As the debate about whether or not to have a second Brexit referendum continues, the form any such process might take remains unclear. Ahead of the launch of his new book on particpatory democracy, Albert Weale argues that caution should be exercised when considering the use of the Alternative Vote system in any future Brexit referendum. 

In a valuable blog on what question might be put to voters in a second Brexit referendum, Jess Sargeant, Alan Renwick and Meg Russell conclude that if a three-way option is put to voters, the alternative vote (AV) system could be the right one to use. The basis for this conclusion is that when three options are involved, the option that receives the single largest number of votes may not receive an overall majority. So some system is needed to find out if there is an all-round winner, and the AV system of voting will do this.

It is certainly true that when voters have to choose among three alternatives, the operation of majority voting gets quite complex. This is one of the reasons why, as I explain in my forthcoming The Will of the People: A Modern Myth, it is relying on a myth to talk about ‘the will of the people’ emerging from a referendum. This does not mean abandoning majority voting. But it does mean that we need to be careful in the way we apply the majority principle.

To see the importance of this, consider the sort of case that Sargeant, Renwick and Russell discuss, which I present in a simplified form in Table 1. Imagine three blocs of voters who have preferences over the three alternatives likely in a second referendum: Remain, a negotiated Deal and No Deal.

Table 1. Possible Preference Orderings

It is easy to see from this table that there is no clear majority preference. No alternative receives at least 50% plus one of the votes. Those who favour Leave are split between Deal and No Deal, so there is no majority winner. Even though 55% favour Leave in some form, there is no agreement among those Leavers as to what that form should be.

How can the principle of majority rule be applied in this case? One attractive answer to this question is to say that the majority winner should be the all-round winner. The all-round winner is the alternative that beats each of the other alternatives in a pair-wise contest. This alternative is also known as the Condorcet-winner, after the eighteenth century French aristocrat and mathematician who first formulated the idea.

In the example given in Table 1, there is a Condorcet-winner. It is the Deal. It beats Remain by 55% to 45%, because it unites blocs B and C. It beats No Deal by 65% to 35%, because it unites blocs A and B. If majority rule means favouring the alternative that beats all of the others in pair-wise voting, then we should ideally choose a voting system that produces a Condorcet-winner. But as Sargeant, Renwick and Russell themselves note, AV might not actually do that. AV invites voters to list the alternatives in their preferred order, eliminating in the first round the alternative that gets the least number of votes. The second preferences of the eliminated alternative would then be redistributed between the other two. Clearly, Deal would be eliminated in the first round, and Remain would win against No Deal. But Remain is not a Condorcet-winner. The reason for this is that AV rests on a sequence of voting in which the alternative that attracts the fewest number of first preference votes is never pitched against the winner at the second round, as shown in Figure 1.

Figure 1. The Sequential Logic of the Alternative Vote

Why, given this, would one think that AV could be the best system to use? Here the argument from Sargeant, Renwick and Russell moves, I think, in the wrong direction. It certainly moves too fast. The argument runs like this. In a system of pair-wise majority voting you cannot guarantee a Condorcet-winner. This possibility is illustrated in Table 2, which shows a modified set of possible preferences over the alternatives. Given those preferences a series of pair-wise votes would lead Remain beating No Deal (A+B is a majority against C), No Deal beating Deal (A+C is a majority against B) and Deal beating Remain (B+C is a majority against A). We go round in a cycle of majority preferences.

Table 2. Preference Orderings with No Condorcet-winner

Sargeant, Renwick and Russell then say that it is impossible to devise a voting system that eliminated the danger of such cycles. This is quite true. Given the pattern of preferences, there may be no determinate winner. However, this is because no voting system can perform the impossible. When you have the pattern of preferences displayed in Table 2, there is no determinate winner by any plausible voting rule. Yet, it is equally true that if there is a Condorcet-winner, then we should want to use a voting system that finds it. And as we have already seen, AV is not such a system. To find the Condorcet-winner, we need a voting system that allows all alternatives to be pitched against all other alternatives.

At this point someone might question the Condorcet test. Is it really the case that the all-round winner is the alternative that should be preferred? AV eliminates in the first round the alternative that secures the fewest number of first preferences in Table 1, which is the Deal, even though it is the Condorcet-winner. AV gives priority to those alternatives that have the highest proportion of first preferences, rather than trying to secure a compromise that a majority can agree on. So, if we want to give as many people as possible their first preference, this may be different from the Condorcet-winner.

But how good is this argument for ignoring the Condorcet-winner if there is one? Why should we give priority in a three-way contest to passionate minorities? One point in favour of giving priority to first preferences is that those favouring the Condorcet-winner may comprise only a very small section of voters, and most people do not want to end up with second-best. Yet, in reply to this point, although it is certainly true that Condorcet-winners may be the first preferences of only a small section of society, it is also true that they may in some cases be a large segment, and we do not know until the voting takes place. For example, in a three-way contest in which one alternative polls 34%, another 33% and a third 32%, AV would eliminate from any further consideration an alternative that had virtually as many first preferences as the other two.

Secondly, even when the Condorcet-winner is only the second preference of many people, it might still be close in their estimation to their first preference. What AV voters are in effect being confronted with is a voting method that gives them a gamble in the second round between a chance of their first preference and a chance of their last preference. Voters may well strongly prefer the certainty of their second-best to a gamble over their best and worst alternatives.

Thirdly, the compromise that the Condorcet test involves goes to the heart of democratic principles. Policy-making is not a horse race in which the punters cheer on their favourites, with those who cheer loudest winning the contest. It is a serious matter in which choices have real-world consequences. Referendums put a crucial element of those choices into the hands of voters. But in a world in which there are sincere and reasonably held differences of view, the practice of compromise becomes a democratic virtue against the principle of a minority ‘winner takes all’ result.

Some might say that a voting system that pitches all alternatives against the others is unnecessarily complicated for the voter. But a system that found the Condorcet-winner, if there is one, is no more complicated than AV. AV requires voters to list at least two alternatives in order of preference. A Condorcet counting system requires no more information. All the complexity is handled at the counting stage, not the voting stage.

Is this just a nerdish worry about voting systems? I think not. If you eliminate the Condorcet-winner, you eliminate that alternative which a majority of the electorate prefers over all the others. So if you end up with an alternative that is not a Condorcet-winner, you end up with a result that may face extensive opposition from a silenced majority, which is not a recipe for social harmony.  If you have a Condorcet cycle, the existence of such a silent majority is inevitable. But in the Brexit case, the likelihood of a cycle is quite low. It requires a large proportion of Remainers to favour No Deal as their second alternative or a large proportion of No Dealers to have Remain as their second alternative – possible but unlikely.

The first Brexit referendum showed us the dangers of stumbling into a democratic and constitutional mess without thinking through the implications. Sargeant, Renwick and Russell have given us some serious work to consider. Let us at least have a thorough discussion of all the implications of a second referendum before we decide what to do next.

About the author

Albert Weale is Emeritus Professor of Political Theory and Public Policy at UCL. His book The Will of the People: A Modern Myth is published by Polity Press.