Monthly Archives: March 2017

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

Wikipedia (a portal and no source) gives an overview of the Dutch general elections of March 15 2017. For the interpretation of the vote, there is this paper: “The performance of four possible rules for selecting the Prime Minister after the Dutch Parliamentary elections of March 2017“.

The abstract of the paper is:

“Economic policy depends not only on national elections but also on coalition bargaining strategies. In coalition government, minority parties bargain on policy and form a majority coalition, and select a Prime Minister from their mids. In Holland the latter is done conventionally with Plurality, so that the largest party provides the chair of the cabinet. Alternative methods are Condorcet, Borda or Borda Fixed Point. Since the role of the Prime Minister is to be above all parties, to represent the nation and to be there for all citizens, it would enhance democracy and likely be optimal if the potential Prime Minister is selected from all parties and at the start of the bargaining process. The performance of the four selection rules is evaluated using the results of the 2017 Dutch Parliamentary elections. Plurality gives VVD. VVD is almost a Condorcet winner except for a tie with 50Plus. Borda and BordaFP give CU as the prime minister. The impossibility theorem by Kenneth Arrow (Nobel memorial prize in economics 1972) finds a crucially different interpretation.” (Paper)

The paper uses the estimate of March 16, and the official allocation of seats presented on March 21 was the same. Here is a letter (in Dutch) to the Speaker of the House with these results and a summary statement.

Relevance for the world

In addition to that paper, let me mention some other points.

  • The Dutch system of proportional representation (PR) with a threshold of 1 seat is most democratic, and is much better than district representation (DR) or the use of high thresholds. (See this other paper.) The low threshold allows the flexible entry and exit of contestants. For example, in Germany, economics professor Bernd Lucke started the originally decent AfD, didn’t get their 5% threshold, and was ousted by extremist members in his party. For the upcoming elections, France and Germany best adopt the Dutch election model, but likely they will not have time to do so.
  • Within the Dutch system, there still is room for even more democracy. Coalitions can be inclusive or exclusive. Politicians tend to think that a minimal majority is most stable, but in all likelihood voters are better served by a larger majority.
  • The news media of the world tended to focus on the Dutch outcome that Geert Wilders didn’t succeed in getting most seats. Incumbent prime minister Mark Rutte got 33 seats and Wilders only 20. This was interpreted as that the threat of populism in Europe might have a turning point. However, Rutte dropped from 41 to 33 and Wilders rose from 15 to 20 seats, so the gap of 26 seats was halved in favour for Wilders. There is also the new right wing lunatic FvD with 2 seats, and the move to the right by other parties feeling the hot breath by Wilders. Overall, the picture is more mixed than the world news media seem to have reported. A bit more background w.r.t. the Dutch reputation of tolerance is in this earlier weblog text.
Some additional findings on turnout

The official results of March 21 2017 allow an additional statement on turnout. The key data are in the following table.

The Dutch House of Commons has 150 seats. With the turnout of 81.9% actually only 120 seats were fully taken. 27 Seats were lost to no-shows, 2 seats were lost to the dispersion of small parties and 1 seat was lost on blank or invalid votes. One might argue that 30 seats should remain unused, so that the parties that were elected in the House would find it tougher to create a coalition of 76 seats or 50%+1. Alternatively, when the 30 seats are still allocated to the elected parties, then one might raise the majority criterion to 94 seats. Instead, however, the elected parties take the 30 seats anyway and still apply the 76 seats majority rule. See this paper for a discussion w.r.t. an earlier election.

A Dutch – Turkish clash

The vote took place while there was a clash between Holland and Turkey – see the scene on Haberturk TV reported on by Euronews. Much has been said about this elsewhere, but here we continue testing the quality of Google Translate: “They protested the Netherlands by squeezing oranges”.

The Turks should however beware that the House of Orange claims Russia, and you wouldn’t want an orange bear on your doorsteps.

Euronews relaying Haberturk TV. “Hollanda’yı portakal sıkarak protesto ettiler…”

Some Dutch had been prepared for this

In the months before, visionary artist Inez Lenders had already created the artistic reply to maltreatment of oranges. In the match on creativity, the score is 1 – 1.

