Such imbalance is not uncommon. At the US election of November 8 2016, the Republicans got 49.1% of the votes and 55.4% of the seats, while the Democrats got 48% of the votes and 44.6% of the seats. At the UK general election of June 8 2017, the Conservatives got 42.2% of the votes and 48.8% of the seats while Labour got 39.9% of the votes and 40.3% of the seats (the wikipedia data of October 16 2017 are inaccurate).

This article clarifies a new and better way to measure this inequality or disproportionality of votes and seats. The new measure is called *Sine-Diagonal Inequality / Disproportionality* (SDID). The new measure falls under descriptive statistics. Potentially it might be used in any area where one matches shares or proportions, like the proportions of minerals in different samples. SDID is related to statistical concepts like *R*-squared and the regression slope. This article looks at some history, as Karl Pearson (1857-1936) created the *R*-Squared and Ronald A. Fisher (1890-1962) in 1915 determined its sample distribution. The new measure would also be relevant for Big Data. William Gosset (1876-1937) a.k.a. “Student” was famously unimpressed by Fisher’s notion of “statistical significance” and now is vindicated by descriptive statistics and Big Data.

Statistics has the triad of Design, Description and Decision.

- Design is especially relevant for the experimental sciences, in which plants, lab rats or psychology students are subjected to alternate treatments. Design is informative but less applicable for observational sciences, like macro-economics and national elections when the researcher cannot experiment with nations.
- Descriptive statistics has measures for the center of location – like mean or median – and measures of dispersion – like range or standard deviation. Important are also the graphical methods like the histogram or the frequency polygon.
- Statistical decision making involves the formulation of hypotheses and the use of loss functione to evaluate that hypotheses. A hypothesis on the distribution of the population provides an indication for choosing the sample size. A typical example is the definition of
*decision**error*(of the first kind) that a hypothesis is true but still rejected. One might accept a decision error in say 5% of the cases, called the level of statistical significance.

Historically, statisticians have been working on all these areas of design, description and decision, but the most difficult was the formulation of decision methods, since this involved both the calculus of reasoning and the more complex mathematics on normal, *t,* chi-square, and other frequency distributions. In practical work, the divide between the experimental and the non-experimental (observational) sciences appeared insurmountable. The experimental sciences have the advantages of design and decisions based upon samples, and the observational sciences basically rely on descriptive statistics. When the observational sciences do regressions, there is an ephemeral application of statistical significance that invokes the Law of Large Numbers, that all error approximates the normal distribution.

This traditional setup of statistics is being challenged in the last decades by Big Data – see also this discussion by Rand Wilcox in *Significance *May 2017. When all data are available, and when you actually have the population data, then the idea of using a sample evaporates, and you don’t need to develop hypotheses on the distributions anymore. In that case descriptive statistics becomes the most important aspect of statistics. For statistics as a whole, the emphasis shifts from *statistical decision making* to *decisions on content.* While descriptive statistics had been applied mostly to samples, Big Data now causes the additional step how these descriptions relate to decisions on content. In fact, such questions already existed for the observational sciences like for macro-economics and national elections, in which the researcher only had descriptive statistics, and lacked the opportunity to experiment and base decisions upon samples. The disadvantaged areas now provide insights for the earlier advantaged areas of research.

The key insight is to transform the loss function into a descriptive statistic itself. An example is the Richter scale for the magnitude of earthquakes. It is both a descriptive statistic and a factor in the loss function. A nation or regional community has on the one hand the cost of building and construction and on the other hand the risk of losing the entire investments and human lives. In the evaluation of cost and benefit, the descriptive statistic helps to clarify the content of the issue itself. The key issue is no longer a decision within statistical hypothesis testing, but the adequate description of the data so that we arrive at a better cost-benefit analysis.

Let us return to the election for the House of Representatives (USA) or the House of Commons (UK). The criterion of *One man, one vote* translates into the criterion that the shares of seats equal the shares of votes. We are comparing two vectors here.

The reason why the shares of seats and votes do not match is because the USA and UK use a particular setup. The setup is called an “electoral system”, but since it does not satisfy the criterion of *One man, one vote, *it does not really deserve that name. The USA and UK use both (single member) districts and the criterion of Plurality per district, meaning that the district seat is given to the candidate with the most votes – also called “first past the post” (FPTP). This system made some sense in 1800 when the concern was district representation. However, when candidates stand for parties then the argument for district representation loses relevance. The current setup does not qualify for the word “election” though it curiously continues to be called so. It is true that voters mark ballots but that is not enough for a real election. When you pay for something in a shop then this is an essential part of the process, but you also expect to receive what you ordered. In the “electoral systems” in the USA and UK, this economic logic does not apply. Only votes for the winner elect someone but the other votes are obliterated. For such reasons Holland switched to equal / proportional representation in 1917.

For descriptive statistics, the question is how to measure the deviation of the shares of votes and seats. For* statistical decision making* we might want to test whether the US and UK election outcomes are statistically significantly different from inequality / proportionality. This approach requires not only a proper descriptive measure anyway, but also some assumptions on the distribution of votes which might be rather dubious to start with. For this reason the emphasis falls on* descriptive statistics*, and the use of a proper measure for inequality / disproportionality (ID).

A measure proposed by, and called after, Loosemore & Hanby in 1971 (LHID) uses the sum of the absolute deviations of the shares (in percentages), divided by 2 to correct for double counting. The LHID for the UK election of 2017 is 10.5 on a scale of 100, which means that 10.5% of the 650 seats (68 seats) in the UK House of Commons are relocated from what would be an equal allocation. When the UK government claims to have a “mandate from the people” then this is only because the UK “election system” is so rigged that many votes have been obliterated. The LHID gives the percentage of relocated seats but is insensitive to how these actually are relocated, say to a larger or smaller party.

The Euclid / Gallagher measure proposed in 1991 (EGID) uses the Euclidean distance, again corrected for double counting. For an election with only two parties EGID = LHID. The EGID has become something like the standard in political science. For the UK 2017 the EGID is 6.8 on a scale of 100, which cannot be interpreted as a percentage of seats like LHID, but which indicates that the 10.5% of relocated seats are not concentrated in the Conservative party only.

Alan Renwick in 2015 tends to see more value in LHID than EGID: “As the fragmentation of the UK party system has increased over recent years, therefore, the standard measure of disproportionality [thus EGID] has, it would appear, increasingly understated the true level of disproportionality.”

The new *Sine-Diagonal Inequality / Disproportionality* (SDID) measure – presented in *this paper* – looks at the *angle* between the vectors of the shares of votes and seats.

- When the vectors overlap, the angle is zero, and then there is perfect equality / proportionality.
- When the vectors are perpendicular then there is full inequality / disproportionality.
- While this angle variates from 0 to 90 degrees, it is more useful to transform it into
*sine*and*cosine*that are in the [0, 1] range. - The SDID takes the
*sine*for inequality / disproportionality and the*cosine*of the angle for equality / proportionality. - With Sin[0] = 0 and Cos[0] = 1, we thus get a scale that is 0 for full inequaliy / disproportionality and 1 for full equality / proportionality.

It appears that the sine is more sensitive than either absolute value (LHID) and Euclidean distance (EGID). It is closer to the absolute value for small angles, and closer to the Euclidean distrance for larger angles. See said paper, Figure 1 on page 10. SDID is something like a compromise between LHID and EGID but also better than both.

When we regress the shares of the seats on the shares of the votes without using a constant – i.e. using Regression Through the Origin (RTO) – then this gives a single regression coefficient. When there is equality / proportionality then this regression coefficient is 1. This has the easy interpretation that this is the diagonal in the votes & seats space. This explains the name of SDID: when the regression coefficient generates the diagonal, then the sine is zero, and there is no inequality / disproportionality.

Said paper – see page 38 – recovers a key relationship between on the one hand the sine and on the other hand the Euclidean distance and this regression coefficient. On the diagonal, the sine and Euclidean distance are both zero. Off-diagonal, the sine differs from the Euclidean distance in nonlinear manner by means of a factor given by the regression coefficient. This relationship determines the effect that we indicated above, how SDID compromises between and improves upon LHID and EGID.

There appears to be a relationship between said regression coefficient and the cosine itself. This allows for a double interpretation as both slope and similarity measure. This weblog text is intended to avoid formulas as much as possible and thus I refer to said paper for the details. Suffice to say here is that, at first, it may seem to be a drawback that such a double interpretation is possible, yet, on closer inspection the relationship makes sense and it is an advantage to be able to switch perspective.

In human psychology there appears to be a distinction between actual differences and perceived differences. This is called the Weber – Fechner law. When a frog is put into a pan with cool water and slowly boiled to death, it will not jump out. When a frog is put into a pan with hot water it will jump out immediately. People may notice differences between low vote shares and high seat shares, but they may be less sensitive to small differences, while these differences actually can still be quite relevant. For this reason, the SDID uses a sensitivity transform. It uses the square root of the sine.

(PM. A hypothesis is that the USA and UK call their national “balloting events” still “elections”, is that the old system of districts has changed so gradually into the method of obliterating votes that many people did not notice. It is more likely though that that some parties recognised the effect, but have an advantage under the present system, and then do not want to change to equal / proportional representation.)

Subsequently, the sine and its square root have values in the range [0, 1]. In itself this is an advantage, but it comes with leading zeros. We might multiply with 100 but this might cause the confusion as if it would be percentages. The second digit might give a false sense of accuracy. It is more useful to multiply this by 10. This gives values like on a report card. We can compare here to Bart Simpson, who appreciates low values on his report card.

Finally, when we compare, say, votes {49, 51} and seats {51, 49}, then we see a dramatic change of majority, even though there is only a slight inequality / disproportionality. It is useful to have an indicator for this too. It appears that this can be done by using a negative sign when such majority reversal occurs. This method of indicating majority reversals is not so sophisticated yet, and at this stage consists of using the sign of the covariance of the vectors of votes and seats.

This present text avoids formulas but it is useful to give the formula for the new measure of SDID, so that the reader may link up more easily with the paper in which the new measure is actually developed. For the vectors of votes and seats we use the symbols *v *and *s, *and the angle between the two vectors give cosine and then sine:

SDID[*v, s*] = *sign *10 √ Sin[*v, s*]

For the UK 2017, the SDID value is 3.7. For comparison the values of Holland with equal / proportional representation are: LHID 3, EGID 1.7, SDID 2.5. It appears that Holland is not yet as equal / proportional as can be. Holland uses the Jefferson / D’Hondt method, that favours larger parties in the allocation of remainder seats. At elections there are also the wasted vote, when people vote for fringe parties that do not succeed in getting seats. In a truly equal or proportional system, the wasted vote can be respected by leaving seats empty or by having a qualified majority rule.

