With the possibility of a second referendum gaining increasing support, what happens if more than two options are on such a ballot paper for voters to rank? Simon Kaye explains the prospect of a Condorcet cycle and considers alternatives. He concludes that, whatever the route taken, there will always be a majority who will find the outcome of the Brexit process to be far from their first choice.

After more than two years of Brexit negotiations, manoeuvres, follow-up polls, and parliamentary debates, the UK has arrived at a political impasse. There are now widely believed to be three potential future outcomes: leave with no deal, remain and forget the whole thing, or approve the government’s Withdrawal Agreement (and its attendant, carefully vague Political Declaration on the UK’s future relationship with the EU). There are frequent attempts by all parties to suggest that one or another of these potentialities is effectively ruled-out or illegitimate, but clearly this has not yet been persuasive.

What is striking is that many polls now indicate that a Condorcet cycle – a paradox of preferences – exists between these three options. This is a paradox which seemingly runs throughout UK politics, from the general public to the dispositions of elected MPs. But what is a Condorcet cycle? Famously explored by Kenneth Arrow, this is about an underlying paradox that can emerge when translating individual preferences into social choices. Imagine three voters choosing between three options.

Voter 1:            A > B > C

Voter 2:            B > C > A

Voter 3:            C > A > B

It is clear that there can be no majority winner among them: the vote is deadlocked, because by pairwise comparisons, each option is at some stage preferable to every other. Different methods of counting these votes will therefore produce varying outcomes, and in any situation a majority of participants will be disappointed by the result. They’ve tried to be democratic, but the final decision will always feel, to some extent, arbitrary.

When people are polled directly on these three options in pairwise comparisons, a similar pattern emerges. At least a few polls now indicate a majority would prefer May’s deal to remaining in the EU outright; almost all polls show that a majority prefers remaining in the EU to leaving; and a subset of these show an even clearer majority preference for remaining over a no-deal exit. Finally, a majority prefers leaving the EU without a deal to the government’s Withdrawal Agreement. Deal > Remain > No-deal > Deal.

To simplify further: the referendum showed that more people oppose remaining than support it. Polling shows that more people oppose the government’s deal than support it, and that more people oppose a no-deal exit than support it. Under all circumstances, a majority will end up displeased.

Reflective of this, at present no outright majority exists in the House of Commons for any of these three outcomes. Even with a strong pro-remain tendency in the average MP, the political mandate created by the 2016 referendum leads to a similar preference-impasse. Intuitively, it makes sense for a median position to be marginally preferred to simply forgetting the whole Brexit idea, among both public and politicians. It also makes sense that the risks associated with no-deal would create a majority preference around remaining if it were the only alternative, and that the compromises and gaps represented by the government’s deal would lead to overarching preferences for that no-deal Brexit in a pairwise comparison.

What can we do in a deadlock situation like this? Here are some possible escape routes:

1. Articulating deeper preferences. Going further than finding out everyone’s first choice through a simple vote – is one. This doesn’t rule out the possibility of Condorcet cycles, as in the example given above, but it does make them more unlikely. See the illustration from a recent Deltapoll survey below. In this poll, the government’s withdrawal agreement emerges as a ‘Condorcet winner’ – a middle option that trumps either outright remain or outright no-deal, as discussed by Simon Fisher. This could be achieved by a referendum using a voting system that allows voters to set out their preferences in order. But this is not uncontroversial: we’re not used to finding ways of endorsing an outcome that’s everybody’s second-favourite. And, perhaps worse, this could ultimately reveal an even deeper level to the Condorcet cycle.

2. Hand-off the decision to a smaller subset of representatives who are empowered to make an informed final decision. Or, more simply – representative democracy. The problem here is that our representatives may be liable to be just as divided as we are, but this is effectively the option that is being trialled now, over days of debating and voting in the Commons. A crucial component that can be added under these circumstances is deliberation, which is impossible to accomplish meaningfully for a whole population, but could make all the difference among 650 legislators. By hearing and offering reasons for the different alternatives, deliberation can sometimes eliminate Condorcet cycles through meta-agreement, where participants find they can agree on the most important benchmarks against which the alternatives should be measured, and/or the emergence of more rational preferences, which will tend to favour the ‘middle’ option in a set of three. This is effectively what the government is counting on. Unfortunately – as any MP can tell you – the Commons is seldom reflective of an ideal deliberative environment, and even if it were, you might nevertheless fail to resolve the deadlock.

3. Formulate a new alternative to disrupt existing preferences or to become a clearer overall preferred option. This could be the product of a deliberative process, or it could be suggested from outside and gather enough support to come into real contention. We might think of the often-mooted Norway model/EFTA Brexit in these terms: a different plan that could end up generating wide appeal among MPs, if not the general public. There are issues here too, though: a new option could simply further complicate the whole picture, leading to a Condorcet paradox between four alternatives instead of three, and further weakening the mandate for any one option.

4. Boil the options down to two alternatives. Condorcet cycles can’t emerge between fewer than three options. If the final decision is between just two options, then a clear winner is likely to emerge. This is probably the simplest response, but also arguably the most arbitrary one. Which two options should be on the table, when at least three seem to be quite equally disliked? How legitimate would a subsequent vote or referendum be in the eyes of those participating? And, of course, even a straight-forward two-way vote can result in near-deadlock. The 2016 referendum itself ended up with a result so close that it has been the backdrop for endless disputes ever since: a more emphatic victory either way would surely have led to a very different political reality now.

Option 2, rolling through the Commons as I write, may yet yield results. Nevertheless, it now seems clear that, however things play out, there will always be a majority in the UK that will find the ultimate outcome of the Brexit process to be very far from their first choice.


About the Author

Simon Kaye is Research Director at the Project for Modern Democracy think tank. He previously worked as an academic at King’s College London.




All articles posted on this blog give the views of the author(s), and not the position of LSE British Politics and Policy, nor of the London School of Economics and Political Science. Featured image credit: Pixabay (Public Domain).

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