The Brexit vote problem

How should the country (through a House of Commons vote or even through a second referendum) decide between several different options? There are two major difficulties.

First, how many and which alternatives should be considered in any vote? One aspect of democratic legitimacy requires that any alternative that commands some support (say 5%, but this can be adjusted) must be on the agenda. This is the open agenda principle. It implies that no alternative that has some support can be excluded on the grounds that it is unfeasible, economically damaging, undemocratic or for some other reason. 

If an alternative has some support, then that alternative is good enough for some people and should, therefore, be among the set of alternatives considered, irrespective of whether others think it is unacceptable. So, even if the claim that ‘No Deal’ is catastrophically costly were valid, this cannot be an argument for excluding the No Deal option from consideration.

Similarly, even if the claim that in 2016 people knew Brexit meant ‘Hard Brexit’ and that any revisiting of the 2016 Brexit decision is undemocratic (as people have already voted) were valid, this cannot be an argument for excluding ‘Soft Brexit’, the ‘Norway option’ or Remain from the agenda as not everyone agrees. 

Hence, it is clear that the open agenda principle implies that all the various types of Brexit as well as Remain should be among the alternatives that the voting constituency (whether the MPs or the people) considers.

Second, what procedure should be used to choose among the alternatives? If there were only two clearly defined alternatives, the democratic choice is easy: one of the alternatives would command a majority and would be chosen. But EU Exit is a complex issue about which we have learnt much over the past 30 months or so, and there are clearly more than two alternatives that command some support (No Deal, Canada++, the Prime Minister’s Deal, Norway+, Remain, etc.).

With three or more alternatives on the same ballot, a single vote is unlikely to result in any one alternative obtaining a strict majority and voters (MPs in a Commons vote or individual people in a referendum) may behave strategically and not vote for their most preferred option. Such a ballot would then not have democratic legitimacy. 

Minimal voting requirements

For any voting procedure to have democratic legitimacy, it should satisfy two minimal requirements. One is that if there exists an alternative, let’s call it A, that is preferred by a majority to any other B, C, D, E etc. in a head‐to‐head vote, the procedure selects alternative A.

This alternative is called the Condorcet winner (CW) after the 18th century philosopher and mathematician, the Marquis de Condorcet. Selecting the CW derives its legitimacy from the fact that it is stable, in the sense that once the CW is selected, there is no other alternative that can win a majority vote against it. 

The second requirement is that the procedure treats all alternatives in the same way. This is the neutrality principle that ensures fairness. It means that how the voting procedure works should not bias the final choice. Thus, voting procedures that treat different alternatives differently by, say, excluding some alternative at some stage of the procedure violate this principle. Violating neutrality would expose the procedure to the accusation that the process was rigged in some way.

Given the possibility of strategic voting, a body of academic research using game theory (including work by one of us; see Bag et al. 2009) shows that procedures designed with only one round of voting (including the single transferable vote) are insufficient to ensure that the CW is selected.

This deficiency undermines the legitimacy of any standard one-round voting procedures. But the deficiency can be overcome by a sequential voting scheme in which in each round, one alternative is eliminated (see Bag et al. 2009 and Rasch 2000 for an overview of voting rules used in various Parliaments).

An example of such a procedure is binary sequential voting in which in each round, voters choose between only two alternatives. But this procedure does not obey the neutrality principle because some alternatives are considered before others, and it can be shown that changing the order in which different alternatives are presented can affect the outcome.

Another example of the above is what we call the weakest link procedure (a bit like the TV show), and it is what we propose. This is a multi-round election in which in each round, voters (MPs or the people) would vote between all remaining alternatives and the one with the least votes would be eliminated. Voting continues until only one alternative is left.

This procedure satisfies the principle of neutrality and can ensure that the CW is selected (if there is one). We believe that there is a strong case for adopting it to resolve the Brexit impasse in the House of Commons, or in any subsequent referendum.

The basic idea can be understood by considering the final round of the weakest link procedure. At that point, there will be two options, for example, the government’s deal and No Deal. At that point the best thing for each voter to do is to vote for his or her preferred option. Hence, whichever of the two is preferred by the majority will be chosen in the showdown vote in the last round.

If we then work backwards to the previous round when there are three alternatives (for example, the government’s deal, No Deal, and Norway++) and it is known that if a CW reaches the last round (with two options), then it will win, as shown above.

So in this penultimate round the rational strategy for a majority of voters is to make sure that the CW is not eliminated and, therefore, to vote for it. So, by a process of backward induction, we can see that the weakest link procedure will get us to the CW.

Voting schemes similar to the weakest link procedure that we propose have become relatively widespread. Practical examples other than The Weakest Link TV show include the selection of the host city for the Olympics, leadership elections in the Conservative party and Strictly Come Dancing(up till the final).

Objection and the next steps

Turning to practical matters: how can the House of Commons adopt our proposed weakest link procedure when their voting system is already fixed? One possibility is for the House to adopt our procedure in order to arrive at an indicative choice, and then proceed to its usual voting procedure of either accepting or rejecting it.

An objection to our procedure is that it may require several ballots. For example, with four alternatives, three ballots are required. While organising multiple ballots in the House of Commons is not difficult, it may be argued that for a public referendum it is too costly.

If the cost of multiple ballots is considered too high, an (imperfect) compromise might be to eliminate more than one alternative in each round; in the extreme limiting the number of rounds to two. Two‐round voting is common (in presidential elections in France and Brazil, for example) and remains preferable to one round of voting.

Another possibility could be to have the weakest link procedure at the level of the House of Commons and then take the last two alternatives to a referendum.

A common argument, repeated over and over again in the media in recent days, is that there is ‘no majority for any alternative in the House of Commons or in the country’. This is misleading and does not undermine our proposal that there is a fair, legitimate and democratic way to come to a decision. It is misleading because the argument that there is no majority for any alternative refers to a situation in which the vote is a choice between the alternative and ‘not that alternative’.

Furthermore, it does not undermine our proposal because the argument does not imply that there is no CW. Requiring an alternative to be preferred by the majority to ‘not that alternative’ is too stringent a requirement in this complex situation as the ‘not that alternative’ includes many possibilities.

What is more appropriate in the given context with multiple alternatives is how each alternative compares against every other alternative. If there is a CW among the set of alternatives considered, then there is an alternative that commands a majority against all the others.

There may, of course, be no alternative that commands a majority against all other alternatives (that there is no CW). What then? While this is logically possible, research shows that such a problem is most acute for small committees and is least acute when many individuals vote on a few alternatives (the situation in a House of Commons vote or a second referendum; see, for example, Jones et al. 1995).

Even if there is no CW, our procedure still has much to recommend it because it ensures that an alternative that commands minority support in head‐to‐head pair-wise voting cannot be selected.