Remarkably, Karl Pearson (1857-1936) also used the cosine when he created *R*-squared, also known as the “coefficient of determination“. Namely:

*R*-squared is the cosine-squared applied to centered data. Such centered data arise when one subtracts the mean value from the original data. For such data it is advisable to use a regression with a constant, which constant captures the mean effect.- Above we have been using the original (non-centered) data. Alternatively put, when we do above Regression Through the Origin (RTO) and then look for the proper coefficient of determination, then we get the cosine-squared.

The SDID measure thus provides a “missing link” in statistics between centered and non-centered data, and also provides a new perspective on *R*-squared itself.

Apparently till now statistics found little use for original (non-centered) data and RTO. A possible explanation is that statistics fairly soon neglected descriptive statistics as less challenging, and focused on statistical decision making. Textbooks prefer the inclusion of a constant in the regression, so that one can test whether it differs from zero with statistical significance. The constant is essentially used as an indicator for possible errors in modeling. The use of RTO or the imposition of a zero constant would block that kind of application. However, this (traditional, academic) focus on statistical decision making apparently caused the neglect of a relevant part of the analysis, that now comes to the surface.

*R*-squared is often mentioned in statistical reports about regressions, but actually it is not much used for other purposes than reporting only. Cosma Shalizi (2015:19) states:

“At this point, you might be wondering just what R-squared is good for — what job it does that isn’t better done by other tools. The only honest answer I can give you is that I have never found a situation where it helped at all. If I could design the regression curriculum from scratch, I would never mention it. Unfortunately, it lives on as a historical relic, so you need to know what it is, and what misunderstandings about it people suffer from.”

At the U. of Virginia Library, Clay Ford summarizes Shalizi’s points on the uselessness of *R*-squared, with a reference to his lecture notes.

Since the cosine is symmetric, the *R*-squared is the same for regressing *y *given *x, *or *x* given *y. *Shalizi (2015, p18) infers from the symmetry: “This in itself should be enough to show that a high *R*² says nothing about explaining one variable by another.” This is too quick. When theory shows that *x *is a causal factor for *y *then it makes little sense to argue that *y *explains *x *conversely. Thus, for research the percentage of explained variation can be informative. Obviously it matters how one actually uses this information.

When it is reported that a regression has an R-squared of 70% then this means that 70% of the variation of the explained variable is explained by the model, i.e. by variation in the explanatory variables and the estimated coefficients. In itself such a report does not say much, for it is not clear whether 70% is a little or a lot for the particular explanation. For evaluation we obviously also look at the regression coefficients.

One can always increase *R*-squared by including other and even nonsensical variables. For a proper use of *R*-squared, we would use the *adjusted R*-squared. *R*-adj finds its use in model specification searches – see Dave Giles 2013. For an increase of *R-adj *coefficients must have an absolute *t*-value larger than 1. A proper report would show how *R-adj* increases by the inclusion of particular variables, e.g. also compared to studies by others on the same topic. Comparison on other topics obviously would be rather meaningless. Shalizi also rejects R-adj and suggests to work directly with the *mean squared error* (MSE, also corrected for the degrees of freedom). Since *R-*squared is the cosine, then the MSE relates to the sine, and these are basically different sides of the same coin, so that this discussion is much a-do about little. For standardised variables (difference from mean, divided by standard deviation), the *R*-squared is also the coefficient of regression, and then it is relevant for the effect size.

*R*-squared is a sample statistic. Thus it depends upon the particular sample. A hypothesis is that the population has a *ρ*-squared. For this reason it is important to distinguish between a regression on fixed data and a regression in which the explanatory variables also have a (normal) distribution (errors in variables). In his 1915 article on the sample distribution of *R*-squared. R.A Fisher (digital library) assumed the latter. With fixed data, say *X, *the outcome is conditional on *X, *so that it is better to write *ρ*[*X*], lest one forgets about the situation. See my earlier paper on the sample distribution of *R-adj. *Dave Giles has a fine discussion about *R*-squared and *adjusted **R-squared. *A search gives more pages. He confirms the “uselessnes” of *R*-squared: “My students are often horrified when I tell them, truthfully, that one of the last pieces of information that I look at when evaluating the results of an OLS regression, is the coefficient of determination (R^{2}), or its “adjusted” counterpart. Fortunately, it doesn’t take long to change their perspective!” Such a statement should not be read as the uselessness of cosine or sine in general.

I am not familiar with the history of statistics, and it is unknown to me what else Pearson, Fisher, Gosset and other founding and early authors wrote about the application of the cosine or sine. The choice to apply the cosine to centered data to create *R*-squared is deliberate, and Pearson would have been aware that it might also be applied to original (non-centered) data. It is also likely that he would not have the full perspective above, because then it would have been in the statistical textbooks already. It would be interesting to know what the considerations at time were. Quite likely the theoretical focus was on statistical decision making rather than on description, yet this for me unknown history would put matters more into perspective.

Part of the history is that R.A. Fisher with his attention for mathematics emphasized *precision* while W.S. Gosset with his attention to practical application emphasized the *effect size* of the coefficients found by regression. Somehow, *statistical significance *in terms of precision became more important than *content significance, *and empirical research has rather followed Fisher than the practical relevance of Gosset. This history and its meaning is discussed by Stephen Ziliak & Deirdre McCloskey 2007, see also this discussion by Andrew Gelman. As said, for standardised variables, the regression coefficient is the *R-*squared, and this is best understood with attention for the effect size. For some applications a low *R-*squared would still be relevant for the particular field.

The new measure SDID provides a better description of the inequality or disproportionality of votes and seats compared to existing measures. The new measure has been tailored to votes and seats, by means of greater sensitivity to small inequalities, and because a small change in inequality may have a crucial impact on the (political) majority. For different fields, one could taylor measures in similar manner.

That the cosine could be used as a measure of similarity has been well-known in the statistics literature since the start, when Pearson used the cosine for centered data to create *R-*square. For the use of the sine I have not found direct applications, but its use is straightforward when we look at the opposite of similarity.

The proposed measure provides an enlightening bridge between descriptive statistics and statistical decision making. This comes with a better understanding of what kind of information the cosine or *R-*squared provides, in relation to regressions with and without a constant. Statistics textbooks would do well by providing their students with this new topic for both theory and practical application.

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Boris Johnson and Nigel Farage have been criticised for spreading false arguments for the June 23 2016 Brexit referendum. Obviously, these two individuals cannot be held accountable for swinging the views of some 45 million voters. I wondered since the referendum whether they had had some help. Apparently Jacob Rees-Mogg had been giving a helping hand.

To clarify Rees-Mogg’s departure from truth, we first must mention some properties of the European Parliament.

The EU Parliament has 751 seats, distributed over 28 member states with 500 million people. The distribution over countries is not proportional to the populations, since countries are units by themselves, and it is felt that this should have some effect. Thus Germany with its population of 82 million has 96 seats (1.17 seats per million), the UK with its population of 65 million has 73 seats (1.12 seats per million), and Malta with its population of 0.5 million has 6 seats (12 seats per million). There is relatively little tension about this apportionment, since the countries fall in comparable classes (large, medium, very small), and the major political differences translate into political parties. The divisions between Christian Democrats, Social Democrats, Liberals, and what have you, apparently are dispersed over countries in similar manner, or, the political parties are able to create alliances over nations. It is part of the wonder of the EU that nationalism is being channeled and that there is more scope for civil democracy. A recent paper of mine on proportional representation is here.

Jacob Rees-Mogg does not explain above democratic solution for dealing with Member States of different sizes. He criticises the EU that Malta is over-represented compared to the UK. It is a fact that Malta has a higher seat-to-vote ratio, but only pointing to this fact obscures the other considerations. He mentions a perhaps older figure of 15 instead of the current 11, but that is irrelevant here. The demagoguery is that many in his audience apparently are not be aware of the key notions in this apportionment, and he apparently takes advantage of their lack of knowledge to win them over to his own closed-mindedness. The demagoguery is that he creates a suggestion as if Malta has 15 times more influence than the UK, as if *6 is 15 times larger than 73* (as, indeed, 6 = 15 * 73).

The quote at the final minute starting at about 11.30 is, with the abusive “proportionally outvote” and the threat of “spectres”:

“So what is this great experiment doing ? It is helping once again the rise of the extreme right, and in some cases the extreme left. That is the threat to democracy that is there, that is coming, that is deeply destructive. But the fundamental problem, the real issue at hand tonight is that there is less democracy in *this* country, because of the European Union. Because, Ladies and Gentlemen, however you vote the next general election, 60% of our laws, and some say higher, is made on the basis of European agreements, where the Maltese proportionally outvote us 15 to 1. Whoever you vote for, matters less than somebody in Malta votes for, about the laws of our country. And if you are unsatisfied with that, and you want it changed, I cannot give you any redress, because the United Kingdom Parliament, the most ancient democratic Parliament in the world, has been made powerless. That is the threat to democracy. It is here, but it is on the continent as well. It is a frightening spectre. The best way to deal with it, is to deal with our relationship with the European Union, to put our own democracy first and foremost, and hope that others follow.”

It is almost silly to protest to this demagoguery:

- The situation w.r.t. the UK and Malta in the EU Parliament has been explained.
- The UK has District Representation (DR) instead of Proportional Representation (PR), which causes that the UK is much less democratic than most countries in the EU or the EU Parliament itself. The PR Gini for the UK of 2017 is 15.6%, but there has been a lot of strategic voting, so that we don’t really know what the first preferences of UK voters are. By comparison, Holland has a PR Gini of only 3.6%, and people in Holland could vote for the party of their first choice. See this weblog text and this paper.
- I tend to think that Rees-Mogg really worries about the state of democracy, while A.C. Grayling rather sees an elitist or even pecunary motive, see this article, as in “follow the money”. Yet Rees-Mogg doesn’t study the topic, and thus he is condemned to repeat an ideology. He studied history but not science. His voting track record apparently shows that he consistently voted against Proportional Representation. Old-fashioned hypocrisy apparently is also part of his old-fashioned style.

