse can be resolved only by going back to the people.

Each of these premises suggests that if the second referendum is to have any political legitimacy, it will need to comprise three questions, which can be loosely characterised as: - Leave with no deal - Leave with the prime minister’s deal - Remain in the EU on unchanged terms (which, thanks to the Court of Justice of the European Union, we now know that Britain can decide to do unilaterally)

This short essay considers the pros and cons of three approaches to organising such a referendum: a two-round referendum, an alternative vote and what is known as a Condorcet vote. The uncomfortable but blunt conclusion is that there is no simple optimum. Serious political work will be needed to explain and legitimise whichever approach is chosen.

A two-round referendum

The decision facing the electorate is essentially sequential: the decision to leave inevitably precedes the implementation of any post-leave arrangements. Economists solve such problems by backward induction, starting with the last step of the actual process (how you leave) and working backwards to determine the best first step (whether to leave). Thus, a rational approach would ask in:

Round 1: “If we leave the EU, would you prefer ‘no deal’ or ‘the prime minister’s deal’?”

And then, when the answer is known, to ask (say, a month later to allow time for campaigning):

Round 2: “Now you know how we would leave, would you prefer to leave or to remain?”

This ordering solves the fundamental flaw of the referendum in June 2016—that no-one knew what “leave”’ meant. It is clearly preferable to the alternative ordering which asks first whether you want to leave and then later (or at the same time) which leave option you prefer.

This process might appear to be complicated, but in fact, this is the way we take decisions every day. Suppose you can spend your evening either going to a concert or having dinner. You see what the best concert is and what the best available restaurant is, and then decide whether to opt for a concert or dinner.

There is, however, a potential problem. If Remainers believe that “remain” will defeat “no deal” in Round 2, they may game the system by voting for “no deal” in Round 1, even though they actually prefer “deal” to “no deal”. Of course, this presupposes that they have huge confidence that they know the outcome of Round 2 and/or that they do not think that “no deal” will be such a disaster. Thus, it may not be such a big problem in practice (as I explain in a recent UKTPO blog post here).

Objections to the two-round model include that it has not previously been used in Britain and that it would take more time than a single vote. However, these concerns should not be decisive if the model is preferred on other grounds.

An alternative vote

This approach has all three questions on the ballot and asks voters to indicate their first choice and, if they wish, a second one. If one option receives 50% of the votes, it wins. Otherwise, the question receiving the fewest first preferences is eliminated and its proponents’ second choices are distributed over the remaining options. The one that has most votes after this is the winner.

The alternative vote method is used in a number of real polls, such as the election for London’s mayor. However, like any electoral system, it is capable of producing perverse results. For example, one can easily imagine outcomes where the winning choice commands the smallest number of first- and second-votes from the whole electorate. Moreover it, too, is open to gaming: voters may use their first choices to engineer the elimination of one option if they believe that the resulting distribution of second choices will achieve the outcome they desire.

The alternative vote tends to favour extreme outcomes over compromise ones (which are no-one’s first choice). This may be an important consideration given the divisions that Brexit has already created.

A Condorcet vote

This proposal is named after an eighteenth-century French philosopher. It consists of a single ballot that runs three, two-way races: A versus B, B versus C and C versus A.

If one option defeats the other two, it is obviously the winner. But one can also get perpetual circles, whereby A beats B, B beats C and C beats A, and in this case one needs a tie-breaker—perhaps the option that receives the largest number of votes in total.

The attraction of this approach is that it requires just a single visit to the voting booth. But, on the other hand, it’s a bit complicated for some people to understand and it’s never been used before (and it’s French).

Is it impossible? Yes, it’s impossible

If it seems impossible to devise a method for aggregating individual preferences among three alternatives that always meet our idea of rationality (ie, that is free of paradoxes), the reason is because it is indeed impossible. This sort of conundrum leading to suboptimal outcomes was explained by Kenneth Arrow, an economist and Nobel laureate, in 1951. And that is, pace David Cameron, one of the reasons to prefer representative over direct democracy for making complex decisions.

Yet Britain must decide, and do so in a way that commands reasonable legitimacy. This requires, at least, that the voting method be properly explained to the electorate. Moreover, if one is concerned about strategic gaming, it would be wise to ban opinion polls for the duration of the campaign. But that opens up a whole other set of controversial questions.

(A deeper explanation of the basic maths behind the voting methods described above appears in an appendix to this commentary here.)

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L. Alan Winters is the director of the UK Trade Policy Observatory at the University of Sussex.