Art and Photography by Inez Lenders, Nijmegen 2017

The Dutch Official News with a false suggestion

The site calculated that the elections generated 5 MP’s with Turkish roots and 8 MP’s with Moroccan roots, and 0 with roots in Suriname. We may include one Turkish-Kurdish MP, so a total of 14 or 9.3% of relatively new immigrants. There are 4 German names, 2 French and 1 Jewish. Thus a total of 21 MP’s or 14% immigrant names.

Notwithstanding such a composition in the new House of Commons, president Tayyip Erdogan fulminated about descendents of nazi’s, though he is right that the Dutch record in World War II is not so good.

When Angela Merkel and other Europeans supported Dutch premier Mark Rutte, then Erdogan presented a statement for which it is important to provide the right translation. Reuters seems to be okay:

“Erdogan warns Europeans ‘will not walk safely’ if attitude persists” (March 22 2017)

This is a fairly decent warning. The age of European imperialism till 1945 is over. In the world population the European share is dwindling. If the world wants to maintain the idea of safe international travel then we need rules and regulations and consistent implementation.

  • Reuters gives a fair representation that Erdogan warns about the effect of arrogance.
  • Dutch national television turned this into a report that Erdogan threatened Europeans. On this NOS website, the official heading and weblink contain the phrase “Erdogan warns” but the picture on that page has the phrase “Erdogan threatens” (Dutch “bedreigt”) (wayback machine).

I have informed NPO Ombudsman Margo Smit about the difference between warning and threatening, but they haven’t changed it yet.

Official Dutch television NOS falsely states that president Erdogan issues a threat that no European in any part of the world can safely walk on the street. In truth he only warns.

[ This is the same text as the former weblog (here), but now we follow Van Hiele’s argument for the abolition of fractions. The key property is that there are numbers xH such that x xH = 1 when x ≠ 0, and the rest follows from there. Thus we replace (y / x) with y xH with H = -1. ]

Robert Siegler participates in the “Center for Improved Learning of Fractions” (CILF) and was chair of the IES 2010 research group “Developing Effective Fractions Instruction for Kindergarten Through 8th Grade” (report) (video).

IES 2010 key advice number 3 is:

“Help students understand why procedures for computations with fractions make sense.”

The first example of this helping to understand is:

“A common mistake students make when faced with fractions that have unlike denominators is to add both numerators and denominators. [ref 88] Certain representa­tions can provide visual cues to help students see the need for common denominators.” (Siegler et al. (2010:32), refering to Cramer, K., & Wyberg, T. (2009))

For a bH “and” c dH kids are supposed to find (a d + b c) (b d)H instead of (a + c) (b + d)H.

Obviously this is a matter of definition. For “plus” we define: a bH + c dH = (a d + b c) (b d)H.

But we can also define “superplus”: a bHc dH = (a + c) (b + d)H.

The crux lies in “and” that might not always be “plus”.

When (a + c) (b + d)H makes sense

There are cases where (a + c) (b + d)H makes eminent sense. For example, when a bH is the batting average in the Fall-Winter season and c dH the batting average in the Spring-Summer season, then the annual (weighted) batting average is exactly (a + c) (b + d)H. Kids would calculate correctly, and Siegler et al. (2010) are suggesting that the kids would make a wrong calculation ?

The “superplus” outcome is called the “mediant“. See a Wolfram Demonstrations project case with batting scores.

Adding up fractions of the same pizza thus differs from averaging over more pizzas.

We thus observe:

  • Kids live in a world in which (a + c) (b + d)H makes eminent sense.
  • Telling them that this is “a mistaken calculation” is actually quite confusing for them.
  • Thus it is better teaching practice to explain to them when it makes sense.

There is no alternative but to explain Simpson’s paradox also in elementary school. See the discussion about the paradox in the former weblog entry. The issue for today is how to translate this to elementary school.