*October 18: In memoriam Daphne Caruana Galizia (1964 – 2017), journalist, killed by a car bomb*.

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Again we can thank YouGov and Anthony Wells for making these data available.

The conclusions do not change, since the estimate apparently was fairly good.

It concerns a very relevant poll, and it is useful to have the uncertainty of the estimate removed.

The earlier discussion on *Proportional Representation versus District Representation* has resulted in these two papers:

*Two conditions for the application of Lorenz curve and Gini coefficient to voting and allocated seats*, MPRA 80297.*Comparing votes and seats with cosine, sine and sign, with attention for the slope and enhanced sensitivity to disproportionality*, MPRA 81389

Brexit stands out as a disaster of the UK *First Past The Post* (FPTP) system and the illusion that one can use referenda to repair disproportionalities caused by FPTP. This information about the real cause of Brexit is missing in the otherwise high quality overview at the BBC.

The former weblog text gave an overview of the YouGov polling data of June 12-13 2017 on the Great Britain (UK minus Northern Ireland) preference orderings on Brexit. The uncertainty of the estimate is removed now, and we are left with the uncertainty because of having polling data. The next step is to use these orderings for the various voting philosophies. I will be using the website of Rob LeGrand since this makes for easy communication. See his description of the voting philosophies. Robert Loring has a website that referred to LeGrand, and Loring is critical about FPTP too. However, I will use the general framework of my book “*Voting theory for democracy*” (VTFD), because there are some general principles that many people tend to overlook.

See the former entry for the problem and the excel sheet with the polling data of the preferences and their weights. LeGrand’s website requires us to present the data in a particular format. It seems best to transform the percentages into per-millions, since that website seems to require integers and we want some accuracy even though polling data come with uncertainty. There are no preferences with zero weights. Thus we get 24 nonzero weighted options. We enter those and then click on the various schemes. See the YouGov factsheet for the definition of the Brexit options, but for short we have** R** = Remain, **S** = Soft / Single Market, **T** = Tariffs / Hard, **N** = No Deal / WTO. Observe that the Remain options are missing, though these are important too.

248485:R>S>T>N

38182:R>S>N>T

24242:R>T>S>N

19394:R>T>N>S

12727:R>N>S>T

10909:R>N>T>S

50303:S>R>T>N

9091:S>R>N>T

22424:S>T>R>N

66667:S>T>N>R

9091:S>N>R>T

36364:S>N>T>R

6667:T>R>S>N

3636:T>R>N>S

12121:T>S>R>N

46667:T>S>N>R

15758:T>N>R>S

135152:T>N>S>R

9697:N>R>S>T

9091:N>R>T>S

8485:N>S>R>T

37576:N>S>T>R

16970:N>T>R>S

150303:N>T>S>R

The basic situation in voting has a Status Quo. The issue on the table is that we consider alternatives to the Status Quo. Only those options are relevant that are Pareto Improving, i.e. that some advance while none lose. Commonly there are more Pareto options, whence there is a deadlock that Pareto itself cannot resolve, and then majority voting might be used to break the deadlock. Many people tend to forget that majority voting is mainly a deadlock breaking rule. For it would not be acceptable when a majority would plunder a minority. The Pareto condition thus gives the minority veto rights against being plundered.

(When voting for a new Parliament then it is generally considered no option to leave the seats empty, whence there would be no status quo. A situation without a status quo tends to be rather exceptional.)

In this case the status quo is that the UK is a member of the EU. The voters for *R* block a change. The options *S, T *and *N *do not compensate the *R. *Thus the outcome remains *R.*

This is the fundamental result. The philosophies in the following neglect the status quo and thus should not really be considered.

PM 1. Potentially though, the *S, T *and *N *options must be read such that the *R *will be compensated for their loss.

PM 2. Potentially though, Leavers might reason that the status quo concerns national sovereignty, that the EU breaches upon. The BBC documentary “*Europe: ‘Them’ or ‘Us’*” remarkably explains that it was Margaret Thatcher who helped abolish the UK veto rights and who accepted EU majority rule, and who ran this through UK Parliament without proper discussion. There seems to be good reason to return to unanimity rule in the EU, yet it is not necessarily a proper method to neglect the rights of *R*. (And it was Thatcher who encouraged the neoliberal economic policies that many UK voters complain about as if these would come from the EU.)

On LeGrand’s site we get Plurality as the first step in the Hare method. *R *gets 35% while the other options are divided with each less than 35%. Thus the outcome is *R.*

(The Brexit referendum question in 2016 was flawed in design e.g. since it hid the underlying disagreements, and collected all dissent into a single Leave, also sandwiching *R *between various options for Leave.)

When we continue with Hare, then *R *remains strong and it collects votes when *S *and *N *drop off (as it is curiously sandwiched between options for Leave). Eventually *R *gets 45.0% and *T *gets 55.0%. Observe that this poll was on June 12-13 2017, and that some 25% of the voters “respect” the 2016 referendum outcome that however was flawed in design. I haven’t found information about preference orderings at the time of the referendum.

Borda generates the collective ranking *S *>* T *>* R *>* N*. This is Case 9 in the original list, and fortunately this is single-peaked.

Using Copeland, we find that *S *is also the Condorcet winner, i.e. wins from each other option in pairwise contests. This means that *S *is also the Borda Fixed Point winner.

The major point of this discussion is that the status quo consists of the UK membership of the EU. Part of the status quo is that the UK may leave by invoking article 50. However, the internal process that caused the invoking of article 50 leaves much to be desired. Potentially many voters got the suggestion as if they might vote about membership afresh without the need to compensate those who benefit from Remain.

Jonathan Portes suggested in 2016 that the Brexit referendum question was flawed in design because there might be a hidden Condorcet cycle. The YouGov poll didn’t contain questions that allows to check this, also because much has happened in 2016-2017, including the misplaced “respect” by 25% of the voters for the outcome of a flawed referendum. A key point is that options for Remain are not included, even though they would be relevant. My impression is that the break-up of the UK would be a serious issue, even though, curiously, many Scots apparently rather prefer the certainty of the closeness to a larger economy of the UK rather than the uncertainties of continued membership of the EU when the UK is Leaving.

It would make sense for the EU to encourage a reconsideration within the UK about what people really want. The Large Hadron Collider is expensive, but comparatively it might be less expensive when the UK switches to PR, splits up its confused parties (see this discussion by Anthony Wells), and has a new vote for the House of Commons. The UK already has experience with PR namely for the EU Parliament, and it should not be too complex to use this approach also for the nation.

Such a change might make it also more acceptable for other EU member states if the UK would Breget. Nigel Farage much benefited from Proportional Representation (PR) in the EU Parliament, and it would be welcome if he would lobby for PR in the UK too.

Nevertheless, given the observable tendency in the UK to prefer a soft Brexit, the EU would likely be advised to agree with such an outcome, or face a future with a UK that rightly or wrongly feels quite maltreated. As confused as the British have been on Brexit, they might also be sensitive to a “stab-in-the-back myth”.

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The UK general election was on June 8 and the poll was taken on June 12-13 so that the persons polled will have had vivid recollections. For this reason, these polling data can be considered quite important.

The poll generated data about confusions in the British electorate. It is useful to belabour the point, for Brexit is a key event and would have quite some impact for the coming decades. I would respect the UK decision to leave the EU but have my doubts when it is not based upon Proportional Representation (PR). A referendum gives proportions but referenda tend to be silly and dangerous, as they are an instrument of populism rather than of representative democracy. Indeed, it appears that the Brexit referendum question was flawed in design. The YouGov poll helps us to observe how confused a major section of the UK electorate is. Let us dig a bit deeper.

The following copies my weblog text of July 11, but now replacing the estimate by the real data.

Let voters consider the options * R = *Remain,

The YouGov poll presents the data in a *ranking matrix,* with the first preferences in the first row, then the second preferences, and so on. For the Brexit referendum outcome of 48% Remain and 52% Leave, for example, we might have the following setup. It is a guess, since the particular ways of Leaving were not included in the referendum question (and neither for Remaining). This example however is the result that you would expect if Remainers and Leavers would have the mentioned consistent orderings.

Observe that each voting weight (take e.g. 48) for a preference order list is put in precisely one place per row and per column, i.e. that it doesn’t occur more times in a single row or column. This explains why the border sums add up to 100.

The YouGov data, that I have been referring to, contain the results of a poll of 1651 adults in Great Britain, i.e. the UK excluding Northern Ireland. From page 13-16 we can collect these data for the whole of Great Britain for 2017. YouGov states that the sample has been weighted for social-economic and political indicators. It is not clear to me how the “Don’t know”s are being handled for this particular issue. See also this discussion by Anthony Wells.

- These are percentages, and both the row sums and the column sums should be 100, except for rounding errors.
- 35% has Remain in the first position, 47% has it in the last position, so that 9 + 8 ≈ 17% (a 1% missing due to rounding) has a confused position, in which Remain is sandwiched between some options for Leaving. We would wonder how such people would vote in a referendum when they are presented with only two options R or L. One cannot say that the referendum was only about the first positions in the rankings, for voters would tend to develop an expectation about what would be the likely kind of Brexit and vote accordingly. Some of these 17% might have voted Remain because they disliked the otherwise expected version for Leave. This might indicate that the outcome for Remain was overstated. Yet we have no information on subdivisions of Remain, that might cause an opposite effect. Some might be okay with Remain as it is but vote for Leave because they fear that the UK otherwise might also join up on the Eurozone or some United States of Europe. The reason why the Brexit referendum question was flawed in design is that it left too much to guess here.
- Remarkably, the split between R and L now in June 2017 would be 35% versus 65% instead of 48% versus 52% in 2016. In one single year Great Britain switched from fairly divided to a seemingly clear preference for Brexit (though divided upon how) ? I very much doubt this distribution, see this discussion on populism and DR. The electoral data still suggest more than 50% for Remain. In the
*July*weblog entry it is discussed that some 26% of the electorate say that they voted for Remain but accept the loss at the referendum, so that they “play along” with the winning side, focusing on what would be the best option for Leave. This seems loyal to some notion of democracy, but it would also be a misplaced loyalty to the flawed Brexit referendum question. (One can respect such loyalty, but it still makes sense to discuss it.)

