The Brexit referendum question was flawed in its design by ignoring Kenneth Arrow’s impossibility theorem. Thomas Colignatus explains why.

Theresa May’s government, with support from the UK Parliament, has adopted Brexit as its policy aim and has invoked Article 50. Yet, economic theory assumes rational agents, and even governments might be open for rational reconsideration.

The unsatisfactory referendum question

Based upon voting theory, the Brexit referendum question can be rejected as technically unsatisfactory. One could even argue that the UK government should have annulled the outcome based on this basis alone. Even more ambitiously, one might imagine that economists and political scientists across Europe take up this issue and hence provide a basis for the EU Commission to negotiate for a proper referendum question. The big question is why the UK procedures didn’t produce a sound referendum choice in the first place.

Renwick et al. (2016) in an opinion in The Telegraph June 14 protested: ‘A referendum result is democratically legitimate only if voters can make an informed decision. Yet the level of misinformation in the current campaign is so great that democratic legitimacy is called into question’.

Their letter complains about the quality of information available to voters (an issue about which the RES has raised complaints with the BBC). It doesn’t make the point that the UK government, by ignoring voting theory, has posed a very misleading question given the complexity of the issue under decision. Quite unsettling is the Grassegger and Krogerus (2017) report about voter manipulation by Big Data, originally on Brexit and later for the election of Donald Trump. But the key point here concerns the referendum question itself.

Image (Wikipedia): Public Domain.

The problem with the question

The question assumes a binary choice — Remain or Leave the EU — while voting theory warns that allowing only two options can easily be a misleading representation of the real choice. When the true situation is more complex, and especially if it is one that arouses strong passions, then reducing the question to a binary one might suggest a political motivation. As a result of the present process, we actually don’t know how people would have voted when they had been offered the true options.

Compare the question: ‘Do you still beat your mother ?’

When you are allowed only a Yes or No answer, then you are blocked from answering:

‘I will not answer that question because if I say No then it suggests that I agree that I have beaten her in the past.’

In the case of Brexit, the hidden complexity concerned:

— Leave, and adopt an EFTA or WTO framework?

— Leave, while the UK remains intact or while it splits up?

— Remain, in what manner?

Voting theory generally suggests that representative democracy — Parliament — is better than relying on referenda, since the representatives can bargain about the complex choices involved.

Deadlocks can lurk in hiding

When there are only two options then everyone knows about the possibility of a stalemate. This means a collective indifference. There are various ways to break the deadlock: voting again, the chairperson decides, flip a coin, using the alphabet, and so on. There is a crucial distinction between voting (vote results) and deciding. When there are three options or more there can be a deadlock as well. It is less well-known that there can also be cycles. It is even less recognised that such cycles are actually a disguised form of a deadlock.

Take for example three candidates A, B and C and a particular distribution of preferences. When the vote is between A and B then A wins. We denote this as A > B. When the vote is between B and C then B wins, or B > C. When the vote is between C and A then C wins or C > A. Collectively A > B > C > A. Collectively, there is indifference. It is a key notion in voting theory that there can be distributions of preferences, such that a collective binary choice seems to result into a clear decision, while in reality there is a deadlock in hiding.

Kenneth Arrow who passed away on February 21 used these cycles to create his 1951 ‘impossibility theorem’. Indeed, if you interpret a cycle as a decision then this causes an inconsistency or an ‘impossibility’ with respect to the required transitivity of a (collective) preference ordering. However, reality is consistent and people do really make choices collectively, and thus the proper interpretation is an ‘indifference’ or deadlock. It was and is a major confusion in voting theory that Arrow’s mathematics are correct but that his own verbal interpretation was incorrect.

Representative government is better than referenda

Obviously a deadlock must be broken. Again, it may be political motivation that reduces the choice from three options A, B and C to only two. Who selects those two might take the pair that fits his or her interests. A selection in successive rounds as in France at the moment is no solution. There are ample horror scenarios when bad election designs cause minority winners. Decisions are made preferably via discussion in Parliament. Parliamentarian choice of the Prime Minister is better than direct election like for the US President.

Voting theory is not well understood in general. The UK referendum in 2011 on Alternative Vote (AV) presented a design that was far too complex. Best is that Parliament is chosen in proportional manner as in Holland, rather than in districts as in the UK or the USA. It suffices when people can vote for the party of their choice (with the national threshold of a seat), and that the professionals in Parliament use the more complex voting mechanisms (like bargaining or the Borda Fixed Point method). It is also crucial to be aware that the Trias Politica model for democracy fails and that more checks and balances are required, notably with an Economic Supreme Court.

The UK Electoral Commission goofed too

The UK Electoral Commission might be abstractly aware of this issue in voting theory, but they didn’t protest, and they only checked that the Brexit referendum question could be ‘understood’. The latter is an ambiguous notion. People might ‘understand’ quite a lot but they might not truly understand the hidden complexity and the pitfalls of voting theory. Even Nobel Prize winner Kenneth Arrow gave a problematic interpretation of his theorem. The Electoral Commission is to be praised for the effort to remove bias, where the chosen words ‘Remain’ and ‘Leave’ are neutral, and where both statements were included and not only one. (Some people don’t want to say ‘No’. Some don’t want to say ‘Yes’.) Still, the Commission gives an interpretation of the ‘intelligibility’ of the question that doesn’t square with voting theory and that doesn’t protect the electorate from a voting disaster.

A test on this issue involves asking yourself: Given the referendum outcome, do you really think that the UK population is clear in its position, whatever the issues of how to leave or the risk of a UK breakup? If you have doubts on the latter, then you agree that something is amiss. The outcome of the referendum really doesn’t give us a clue as to what UK voters want. Scotland wants to remain in the EU and then break up? This is okay for the others who want to Leave? (And how?) The issue can be seen as a statistical enquiry into what views people have, and the referendum question is biased and cannot be used for sound conclusions.

In an email to the author in July 2016 a spokesman for the Electoral Commission said its role: ‘… is to evaluate the intelligibility of referendum questions in line with the intent of Parliament; it is not to re-evaluate the premise of the question. Other than that, I don’t believe there is anything I can usefully add to our previously published statements on this matter.’

Apparently the Commission knows the ‘intent of Parliament’, while Parliament itself might not do so. Is the Commission only a facilitator of deception, and don’t they have a mission to put voters first? At best the Commission holds that Whitehall and Parliament fully understood voting theory and therefore intentionally presented the UK population with a biased choice, so that voters would be compelled to neglect the complexities of leaving or even a break-up of the Union. Obviously the assumption that Whitehall and Parliament fully grasp voting theory is dubious. The better response by the Commission would have been to explain the pitfalls of voting theory and the misleading character of the referendum question, rather than facilitate the voting disaster.

Any recognition that something is (very) wrong here, should also imply the annulment of the Brexit referendum outcome. Subsequently, to protect voters from such manipulation by Whitehall, one may think of a law that gives the Electoral Commission the right to veto a biased Yes / No selection, which veto might be overruled by a 2/3 majority in Parliament. Best is not to have referenda at all, unless you are really sure that a coin can only fall either way, and not land on its side (by a hidden deadlock).

This article first appeared in the Newsletter of the Royal Economic Society, it gives the views of the author, and not the position of LSE Brexit, nor of the London School of Economics.

Thomas Colignatus is the science name of Thomas Cool, an econometrician (Groningen 1982) and teacher of mathematics (Leiden 2008), Scheveningen, Holland.