[ Some readers may not be at home in statistics. Let the weight of b be w = b (b + d)H. Then the weight of d is 1 – w. The weighted average is (a bH) w + (c dH) (1 – w) = (a + c) (b + d)H. ]

Cats and Dogs

Many examples of Simpson’s paradox have larger numbers, but the Kleinbaum et al. (2003:277) “ActivEpi” example has small numbers (see also here). I add one more to make the case less symmetrical. Kady Schneiter rightly remarked that an example with cats and dogs will be more appealing to students. She uses animal size (small or large pets) as a factor, but let me stick to the idea of gender as a confounder. Thus the kids in class can be presented with the following case.

  • There are 17 cats and 16 dogs.
  • There are 17 pets kept in the house and 16 kept outside.
  • There are 17 female pets and 16 male pets (perhaps “helped”).

There is the phenomenon – though kids might be oblivious why this might be “paradoxical”:

  1. For the female pets, the proportion of cats in the house is larger than the proportion for dogs.
  2. For the male pets, the proportion of cats in the house is larger than the proportion for dogs.
  3. For all pets combined, the proportion of cats in the house is smaller than the proportion for dogs.
The paradoxical data

The paradoxical data are given as follows. Observe that kids must calculate:

  • For the cats: 6 7H = 0.86, 2 10H = 0.20 and (6 + 2) (7 + 10)H = 0.47.
  • For the dogs: 8 10H = 0.80, 1 6H = 0.17 and (8 + 1) (10 + 6)H = 0.56.

A discussion about what this means

Perhaps the major didactic challenge is to explain to kids that the outcome must be seen as “paradoxical”. When kids might not have developed “quantitative intuitions” then those might not be challenged. It might be wise to keep it that way. When data are seen as statistics only, then there might be less scope for false interpretations.

Obviously, though, one would discuss the various views that kids generate, so that they are actively engaged in trying to understand the situation.

The next step is to call attention to the sum totals that haven’t been shown above.

It is straightforward to observe that the F and M are distributed in unbalanced manner.

The correction

It can be an argument that there should be equal numbers of F and M. This causes the following calculations about what pets would be kept at the house. We keep the observed proportions intact and raise the numbers proportionally.

  • For the cats: 0.86 * 10 ∼ 9, and (9 + 2) (10 + 10) H = 0.55.
  • For the dogs: 0.17 * 10 ∼ 2, and (8 + 2) (10 + 10) H = 0.50.

And now we find: Also for all pets combined, the proportion of cats in the house is larger than the proportion for dogs. Adding up the subtables into the grand total doesn’t generate a different conclusion on the proportions.

Closure on causality

Perhaps kids at elementary school should not bothered with discussions on causality, certainly not on a flimsy case as this. But perhaps some kids require closure on this, or perhaps the teacher does. In that case the story might be that the kind of pet is the cause, and that the location where the pet is kept is the effect. When people have a cat then they tend to keep it at home. When people have a dog then are a bit more inclined to keep it outside. The location has no effect on gender. The gender of the pet doesn’t change by keeping it inside or outside of the house.

Vectors in elementary school

Pierre van Hiele (1909-2010) explained for most of his professional life that kids at elementary school can understand vectors. Thus, they should be able to enjoy this vector graphic by Alexander Bogomolny.

Van Hiele also proposed to abolish fractions as we know them, by replacing y / x by y x^(-1). The latter might be confusing because kids might think that they have to subtract something. But the mathematical constant H = -1 makes perfect sense, namely, check the unit circle and the complex number i. Thus we get y / x = y xH. The latter would be the better format. See A child wants nice and no mean numbers(2015).


Some conclusions are:

  • What Siegler & IES 2010 call a “common mistake” is the proper approach in serious statistics.
  • Teaching can improve by explaining to kids what method applies when. Adding fractions of the same pizza is different from calculating a statistical average. (PM. Don’t use round pizza’s. This makes for less insightful parts.)
  • Kids live in a world in which statistics are relevant too.
  • Simpson’s paradox can be adapted such that it may be tested whether it can be discussed in elementary school too.
  • The discussion corroborates Van Hiele’s arguments for vectors in elementary school and the abolition of fractions as we know them (y / x) and the use of y xH with H = -1. The key thing to learn is that there are numbers xH such that x xH = 1 when x ≠ 0, and the rest follows from there.

PM. The excel sheet for this case is: 2017-03-03-data-from-kleinbaum-2003-adapted