Using techniques of apportionment we estimated the number of people per cell in the poll. However, we now have the actual data (rounded to one digit from multi-digit percentages times 1651):

With 4 options there are 4 possibilities for a first place, 3 remaining for the second place, 2 remaining for the third place, and then the final one follows. Thus there are 4 x 3 x 2 x 1 = 24 permutations for possible rankings. We already saw two of these: *R *> *S *> *T *> *N* and its reverse. Above ranking matrix is actually based upon these 24 possibilities.

Some of these 24 possibilities will be rather curious. It is not clear what to think about *R *> *N* *> **S *> *T *for example (Case 5 below). This would be a Remainer who would rather prefer No Deal to the EEA or some agreement not to have a trade war on tariffs. A tentative explanation is that this voter has a somewhat binary position, as *Remain* versus *No Deal At All*, while the other options are neglected.

Policy options can also be sorted in logical order. This gives rise to the theory of Single Peakedness. For the topics of *R, S, T *and *N *there is a logical scale from left to right. An example of single-peakedness is Case 7 below, with a ranking *S *> *R* > *T* > *N. *See the graph below. The 1st rank gets utility level 4, the 2nd rank gets utility level 3, the 3rd rank gets utility level 2, and the 4th rank gets utility level 1. The utility levels are just the reversed of the ranks, but then the case must be reordered to the logical order.

Voting theory has a core that assumes that voters are both autonomous and rational, so that any preference would have some logic. The logical order *R, S, T *and *N *might seem arbitrary to some voters who may think otherwise. We do not impose that order but invite voters who think otherwise to explain why they choose a different order. Potentially each voter has his or her own criteria so that the best is on top, and all other options follow in proper order. Voters with multiple peaks in their preferences would have more to explain to us to understand them than voters with a single peak. Without a good explanation, we cannot reject the possibility that there is some confusion.

The following are the YouGov data for the preferences orderings that underlie above YouGov results on percentages. See the excel sheet in the Appendix. This table shows only the percentages and not the numbers of people in the poll (that add up to above table), since the percentages are the main finding. Single dots are zero’s. The ConR / L and LabR / L subdivisions concern the voters in the poll who voted R or L in the 2016 Brexit referendum and who voted Con or Lab in 2017. They form only a part of the sample, so their sum doesn’t add up to the total on the left.

Some observations are:

- The YouGov summary ranking matrix already showed a rather even split on
*S, T*and*N,*but the data give a landscape with even more diversity in opinions. - Only 24.8% has the preference
*R*>*S*>*T*>*N*and only 15.0% its reverse, so that 60.1% has some mixture.

Above results for GB can be split up in on the peaks and sandwich. The combinations give the following percentages:

- The mentioned 60.1% split up again in 33.3% who are single peaked, and 26.8% who have multiple peaks.
- The sandwich of 17.3% splits up into 8.5% with a single peak and 8.8% with multiple peaks.
- Of the 26.8% with multiple peaks there are 10.5% who can join the Remainers with a first preference and there are 7.4% who can join the leavers with Remain in the last position (but various ways how to Leave).

The 8.8% would be a relevant section of the vote. They all voted Leave, but divided on *S, T* and *N*. Potentially the outcome of the 2016 Brexit referendum has been decided by the 8.8% GB voters who have Remain neither in the first or last position, and who do not follow the standard logical order on the options.

The division of ConR / L and LabR / L is losing its relevance because it are dwingling groups, they are changing loyalties, and their 2016 votes are becoming history while there are new issues. Yet, the 2016 referendum question was flawed, and it is relevant to see how sizeable parts of the UK electorate deal with the logical conundrum that they took part in.

- The 17.3% of the votes with Remain sandwiched can be found in the subdivisions in similar proportions.
- 28.6% of ConR voters and 55.2% of LabR voters are united on the preference
*R*>*S*>*T*>*N*. Presumably this was also the case in 2016, or there must be factors that increased or reduced consistency or confusion. - 30.8% of ConL and 22.4% of LabL are united on the preference
*N*>*T*>*S*>*R*. Presumably this was also the case in 2016, or there must be factors that increased or reduced consistency or confusion. - One might expect that ConR / L and LabR / L voters of 2016 would have the benefit of a party preference and thus show more consistency, yet the distribution of views is quite as much, and the sandwich with multiple peaks is quite present.
- The 2016 Conservative Remainers are loyal for 45.2% to the old point of view, but still vote for a Conservative party that is set on Leave. Part will be the misplaced loyalty for the flawed referendum. Alternatively, they voted for a minority in this party that still tries to bring balance ? (A good poll requires a focus group.) (And there is more in the world than just Brexit.)
- The 2016 Labour Remainers are 76.1% loyal to the old point of view. Yet Labour leader Corbyn also prefers a Brexit. It might be the pecularities of the British system of District Representation (DR) that caused these voters not to switch to LibDem. (But the LibDem also have a liberal policy that many voters for Labour dislike. The system of DR doesn’t favour the entry of new political competitors.)
- The 2016 Leavers have a high loyalty to the old view, ConL 88.2% and LabL 73.3%. Yet this doesn’t diminish the diversity of opinion about how to Leave.

- The ranking matrix is a fine way to summarize results, yet the preference ordering are more accurate on the underlying and relevant orders. The ranking matrix is merely a matter of presentation by the statistical reporter. A person in a poll who can answer on a ranking matrix in fact gives the personal preference ordering. The statistician can compound these data while not losing information on the permutations. From the permutations it is always possible to create a ranking matrix, yet the reverse requires estimation techniques which generate needless uncertainty.
- Asking for voter preference orderings in a poll is a useful exercise. It is not intended to propose this for general elections. For general elections it suffices that voters exercise a single vote for a party of choice. The condition however is Proportional Representation, otherwise there are serious distortions, see the earlier discussion on this weblog.
- The information on the rankings and implied preference orderings suggest a rather large state of confusion in the electorate of Great Britain. The notion of single-peakedness appears to be quite useful in highlighting the issue of the preference order. Perhaps we cannot quite call this “confusion” since voters might have their own logic to order the four options. Until there is more clarity on what strikes one as illogical, the term “confusion” seems apt though.
- It must be greatly appreciated that YouGov and Anthony Wells made these data available, since they provide a key insight in the state of opinion in Great Britain close to the general election of June 8 2017.

The excel workbook with the full YouGov data and the earlier estimate is: 2017-09-18-YouGov-Rankings-full-data

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Brexit stands out as a disaster of the UK *First Past The Post* (FPTP) system and the illusion that one can use referenda to repair disproportionalities caused by FPTP. This information is missing in the otherwise high quality overview at the BBC.

In the earlier *Puzzle on the YouGov poll* I estimated Brexit preference orderings from a summary statistic published by YouGov. The next step is to use these orderings for the various voting philosophies. I will be using the website of Rob LeGrand since this makes for easy communication. See his description of the voting philosophies. Robert Loring has a website that referred to LeGrand, and Loring is critical about FPTP too. However, I will use the general framework of my book “*Voting theory for democracy*” (VTFD), because there are some general principles that many people tend to overlook.

See the Puzzle weblog text for the problem and the excel sheet with the estimate of the preferences and their weights. LeGrand’s website now requires us to present the data in a particular format. It seems best to transform the percentages into per-millions, since that website seems to require integers and we want some accuracy even though the estimate is tentative. We can also drop the preference rankings with zero weights. Thus we get 14 nonzero weighted options. We enter those and then click on the various schemes. See the YouGov factsheet for the definition of the Brexit options, but for short we have** R** = Remain, **S** = Soft / Single Market, **T** = Tariffs, **H** = Hard / WTO. Observe that the Remain options are missing, though these are important too.

261841:R>S>T>H

53499:R>S>H>T

38386:R>T>H>S

60161:S>R>T>H

30087:S>R>H>T

44443:S>T>R>H

34960:S>T>H>R

22354:S>H>T>R

24777:T>S>H>R

15640:T>H>R>S

181873:T>H>S>R

49951:H>S>T>R

20475:H>T>R>S

161553:H>T>S>R

The basic situation in voting has a Status Quo. The issue on the table is that we consider alternatives to the Status Quo. Only those options are relevant that are Pareto Improving, i.e. that some advance while none lose. Commonly there are more Pareto options, whence there is a deadlock that Pareto itself cannot resolve, and then majority voting might be used to break the deadlock. Many people tend to forget that majority voting is mainly a deadlock breaking rule. For it would not be acceptable when a majority would plunder a minority. The Pareto condition thus gives the minority veto rights against being plundered. (When voting for a new Parliament then it is generally considered no option to leave the seats empty, whence there would be no status quo. A situation without a status quo tends to be rather exceptional.)

In this case the status quo is that the UK is a member of the EU. The voters for *R* block a change. The options *S, T *and *H *do not compensate the *R. *Thus the outcome remains *R.*

This is the fundamental result. The philosophies in the following neglect the status quo and thus should not really be considered.

PM 1. Potentially though, the *S, T *and *H *options must be read such that the *R *will be compensated for their loss.

PM 2. Potentially though, Leavers might reason that the status quo concerns national sovereignty, that the EU breaches upon. The BBC documentary “*Europe: ‘Them’ or ‘Us’*” remarkably explains that it was Margaret Thatcher who helped abolish the UK veto rights and who accepted EU majority rule, and who ran this through UK Parliament without proper discussion. There seems to be good reason to return to unanimity rule in the EU, yet it is not necessarily a proper method to neglect the rights of *R*. (And it was Thatcher who encouraged the neoliberal economic policies that many UK voters complain about as if these would come from the EU.)

On LeGrand’s site we get Plurality as the first step in the Hare method. *R *gets 35% while the other options are divided with each less than 35%. Thus the outcome is *R.*

(The Brexit referendum question in 2016 was flawed in design e.g. since it hid the underlying disagreements, and collected all dissent into a single Leave, also sandwiching *R *between various options for Leave.)

When we continue with Hare, then *R *remains strong and it collects votes when *S *and *H *drop off (as it is curiously sandwiched between options for Leave). Eventually *R *gets 44.4% and *T *gets 55.6%. Observe that this poll was on June 12-13 2017, and that some 25% of the voters “respect” the 2016 referendum outcome that however was flawed in design. I haven’t found information about preference orderings at the time of the referendum.

Borda generates the collective ranking *S *>* T *>* R *>* H*. This is Case 9 in the original list (including zero weights), and fortunately this is single-peaked.

Using Copeland, we find that *S *is also the Condorcet winner, i.e. wins from each other option in pairwise contests. This means that *S *is also the Borda Fixed Point winner.

