The French general elections for the Legislative were held on June 11 and 18 2017. The results provided by the French government are presented more accessible in wikipedia (a portal and no source), and have been used in this 2017-France-Lorenz-Gini excel sheet to determine the Lorenz curve and Gini coefficient.

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).

Two rounds mean two sets of data

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 and Gini

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.

Data on turnout

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.

An example of the inequality

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.

In the former weblog entry, I reported on a rather important YouGov poll. The UK general election was on June 8 and the poll was taken on June 12-13 so that we may assume that persons polled had still vivid recollections. 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 decennia. 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.

Representation of preferences via a ranking matrix

Let voters consider the options R = Remain, S = European Economic Area (EEA) a.k.a. Single Market a.k.a Soft, T = Tariffs, H = World Trade Organisation (WTO) a.k.a. Hard. A consistent Remainer would tend to have the ranking R > S > T > H, and a consistent Leaver would tend to have this in reverse.

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 poll of June 12-13 2017

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.

We can observe:

  • 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):

Possible permutations of rankings

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 > H > S > 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.

Presentation of preferences via preference orderings

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.

Discussion on GB

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.

Discussion on ConR / L and LabR / L

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).
Comment on uncertainty in this estimate

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.
Appendix July 15 2017

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

Before the UK Brexit referendum of June 23 2016 I warned that referenda can be silly and dangerous, see here in April 2016. I clarified that the Brexit referendum question was flawed in design. I did not look deeper into this, since, like so many others, I had been put on the wrong foot by the 2016 poll average that suggested a continuation of the status quo. After the surprise outcome, I advised the youngsters in the UK to focus attention on this design flaw, as this is the clearest issue and proper reason to argue that the outcome should be annulled, see here in June 2016. When Kenneth Arrow passed away in early 2017, this was an occasion to have a summary text published in the RES Newsletter, which was republished on the LSE Brexit blog. Now, with the 2017 UK general election, I have been looking a bit deeper at these UK election issues.

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.

A main confusion

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.

The YouGov data of June 12th – 13th 2017

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.

More on the design flaw of the Brexit referendum question

A few days ago, I rephrased one aspect as follows: 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 (Leave) then there arises the claimed binary choice between R and L. Voters who are in the categories S > R > H or H > R > 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 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)

There is a weird brew of populism and District Representation (DR). You might think that countries with Proportional Representation (PR) would be most sensitive to populism, but it is rather the reverse. Countries with DR appear to be much more affected, like the France, UK and USA that use First Past The Post (FPTP).

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.

Limitations to current representation

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 data on Remain or Leave

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.

Transforming these data in party flows

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.

Results of these considerations

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.

Northern Ireland

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.

Splitting up parties ?

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.

Annette Bongardt and Francisco Torres (ABFT) argued about the UK Parliament: “Parliament has a strong and clear mandate for Brexit, Remainers and EU politicians shouldn’t question it“. Their reasoning can be rejected on key steps.

(1) Parliament already has a mandate

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.
(2) Don’t confuse a general vote and a single issue

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.

(3) The Brexit referendum result doesn’t provide evidence

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.

(4) The role of scientists

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.

(5) Democracy versus populism

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.

(6) Proportional Representation (PR) versus District Representation (DR)

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.

  1. 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.
  2. 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.
  3. 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 2q, since s / 2 * 2q = 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 / (2q) = 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.
(6) Conclusion

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 > would face a hard question, and might decide not to turn out, or develop assumptions about what the 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.

The former two weblog texts discussed British and Scottish incomprehension of democracy. Our discussion used numbers though. A picture says more than one thousand words. Lorenz curves are a nice way to display inequality. The data, calculations and following charts are in this excel sheet: 2015-2017-UK-Holland-NIreland-Lorenz-Gini

The UK General Elections in 2015 and 2017

The following two charts show the results in the UK General Elections of 2015 and 2017.

The horizontal axis gives the cumulative percentage of the popular vote. The vertical axis gives the cumulative percentage of the seats. If there is proportional allocation of seats, then the blue line of the seats would cover the pink diagonal.

