Voting theory and the Brexit referendum question

The Theresa May government has adopted Brexit as its policy aim and has received support from the Commons. Yet, economic theory assumes rational agents, and even governments might be open for rational reconsideration, even at the last moment.

Scientifically unwarranted referendum question

Based upon voting theory, the Brexit referendum question can be rejected as scientifically unwarranted. My suggestion is that the UK government annuls the outcome based upon this insight from science, and upon this insight alone. Let me invite (economic) scientists to study the argument and voting theory itself, so that the scientific community can confirm this analysis. This study best be done all over Europe, so that also the EU Commission might adopt it. Britons might be wary when their government or the EU Commission would listen to science, but then they might check the finding themselves too. A major worry 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.”

Curiously, however, their letter doesn’t make the point that the referendum neglects voting theory, since the very question itself is misleading w.r.t. the complexity of the issue under decision. Quite unsettling is the Grassegger & 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.

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 be a misleading representation. When the true situation is more complex, then it may be political manipulation to reduce this to a binary one. 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 as EFTA or WTO ?
  • 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 lesser known that there can also be cycles. It is even lesser known that such cycles actually are 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 (1921-2017) who passed away on February 21 used these cycles to create his 1951 “impossibility theorem”. Indeed, if you interprete a cycle as a decision then this causes an inconsistency or an “impossibility” w.r.t. 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, see my VTFD Ch. 9.2.

Representative government is better than referenda

Obviously a deadlock must be broken. Again, it may be manipulation to reduce 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 rounds like in France 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 Proportional Representation (PR) 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 complexer 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 is 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 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 me a clue as to what UK voters really 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 statistics.

In an email to me 2016-07-11:

“The Electoral Commission’s 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 they don’t have the mission to put voters first ? At best the Commission holds that Whitehall and Parliament fully understood voting theory therefor deliberatedly presented the UK population with a biased choice, so that voters would be seduced to neglect complexities of how to Leave or the risks of a UK breakup. 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 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.

Addendum March 31

  • The UK might repeal the letter on article 50 – see this BBC reality check. Thus science might have this time window to clarify to the general public how the referendum question doesn’t comply with voting theory.
  • The recent general elections in Holland provide another nice example for the importance of voting theory and for the meaning of Arrow’s Impossibility Theorem, see here.

BBC (2017), “Article 50: May signs letter that will trigger Brexit“, March 29

Carrell, S. (2017), “Scottish parliament votes for second independence referendum“, The Guardian, March 28

Colignatus (2001, 2004, 2011, 2014), “Voting theory for democracy” (VTFD), pdf online,

Colignatus (2010, “Single vote multiple seats elections. Didactics of district versus proportional representation, using the examples of the United Kingdom and The Netherlands”, May 19 2010, MPRA 22782,

Colignatus (2011a), “The referendum on PR“, Mathematics Teaching 222, January 5 2011, also on my website

Colignatus (2011b), “Arrow’s Impossibility Theorem and the distinction between Voting and Deciding”,

Colignatus (2014), “An Economic Supreme Court”, RES Newsletter issue no. 167, October 2014, pp.20-21,

Colignatus (2016), “Brexit: advice for young UK (age < 50 years), and scientific outrage for neglect of voting theory“, weblog text June 29

Colignatus (2017), “The performance of four possible rules for selecting the Prime Minister after the Dutch Parliamentary elections of March 2017″, March 17, MPRA 77616

Grassegger, H. and M. Krogerus (2017), “The Data That Turned the World Upside Down”,

Renwick, A. e.a. (2016), “Letters: Both Remain and Leave are propagating falsehoods at public expense“, The Telegraph, Opinion, June 14

From the BBC website

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