The major point of this discussion is that the status quo consists of the UK membership of the EU. Part of the status quo is that the UK may leave by invoking article 50. However, the internal process that caused the invoking of article 50 leaves much to be desired. Potentially many voters got the suggestion as if they might vote about membership afresh without the need to compensate those who benefit from Remain.

Jonathan Portes suggested in 2016 that the Brexit referendum question was flawed in design because there might be a hidden Condorcet cycle. The YouGov poll didn’t contain questions that allowed to check this, also because much has happened in 2016-2017, including the misplaced “respect” for the outcome of a flawed referendum. A key point is that options for Remain are not included, even though they would be relevant. My impression is that the break-up of the UK would be a serious issue, even though, curiously, many Scots apparently rather prefer the certainty of the closeness to a larger economy of the UK rather than the uncertainties of continued membership of the EU when the UK is Leaving.

It would make sense for the EU to encourage a reconsideration within the UK about what people really want. The Large Hadron Collider is expensive, but comparatively it might be less expensive when the UK switches to PR, splits up its confused parties, and has a new vote for the House of Commons. The UK already has experience with PR namely for the EU Parliament, and it should not be too complex to use this approach also for the nation. Such a change might make it also more acceptable for other EU member states if the UK would Breget. Nigel Farage much benefited from Proportional Representation (PR) in the EU Parliament, and it would be welcome if he would lobby for PR in the UK too.

Nevertheless, given the observable tendency in the UK to prefer a soft Brexit, the EU would likely be advised to agree with such an outcome, or face a future with a UK that rightly or wrongly feels quite maltreated. As confused as the British have been on Brexit, they might also be sensitive to a “stab-in-the-back myth”.

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The earlier discussion on Lorenz curve and Gini was about the Dutch and UK general elections.

Both UK and France have district representation (DR) with a *First Past the Post* rule. In the UK this causes strategic voting, in which a voter may not vote for the candidate of first choice, but tries to block a candidate who might win but would be worst. France has elections in two rounds so that there is less need for such a strategy. The second round is between the two top candidates in the district, and thus one might try to get at least one good candidate in that position.

Proportional representation (PR) may allow a larger (but fairer) share of the seats for the more extreme parties, like the party of Geert Wilders in Holland, yet PR also allows more stability for the center. Thus PR tends to avoid the swings between extremes that might happen in systems of district representation (DR).

The French system seems to make it more difficult to determine the Lorenz curve and Gini coefficient. There are two rounds, and thus there is the question what data to take. However, the following choice suggests itself:

- The data of the first round provide the first preferences, and thus provide the votes.
- The data of both rounds provide the seats.

This choice finds support in the data. The first round has a turnout of 48.7% and 0.5 million invalid or blank votes. In the second round, more people remain at home, with a turnout of 42.6%, while those who vote produce almost 2 million invalid or blank votes, who apparently disprove of the available candidates or the system itself. Thus the higher turnout and lower blanks in the first round suggest that these indeed present the first preferences (with some limited level of strategy).

The Lorenz curve shows a rather surprising level of inequality, with a Gini of 41.6%. Compare the value of Holland with a Gini of 3.6%. If the blue line would cover the pink diagonal then there would be full proportionality.

The following table gives the data on turnout for the first round. The votes for *“Elected in the House”* is for parties that eventually got elected in the Legislative. The votes for *“Not in the House”* is for a radical leftist party that got votes in the first round but got no seat in none of the rounds.

The* wasted vote* consists of the invalid and blank votes and the latter “Not in the House”, to a total of almost 3%. A standard majority would be 289 seats of a House of 577 seats. If one would keep account of the wasted vote, then one might leave seats empty, or use a qualified majority of 298 seats, thus 9 more than usual.

When we divide the electorate by the number of votes per seat, then the Legislative would require 1222 rather than 577 seats. A majority would require 611 seats, which is more than the actual number of seats used. If one would want to keep account of the voters who did not turn out, then 51.3% or 296 of the 577 seats would be empty, or one would use the 611 seats as a qualified majority.

The new French President Emmanuel Macron had the highest score of 24% of the vote in the first round of the Presidential elections of 2017, with runner-up Marine Le Pen with 21.3%. Macron then won the second round with 66.1% (20.7 million) against Marine Le Pen with 33.9% (10.6 million) of the vote.

For the Legislative, Macron’s party REM got 27.6% while the Front National (FN) got 12.9% in the first round. For the Legislative Le Pen managed to get only 3 million votes, compared to the potential of 10.6 million at the presidential elections. With both rounds REM got 308 seats and FN got 8 seats.

These ratios would turn, if Le Pen would manage to motivate the voters of the presidential race to also support her for the Legislative. If the other parties would have a divided vote then Le Pen would benefit from First Past The Post.

For the UK in 2017 we calculated a Gini of 15.6% but this was a very tentative number since we had no estimate about the amount of strategic voting involved. For France we have an indication of the first preferences, namely from the first round.

France appears to have a surprisingly high Gini of 41.6%, which can be compared to the system of proportional representation (PR) in Holland that generates 3.6%.

This political inequality doesn’t bode well for the feelings amongst the French electorate about whether they are represented. The low turnout seems to reflect dissatisfaction rather than satisfaction. Such dissatisfaction might also translate into a protest vote over 4 years, especially when Macron doesn’t deliver.

Many observers in Europe seem to be happy with the election of Macron and his party REM, but the outcome is quite disproportional. If this disproportionality can happen for one party then it might also happen for another party – that one doesn’t like as much.

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Let voters consider the options * R = *Remain,

The YouGov poll presents the data in a *ranking matrix,* with the first preferences in the first row, then the second preferences, and so on. For the Brexit referendum outcome of 48% Remain and 52% Leave, for example, we might have the following setup. It is a guess, since the particular ways of Leaving were not included in the referendum question. This example however is the result that you would expect if Remainers and Leavers would have the mentioned consistent orderings.

Observe that each voting weight (take e.g. 48) for a preference order list is put in precisely one place per row and per column, i.e. that it doesn’t occur more times in a single row or column. This explains why the border sums add up to 100.

The YouGov data, that I have been referring to, contain the results of a poll of 1651 adults in Great Britain, i.e. the UK excluding Northern Ireland. From page 13-16 we can collect these data for the whole of Great Britain for 2017. YouGov states that the sample has been weighted for social-economic and political indicators. It is not clear to me how the “Don’t know”s are being handled for this particular issue. See also this discussion by Anthony Wells.

- These are percentages, and both the row sums and the column sums should be 100, except for rounding errors.
- 35% has Remain in the first position, 47% has it in the last position, so that 9 + 8 ≈ 17% (a 1% missing due to rounding) has a confused position, in which Remain is sandwiched between some options for Leaving. We would wonder how such people would vote in a referendum when they are presented with only two options R or L. One cannot say that the referendum was only about the first positions in the rankings, for voters would tend to develop an expectation about what would be the likely kind of Brexit and vote accordingly. Some of these 17% might have voted Remain because they disliked the otherwise expected version for Leave. This might indicate that the outcome for Remain was overstated. Yet we have no information on subdivisions of Remain, that might cause an opposite effect. Some might be okay with Remain as it is but vote for Leave because they fear that the UK otherwise might also join up on the Eurozone or some United States of Europe. The reason why the Brexit referendum question was flawed in design is that it left too much to guess here.
- Remarkably, the split between R and L now in June 2017 would be 35% versus 65% instead of 48% versus 52% in 2016. In one single year Great Britain switched from fairly divided to a seemingly clear preference for Brexit (though divided upon how) ? I very much doubt this distribution, see the pre-former weblog discussion. The electoral data still suggest more than 50% for Remain. In the former weblog entry it is discussed that some 26% of the electorate say that they voted for Remain but accept the loss at the referendum, so that they “play along” with the winning side, focusing on what would be the best option for Leave. This seems loyal to some notion of democracy, but it would also be a misplaced loyalty to the flawed Brexit referendum question. (One can respect such loyalty, but it still makes sense to discuss it.)

Using techniques of apportionment we can estimate the actual number of people per cell in the poll. My estimate is (and YouGov would have the true numbers):

With 4 options there are 4 possibilities for a first place, 3 remaining for the second place, 2 remaining for the third place, and then the final one follows. Thus there are 4 x 3 x 2 x 1 = 24 permutations for possible rankings. We already saw two of these: *R *> *S *> *T *> *H* and its reverse. Above ranking matrix is actually based upon these 24 possibilities.

Some of these 24 possibilities will be rather curious. It is not clear what to think about *R *> *H* *> **S *> *T *for example (Case 5 below). This would be a Remainer who would rather prefer a Hard Brexit to the EEA or some agreement not to have a trade war on tariffs. A tentative explanation is that this voter has a somewhat binary position, as Remain versus Hard Brexit, while the other options are neglected.

Voting theory may assume voters that are both autonomous and rational, so that any preference would have some logic. This gives rise to the theory of Single Peakedness. Potentially each voter has his or her own criteria so that the best is on top, and all other follow in proper order. However, for the topics of *R, S, T *and *H *there is a logical scale from left to right. Voters with multiple peaks in their preferences have more to explain than voters with a single peak. An example of single-peakedness is Case 7 below, with a ranking *S *> *R* > *T* > *H. *See the graph below. The 1st rank gets utility level 4, the 2nd rank gets utility level 3, the 3rd rank gets utility level 2, and the 4th rank gets utility level 1. The utility levels are just the reversed of the ranks, but then the case must be reordered to the logical order.

The following are estimates for the preferences orderings that would underlie above YouGov results. The estimate minimises the sum of squared error on that ranking matrix, with a weight of 10 for the error on the first preferences. See the excel sheet in the Appendix. This table shows only the percentages and not the numbers of people in the poll (that add up to above table), since the percentages are the main estimation result. Single dots are zero’s. Some have been caused by explicitly setting the possibility of such a preference ordering to zero, see the “comment” keyword for the reason. (A technical reason are also the degrees of freedom.) The ConR / L and LabR / L subdivisions concern the voters in the poll who voted R or L in the 2016 Brexit referendum and who voted Con or Lab in 2017. They form only a part of the sample, so their sum doesn’t add up to the total on the left. The percentages have a decimal to allow easier identification, not for claimed accuracy.

Some observations are:

- The YouGov summary ranking matrix already showed a rather even split on
*S, T*and*H,*but the estimate generates a landscape with even more diversity in opinions. - Only 26.2% has the preference
*R*>*S*>*T*>*H*and only 16.2% its reverse, so that 57.7% (addition effect) have some mixture.

Above results for GB can be split up in on the peaks and sandwich. The combinations give the following percentages:

- The mentioned 57.7% split up again in 34.6% who are single peaked, and 23.1% who have multiple peaks.
- The sandwich of 17.1% splits up into 10.4% with a single peak and 6.7% with multiple peaks.
- Of the 23.1% with multiple peaks there are 9.1% who can join the Remainers with a first preference and there are 7.3% who can join the leavers with Remain in the last position (but unclear how to Leave).

The 6.7% would be a relevant section of the vote. They all voted Leave, but divided on S, T and H. Potentially the outcome of the 2016 Brexit referendum has been decided by the 6.7% GB voters who have Remain neither in the first or last position, and who do not follow the standard logical order on the options.

The division of ConR / L and LabR / L is losing its relevance because it are dwingling groups, they are changing loyalties, and their 2016 votes are becoming history while there are new issues. Yet, the 2016 referendum question was flawed, and it is relevant to see how sizeable parts of the UK electorate deal with the logical conundrum that they took part in.

- The 17% of confused votes on the first preference can be found in the subdivisions in similar proportions.
- 33.5% of ConR voters and 61.1% of LabR voters are united on the preference
*R*>*S*>*T*>*H*. Presumably this was also the case in 2016, or there must be factors that increased or reduced consistency or confusion. - 28.9% of ConL and 19.4% of LabL are united on the preference
*H*>*T*>*S*>*R*. Presumably this was also the case in 2016, or there must be factors that increased or reduced consistency or confusion. - One might expect that ConR / L and LabR / L voters of 2016 would have the benefit of a party preference and thus show more consistency, yet the distribution of views is quite as much, and the sandwich with multiple peaks is quite present.
- The 2016 Conservative Remainers are loyal for 45.5% to the old point of view, but still vote for a Conservative party that is set on Leave. Part will be the misplaced loyalty for the flawed referendum. Alternatively, they voted for a minority in this party that still tries to bring balance ? (A good poll requires a focus group.) (And there is more in the world than just Brexit.)
- The 2016 Labour Remainers are 76.1% loyal to the old point of view. Yet Labour leader Corbyn also prefers a Brexit. It might be the pecularities of the British system of District Representation (DR) that caused these voters not to switch to LibDem. (But the LibDem also have a liberal policy that many voters for Labour dislike. The system of DR doesn’t favour the entry of new political competitors.)
- The 2016 Leavers have a high loyalty to the old view, ConL 86.8% and LabL 74.4%. Yet this doesn’t diminish the diversity of opinion about how to Leave (though
*T*gets more votes than*H*).

For n = 4, there are n! = 24 variables, (n-1)^2 = 9 independent equations within the matrix, and there is the addition constraint 1, so that the degrees of freedom are 14. Yet we cannot randomly set weights to zero. If there would be nonzero weights for single-peaked preferences only, then the YouGov ranking matrix would show zeros, which it doesn’t. Thus it takes some arbitration which weights to exclude. There are quite a lot of possibilities, and I can only hope that my choice was wisest. As said, the percentages provided by YouGov have been scaled up to the table given above, and this allows us to determine the error in the estimate. Due to degrees of freedom the calculated error is quite low. The use of an error measure is limited to comparing estimates and not something that is useful to mention here. As said, YouGov have the proper data, and it must be hoped that they will look into this.

- The ranking matrix is a fine way to summarize results, yet the preference ordering are more accurate on the underlying and relevant orders. This is merely a matter of presentation by the statistical reporter. A person in a poll who can answer on a ranking matrix in fact gives the personal preference ordering. The statistician can compound these data while not losing information on the permutations. From the permutations it is always possible to create a ranking matrix, yet the reverse requires estimation techniques which generate needless uncertainty.
- Asking for voter preference orderings in a poll is a useful exercise. It is not intended to propose this for general elections. For general elections it suffices that voters exercise a single vote for a party of choice. The condition however is Proportional Representation, otherwise there are serious distortions, see the earlier discussion on this weblog.
- The information on the rankings and implied preference orderings suggest a rather large state of confusion in the electorate of Great Britain. The notion of single-peakedness appears to be quite useful in highlighting the issue of the preference order.

I slightly revised the manner of rounding and included case 16 for all columns. The polished up excel workbook is: 2017-07-15-YouGov-Rankings

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One result has been the use of the Lorenz curve and Gini coefficient to show the disproportionality in the UK between votes and seats. Almost all EU members have Proportional Representation (PR) with clear exceptions of the UK and France that have District Representation (DR). Apparently, this is a main reason for the influence of populism in the latter two countries. DR allows that politicians are elected with a minority of the vote, which causes a gap with the majority. Politicians like David Cameron can use a referendum to introduce an element of proportionality. Yet referendum questions are quickly flawed.

Another surprise for me was the existence of the *Re-Leavers* who make up some 23% of the electorate, and who are very likely also a major section in the House of Commons that supported the invoking of article 50.

Apparently many British voters are awfully respectful of democracy, and while they voted for Remain, they accept the referendum outcome, and let their voting behaviour now be guided by Leave. In other words: they no longer operate as *voters *who are supposed to express their first preference, but they operate as *politicians *who develop policy using such preferences.

Voters are better not confused about the following angles:

- It is one thing to accept the Brexit referendum outcome as a fact. Please accept facts.
- It is another thing to discuss the consequences of that fact.
- There is always the distinction between your first preference and dealing with new developments.
- Your first preference can change, but rather only because of arguments, and not just because of a majority view.

For me, it is easy to say this, in a country that is used to PR. In the UK case of DR it may well be that strategic voting requires voters to run with with herd. Nevertheless, the Re-Leavers cause quite a confusion in the voting record. Also for the general elections of 2017 we now can observe that we don’t know what people really want.

Anthony Wells provided and discussed these data that show the impact of the Re-Leavers. Let me quote the main part, and for this quotation I also moved their copyright sign up.

These early June data are most relevant for judging the June 8 2017 UK general election. Apparently 26% of all adults in Great Britain (UK excl. Northern Ireland),* but also 53% of the voters who voted Remain in 2016,* reason as follows:

I did not support Britain leaving the EU, but now the British people have voted to leave the government has a duty to carry out their wishes and leave.

I consider this an illogical and rather undemocratic statement.

- Logic would require the annulment of the referendum outcome, and not to take it seriously.
- In representative government, it is Parliament that determines policy, not the people by some referendum.

Most of the EU has PR and thus the notion of representative government. The 2016 Remain voters want to remain in the EU, but, 53% apparently also reject the EU notion of representative government, and instead they appeal to the populism of referenda.

A few days ago, I rephrased one aspect as follows: With *R *for Remain, *S *for Soft (EEA), *T *for some Tariffs, and *N *for No Deal (WTO), there are 6 possible strict preferences for a deal, from *R *> *S *> *T* (Theresa May before the referendum) to *T *> *S *> *R *(Theresa May after the referendum). If *S *and *T *are collected in *L *(Leave) then there arises the claimed binary choice between *R *and *L. *Voters who are in the categories *S *> *R *> *T* or *T *> *R *> *S *would face a hard question. If they expect that *R *might win, but also that their own preferred option might not win, should they still go out and vote ? They might decide not to turn out, or develop assumptions about what *L *actually might become, given what what they think about future developments. Similarly for the versions of *R.* See the voting theory about single peaked preferences (and these are not single peaked but double peaked). Overall it is a fallacy that there is a binary choice. Lawyers can argue that one either invokes article 50 or doesn’t invoke it, yet the referendum isn’t such a legal case, for it is an issue of policy preferences.

In fact, above YouGov poll provides us a bit more information on this issue. Look at this section on their page 15:

Look at the column of the total (with 1651 people in the weighted sample). 35% are clearly for Remain, in their first rank. 47% are clearly against Remain, in their last rank. Thus the middle 8 + 9 ≈ 18% (rounding error) is rather confused, for they put Remain between one of the Leave options. How would they have to vote at a referendum that only allows R or L ? We find similar percentages for the subgroups on the right hand side.

The discussion in the UK would be served by greater awareness of these distinctions:

- The difference between voting for your first preference (setting the target) and trying to second-guess politicians (as if you are in the driver seat).
- A valid question and an invalid or flawed one, like the Brexit referendum question.
- The crucial differences between Proportional Representation (PR) and District Representation (DR), linked to the distinction between representative democracy (mostly PR) and populism (mostly DR).
- There is also something not discussed in the above, but that is the difference between the failing Trias Politica and improved democracy with an Economic Supreme Court.

(Updated July 11 2017) (September 18 2017: Changed “Hard, H” in “No deal, N”)

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Countries with DR run the risk that the seats in the House of Commons do not reflect the popular vote, and then they might try to repair this with a referendum, that is proportional. In countries with PR there would be no need for referenda.

The UK had the Brexit referendum of June 23 2016, that generated a relatively high turnout of 72.2%, with 51.9% Leave and 48.1% Remain. The Leave vote concerned only 72.2% * 51.9% = 37.5% of the electorate, and apparently the Leavers were quite motivated to turn out.

Many wonder how the UK general election of June 8 2017 squares with this referendum. In the previous weblog text, I already discussed caution. Some key aspects for digging deeper are:

- The election was on many more issues than only Brexit, and had a turnout of 68.8%. A poll of YouGov showed that 17% who voted at the referendum didn’t vote in 2017.
- Turnout is not proportional in the Remain / Leave segments.
- Though the UK is set to Leave, it matters what you call “Leave”. The UK has a discussion about Hard and Soft Brexit, see these options. The political parties tend to be ambiguous about what they want, and voters thus have to guess.

PM. The events after the June 8 elections show the complexity of the situation. Labour leader Jeremy Corbyn apparently favours a hard Brexit like Conservative PM Theresa May. The evening of June 29 showed a House rejection by 227 abstaining, 322 against and 101 for a Queen Speech motion to target for remaining in the single market and customs union (regarded as a soft Brexit). Yes voters were Lab 49, LD 12, SNP 34, PC 4, Green 1, Hermon 1. It might still be that some prefer Remain but only vote for a soft Brexit to avoid a hard one. Labour MPs who abstained might have done so only tactically. The whip rule was not to vote for this particular amendment but to vote for the Labour amendment, to deliver a Brexit that prioritises jobs and delivers the “exact same benefits” of the European single market and customs union. The latter Labour motion is quite an incongruity. Yet Corbyn demoted some frontbenchers for not sticking to the party whip.

For both PR and DR alike, there is the problem that the House doesn’t represent (i) who didn’t vote and (ii) whose votes got wasted. There is also the distinction between the registered electorate and the unregistered.

See the case of Holland 2017 with PR, here.

The wasted votes in the UK amount to 3.5% of the votes. A solution might be to leave 3.5% of the 650 seats empty. Alternatively, the standard majority of the House of 326 seats is replaced by a qualified majority of 337 seats.

The 14.5 million electors who didn’t turn out are more than the 13.6 million who voted for the Conservatives, who got 317 seats. An alternative might be to leave 202 seats empty or use a qualified majority of 489.

The present situation thus means that the Conservatives with their 317 seats seem to be overpresented with their view on Brexit. We don’t really know, since the Unregistered and Don’t Vote did not come to vote to show their opinion.

YouGov provides us with (at least) two polls for 2015 and 2017 that tell us how parties are divided on Remain or Leave. In this, the 18 seats for Northern Ireland tend to be excluded, so that the data concern Great Britain, with England, Wales and Scotland.

The first poll is from 2015, and let me quote their graph. Observe that the Undecideds are not in the graphs. Apparently 7% of the Ukippers did not want to Leave immediately.

The second poll is after the 2017 election, and asks how those voted in 2017 also voted in 2015 and the referendum. Let me again quote from their graph, and see their website for the *data sheet*. Observe that the Undecideds are now excluded, since the Undecideds obviously did not vote either Yes or No.

The challenge is to transform these data into party flows. Considerations are:

- The data on 2015 are rather rough while the data on 2017 benefit from the 2016 referendum.
- The poll of 2015 however is still more useful since many voters in 2015 may not have participated in the referendum. Thus, using the 2015 poll, the Undecideds of 2015 can be allocated to Remain or Leave, based upon party flavour.
- From the voters of the referendum in 2016, we can subtract the deceased, then allocate the transition flows as observed in the poll of 2017, with e.g. the key information that 15% of Remain and 26% of Leave in 2016, thus on average 20.7%,
*did not vote*in 2017. Thus we should not be surprised if the 2017 outcome might be relatively more in favour of Remain. - For the transition flows, we assume that the
*dispositions*to Remain or Leave do not change. Check this YouGov poll in March 2017. For example, when a Conservative Leaver switches to Labour, then this indicates a shift from Hard to Soft, and not a shift from Soft to Remain. - The above gives us the divisions in the 2016 electorate that also voted in 2017.
- We allocate the new voters in same proportions: not only the attainers (coming of age, turnout of 57%) but also those who didn’t vote at the referendum, and the new registrations for the electorate (who we might assume that they only register to actually go and vote). These data are rough, namely measured over the full year, so that we must assume e.g. that the deaths in the last half of 2016 are about the same as those in the first half of 2017.

The voter dynamics are a bit remarkable. Of the vote of about 32 million there is a stable core of only 26 million. The 26% leavers who do not vote at the general election in 2017 amount to 4.4 million voters. The surviving non-voters at the referendum amount to 12.3 million, of which 5 million or 41.4%, decide to vote at the general election in 2017, while the other 7.2 miljoen do not show up at both occasions.

Northern Ireland and other parties have been included again, so we leave GB and return to the UK.

- For the minor parties we may assume (see also the BBC on MPs) that Remain are Plaid Cymru, Sinn Fein, Independent Unionist (Hermon), and Leave is DUP. The popular vote also has parties not represented. UKIP will be Leave, and the Speaker and the remaining wasted vote will be Blank.
- Percentages of the popular vote will include the wasted vote of 3.5% (including UKIP) in the denominator.

Remain is horizontal, Leave is vertical. The diagonal or 45º line gives the split. Below the diagonal a party has more Remain votes, above the diagonal it has more Leave votes. 2015 has Squares and 2017 has Triangles.

The axes are in millions of voters. For example, UKIP (on the bottom left) in 2015 had almost 4 million votes, of which apparently 7% Remain. UKIP in 2017 lost almost all of its votes, mostly to the Leave part of the Conservatives. For example, LD (LibDem) is stationary.

The votes for the House of Commons are a sum and would not fit in the graph, and thus 11 million votes have been subtracted on both axes.

- The division in the House in 2015 was {Remain, Leave} = {17.3, 12.8} but depicts {6.3, 1.8}.
- In 2017 this became {16.5, 15.2} which depicts {5.5, 4.2}.

A major explanation for this huge shift in public opinion is that many Remainers apparently want to* respect* the Brexit referendum outcome. The vote on June 8 also was after invoking article 50. Voters regard Brexit as a given, and voted for parties to make the best of it. See this YouGov poll on the Re-Leavers.

The following is a chart for the division in 2017, again on votes and not seats. Remain still seems to have the larger share of 51.0% compared to Leave at 47.2%. Observe that the latter includes UKIP though it didn’t get a seat. The remaining 1.8% Blank are for the Speaker and the remainging wasted vote. These parties may well have an opinion, or their own subdivisions, but those do not show here because they don’t show in above polls.

Prime Minister May made a deal with DUP, including that DUP must respect the Brexit deal that May will achieve with the EU. The 317 Conservative seats and the 10 seats for DUP generate a majority of 327, just 1 above the standardly required 326.

Let us look closer at this. The following data have been retrieved from the BBC.

In 2015 DUP had 8 seats. DUP in 2017 managed to get 2 more seats at the cost of others, with a huge wasted vote of 32.6%. A seat in the UK House of Commons required only 30,410 votes while the UK average is 47,901. At that rate NI should have 41 seats if the total electorate should be represented. Then a majority of 50% would require 21 seats, more than there actually are.

The Lorenz curve for Northern Ireland looks quite like the unequal Lorenz curve of the UK in 2015, when UKIP got only 1 seat, see the graphs in an earlier weblog text. There is a PR Gini of 36.7%, while Holland has a PR Gini of 3.6. The excel sheet for Northern Ireland is: 2017-Northern-Ireland-Lorenz-Gini

PM 1. Sinn Fein apparently never visits the sessions of the UK Parliament. This doesn’t seem to be relevant for this discussion that concerns the popular vote.

PM 2. In the EU Parliament smaller countries like Holland have a disproportionate number of seats compared to Germany. The situation is similar for NI in the UK. This would tend not to be relevant for the present discussion, except that the low value in NI in 2017 derives from the huge wasted vote.

Why doesn’t the UK split up the parties along the Remain / Leave divide ? Thus we would get ConR, ConL (Hard), LabR, LabL (Soft), and so on. With proportional representation (PR), then we would not have to rely on polling, but would see the proper allocations directly in the House of Commons.

There might be other (attractive or unattractive) features. When Mrs. May would have been part of the ConR party in 2015, then this would have made it more difficult for her to start leading the ConL party after the Brexit referendum outcome. Or when Corbyn would have been part of the LabL party in 2015 then he could not have easily gotten a hold on the LabR as happens now.

There was an entertaining poll on this very idea itself by YouGov in 2015, here. That poll isn’t quite convincing given the wrong forecast for the Brexit referendum outcome, which suggests some misrepresentation.

However, the system of DR tends to make such break-ups unattractive for the parties involved. Smaller parties are destined to lose in the FPTP system. Thus another argument for PR would be that it provides voters with more clarity on party positions. The final result may well be a compromise, but this can be bargained by the professionals in Parliament, and better is not tried by the voters in the isolation of the ballot box.

Also, I have argued before that the Brexit referendum question was flawed in design. We don’t know what the voters really want. We might need to further split up the parties, e.g. in “Labour Leave, as UK” and “Labour Leave, but Scotland Independent so that Scotland can Remain” and so on. In countries with PR like Holland, one tends to think that parties divide up along issues that people tend to think of as fundamental, in which ideology tends to be of key importance.

The issue of Brexit might not be as fundamental as many people in the UK seem to think. The issue of immigration got translated into the question on Brexit, and this might have to do with the DR structure, see also France. This might be a topic to look closer into.

Conclusions are:

- Above calculations obviously are fairly rough, and based upon perhaps arbitrary assumptions (but that is what arbitrage is). They basically set the stage for a better opinion poll that replaces these uncertainties by normal poll variety.
- Above estimate suggests that the popular vote would be still very close to a majority on Remain.
- The above doesn’t look at the views of the 1.8% of the vote that hasn’t been in the YouGov polls that were used here or that could not easily been determined by the other reports.
- The share for Leave is a mixed bag, with Hard and Soft.
- The Conservatives have a disproportionate share of the seats, which causes that their view on Brexit finds disproportion too (and their view might be the view of their leadership).
- Northern Ireland has a huge disproportion in 2017, similar to the UK House in 2015. The Conservative + DUP majority in 2017 thus is of dubious value.
- When the UK electorate better understands that the Brexit referendum question was flawed in design (see here), then many voters will lose the tendency to vote for Leave only out of respect for the Brexit referendum result, and then vote for their own opinion.

The main point is: We see the weird brew of populism and DR.

- In 2015, UKIP with 12.5% of the vote got only 1 seat, and the unrest in the Conservative party caused David Cameron to call the referendum: which is populism, since a democratic response would have been to call for PR.
- Now in 2017 a disproportionality in Northern Ireland seems to facilitate the Hard Brexit that Theresa May now seems to want.

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Parliament has a mandate, potentially provided by the Magna Carta of 800 years ago, though historians and legal experts will discuss aspects.

- There is no reason to argue that this mandate would vary with a particular issue.
- When someone questions Brexit then this doesn’t put Parliament’s mandate into doubt.

It might be possible for political scientists to look at a particular issue and argue that the mood in Parliament concurs with the public mood. This would require a poll targeted on that issue, both for Parliament and the public, so that we can do a statistical test on the difference of the averages. ABFT provide no such a poll. They refer to the general election, but the election concerned more issues. It is too simple to argue that the positions of the parties and candidates were clear on Brexit and that the popular votes concurred with such views.

ABFT refer to the Brexit referendum of 2016 as one way how the electorate has expressed its view. However, scientists should clarify that the Brexit referendum question was flawed in its design. I explained this in an earlier LSE Brexit blog text of May 17, see here. My text relies upon a sound theoretical foundation. Representative democracy should not be confused with populist referenda, since voters in the ballot box cannot bargain.

There is no need to repeat my text here. The discussion on the weblog however was helpful in highlighting the distinction between the legal situation and the moral issue. Whether one legally invokes article 50 is a binary Yes / No issue. The Brexit referendum question was not about this legality but about policy preferences.

The referendum question wasn’t:

- “Britain has to invoke article 50 in 2017 whatever happens”
- versus “It is not so that Britain must invoke article 50 in 2017 whatever happens”.

For policy, there are various options, and electors have (conditional) expectations. With this ambiguity, the referendum question generated a situation of “*Garbage in, garbage out*“. We really don’t know what the electorate wants.

For example, some Scots who voted for *both independence and Remain* might now decide to stay in the UK with Brexit, which means *both dependence and Leave* and thus is quite opposite to their original views. Apparently they fear many years in the wilderness when out of both EU and UK. Clearly these voters have had conditional expectations and they have not been invoking legal rules. The Brexit referendum question clearly doesn’t provide the clarity that many attach to it.

Article 50 has been invoked and Brexit will happen, potentially abruptly on March 29 2019 when there is no agreement. One might argue that it is silly to continue to mention the problem of the Brexit referendum question.

However, my impression is that the 27 EU members may well be sensitive to the scientific observation that the Brexit referendum question was flawed in design. An international agreement on this observation, potentially also at the ECJ, might create an option for the UK to propose to annul the invoking of article 50.

If the UK should wish to annul Brexit before March 29 2019 then this can only be done with agreement of the 27 EU members. These members will not be swayed easily when the UK would use economic and financial arguments to explain its regret. But the issue of the referendum question is clear and impartial. The diagnosis may well be that the UK has fallen victim to a populist mood, and up to now lacks insufficient checks and balances to correct this. It fits the role of scientists to clarify this, which in itself is a check and balance.

ABFT propose that scientists defend the Brexit referendum question as sound, or at least want them to be silent, given that article 50 has been invoked and Parliament apparently has no intention of trying to annul it. I would hope that they are open to the idea that scientific criticism still would be possible.

They present events as a rational story from referendum to Parliamentarian mandate, while the events might well be diagnosed as an accidental course of history, in which the UK happened to take a one-way street because a street sign was flawed.

A YouGov poll also showed that people show loyalty to a seeming majority:

“There is a third group who change the dynamics of EU-related arguments – the “Re-Leavers.” These are people who voted to Remain in the EU and many still think that leaving was the wrong decision, but crucially now believe the government has a duty to carry out the will of the British people.”

“When taking this into account, we can split the country into three groups instead of two: The Hard Leavers who want out of the EU (45%); the Hard Remainers who still want to try to stop Brexit (22%); and the Re-Leavers (23%). The other 9% don’t know.” (YouGov May 12 2017)

Also the UK audience would be served with a better understanding that the Brexit referendum question was flawed.

My comments w.r.t. Brexit are scientifically warranted, yet ABFT in their final paragraph suggest that this criticism is “subverting the popular mandate”, “undermining the position of the EU chief negotiator” and “a subversion of the democratic process that led to the country’s decision to exit the EU”. Referring to the Brexit referendum result they ask: “Were there no lessons learnt from the rise of populism?”

ABFT are confused on the distinction between representative democracy (Parliament) and populism (referenda, neverendum). David Cameron’s decision to have a referendum was populism itself, while a democrat would have adopted proportional representation (PR), see below. See this interview by Protesilaos Stavrou on the distinction. Potentially one might use a referendum to ratify a constitutional change, but then one would tend to require that at least 1 / 2 of the population would decide, say a 3 / 4 majority of a turnout of 2 / 3. In Holland, a constitutional change, adopted (in proposal) by one parliament, requires ratification by another parliament after new elections. The Brexit referendum did not meet such standards (a turnout of 72% and a majority of 52%), apart from the design flaw on the question. It is a bit too simple to hold that there are now two UK Parliaments who are on a course towards Brexit, since article 50 was invoked by only one Parliament (referring to that flawed referendum).

ABFT are also confused on subversion. When there are serious points on content then those have to be dealt with, and it doesn’t help to neglect or denounce valid criticism.

At first it seems as if proportional representation (PR), differing from current district representation (DR), doesn’t matter, and would be another topic than Brexit. It is a dead topic too, as John Cleese already showed in 1987, here. The ABFT article however clarifies that they entertain some assumptions that better be discussed too.

- The use of DR rather than PR is part of the
*lack of checks and balances*that explain the populist drift in the UK. While DR strengthens existing powers, PR allows easy entry of challengers, which is the competition that economists like to see. Brexit would quite likely never have happened if the UK had had PR. In 2015 UKIP got 12.5% of the votes and only 1 seat. With a similar share of seats, UKIP would have become just one of the parties instead of the nightmare of the mainstream parties. In Holland Geert Wilders with a similar percentage has a marginal existence. If needed, mainstream politicians can agree on a Grand Coalition, like in Germany. - ABFT suggest that support for Conservatives or Labour might mean support for either Hard or Soft Brexit but then they neglect the logic of DR, that voters must vote strategically.
- It is better to discuss annulment of Brexit within an environment with PR than one with DR. Britain might have a year to redesign its electoral system and have proper elections.

PR is the key design criterion for democracy, and districts are mainly a historical hangup. Consider *n* voters and *s* = 650 seats, so that the quota is *q* = *n* / *s*. This allows *s* / 2 = 325 districs of size 2*q*, since *s* / 2 * 2*q* = *n* again. With 45 million voters there might be an average distict size of about 140,000, correcting for turnout. Let a district candidate be elected when he or she has more than the quota *q*. Normally this amounts to *q* / (2*q*) = 50% of the district. One might allow a lower turnout in districts, and let a candidate be elected with at least 50% of the local vote. The unfilled seats can be filled by the nationwide PR criterion. Thus, a fundamental choice for PR still allows a feature of districts if so desired.

In France, the UK, USA and India DR is the key design feature whatever happens to PR. They haven’t overcome the historical hangup yet.

The distance of DR to PR can be expressed in the PR Gini coefficient, see the excel sheet and graphs. David Cameron in 2015 had a coefficient of 29.7%. In 2017 Theresa May has a PR Gini coefficient of 15.6% while Holland has 3.6%. The UK thus is strikingly disproportional.

The major non-PR impact in Holland is from the 2% wasted votes for small parties that get zero seats. In 2017 the UK had 3.5% wasted votes for parties that got no seats. One might argue that the wasted votes should be omitted from the Gini, yet, they rather stand out as a sore spot in current representation. A proportional representation of the wasted vote *w* in total* n* is possible by leaving seats empty or by filling the seats and taking a qualified majority *f* = 1/2 / (1 – *w* / *n*). In 2017 *f* = 50% / 96.5% = 51.8%. A representative majority in a full House of 650 seats then requires 337 seats, and not 326. See here.

There is a curious miscomprehension in the UK about what democracy actually is.

**(a)** This miscomprehension also exists in the* Electoral Reform Society* (ERS). There is a ranked system called “Single Transferable Vote” (STV) that has some PR properties when applied to the whole country. ERS however applies it to districts, and DR destroys nationwide PR again. Incongruously, the ERS keeps saying that their STV would be PR while it isn’t. See my criticism and counterexample here. The continued support of ERS for districts is part of the confusion that blocks the change to PR. (And with 4 candidates per district, their districts might need to have some 500,000 voters, curiously undermining the reason of “closeness” to have a district.)

**(b)** The 2011 referendum on PR was actually a referendum on the method of “Alternative Vote” (AV), which is a suboptimal suggestion too.

**(c)** It is apparently a misconception and dogma amongst UK voting theorists too that the (more complex) methods of ranked voting like STV or AV must be used at the level of voters. There is no need for this. It would suffice if the UK would adopt the Dutch system of Open Party Lists. The (more complex) methods of ranked voting can be applied by the professionals in Parliament, if needed, and those methods would still be proportional if the party weights were proportional.

**(d)** There is now an initiative *“Make Votes Matter*” (MVM) that strictly targets PR but that suffers from such conceptual problems too.

- ERS signed their declaration though ERS, as said, also proposes a system of STV with districts that is not PR.
- MVM might generate a discussion on ranked systems while such complexity might block PR again. It is better to clearly state a preference for “PR in Open Party Lists”, for otherwise we might end up with another situation that it is unclear what people signed up or voted for.

Parliament’s mandate comes rather from Magna Carta, not Brexit, and scientists have every reason to question both Brexit and the lack of proportional representation (PR).

I now notice the contribution by Jonathan Portes on the LSE Brexit blog about the influence of the Condorcet paradox (June 15 2016, that appeared on the NIESR blog on June 6 2016). I referred to voting theory too, though after the referendum of June 23, in this comment of June 29 2016, that was later summarized on the LSE Brexit blog in May 2017.

- Portes suggests that there is a Condorcet paradox, while my argument only mentions that there might be one, and that we basically don’t know, even after the Brexit referendum “outcome”. The argument by Portes is less strong since he provides a speculation, and others might argue that it is only a speculation.
- My main reason to refer to the Condorcet paradox was to explain that many voting theorists themselves do not properly understand voting theory. See the argument about the distinction between voting and deciding. Indeed, Portes repeats the confusion: “We may not be inconsistent individually but we can be so collectively.” Potentially a collective might be inconsistent, of course, as an individual might be. Yet, the point concerns the distinction between intransitive vote counts and the transitive decisions (e.g. indifference) based upon such vote counts.

For the Brexit referendum question, the problem might likely be a bit different. With *R *for Remain, *S *for Soft (EEA) and *H *for Hard (WTO), there are 6 possible strict preferences, from *R *> *S *> *H* (Theresa May before the referendum) to *H *> *S *> *R *(Theresa May after the referendum). If *S *and *H *are collected in *L *then there arises the claimed binary choice between *R *and *L. *Voters who are in the categories *S *> *R *> *H* or *H *> *R *> *S *would face a hard question, and might decide not to turn out, or develop assumptions about what the *L *actually might mean given what what they think about future developments. Similarly for the versions of *R. *

Overall, the argument suffices that Proportional Representation is the proper method for modern democracy, and that referenda are an instrument of populism, notably within systems with District Representation. See the next weblog entry.

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