The parties in the line-up have been ordered by mismatch. Conforming to the Sainte Laguë / Webster criterion (see this discussion by Alan Renwick), the mismatch is determined by ((the ratio of % seats to the % of votes) minus 1). For example in 2015, UKIP with 12.5% of the votes got only 0.2% of the seats, namely 1 seat for Nigel Farage himself. Their mismatch is 0.2 / 12.5 – 1 = 0.012 – 1. LibDem got 7.8% of the votes but only 1.2% of the seats, a mismatch of 0.158 – 1 (with rounding). Another mismatch are the parties that got no seats: the “Others” still got 2.1% of the votes, which means a mismatch of 0 / 2.1 – 1 = -1.

One might argue that the wasted votes should be omitted from the graph and 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 the wasted vote was 3.5% and then f = 50% / 96.5% = 51.8%. A representative majority in a full House of 650 seats then requires 337 seats, and not 325. See here.

The situation in 2017 has improved mainly because UKIP no longer really participated. The LibDem still got 7.3% of the vote and 1.8% of the seats, which is a marginal improvement.

The Gini coefficients are 29.7% in 2015 and 15.6% in 2017.

The graphs and coefficients are inaccurate because of strategic or tactical voting. A voter who favours a Conservative candidate but sees a loss against a Labour candidate might vote for the LibDem, reflecting a {Conservative, LibDem, Labour} preference ordering. There will be some averaging out, but the official votes will likely not reflect the true proportions of the first choices.

The nice thing of these graphs also is that one can recognise some of the parties. In both graphs the Conservatives are on the right hand side, with 36.4% of the votes and 50.8% of the seats in 2015, and 42.2% of the votes and 48.8% of the seats in 2017.

(These percentage take the wasted votes including the invalid votes as part of the denominator. Other sources may report that e.g. the Conservatives got 43.8 of the vote, looking only at the parties that got seats.)

The Dutch General Elections of 2017

The PR Gini for the UK shows that it is strikingly disproportional. Holland provides a useful point of reference. Holland had general elections in 2017 too, and its PR Gini is 3.6%. The major impact in Holland are the 2% of voters for small parties who got no seats. The Dutch qualified majority is f = 50% / 98% = 51%. In a PR system there will be strategic voting too, for example w.r.t. the coalition government. This however is no good reason to adjust the PR Gini coefficient, since such considerations are not quite those of proportionality, see also the discussion in the Appendix.


Conclusions are:

  1. The UK is at an alarming distance from proportional representation. This is detrimental for: (a) The ability to work together, compromise, form coalitions, and respect the opposition, (b) The possibility for smaller parties to partake in government and responsibility. (c) The entry and exit of new parties. (d) The notion among the electorate that they are represented.
  2. The Lorenz graph is a useful tool to show proportionality. The graph and Gini coefficient are not difficult to make. The ordering via the Sainte Laguë / Webster criterion gives (slightly) higher Gini coefficients (less proportionality) than ordering by the difference between % seats and % votes.

Technical appendix

The calculation of the Gini is straightforward. Each step, from one party to the next one, generates a small trapezium, with the area h (a + b) / 2. The height h is in this case the horizontal distance, given by the vote share of the next party. The sides a and b are the differences (on the left and right) between the diagonal and the cumulated seat curve. Summation of all these areas gives a total A. The Gini is equal to 2A, since the whole area of the square is 1. (The formulas for the Gini in wikipedia are more complex than needed for this piecewise linear application.)

Once I had decided to use the Lorenz graph, a google generated some predecessors.

Orit Kedar, Liran Harsgor and Raz A. Sheinerman (2013) refer on page 5 to Taagepera and Shugart (1989) Seats and Votes. Let me reproduce the quote from the first authors quoting the second authors:

“They note that ‘an alternative [to the measure of deviation from PR which they use] is the Gini index of inequality, which has theoretical advantages but is more complex to calculate’ (p. 204). They add that ‘the Gini index is the most widespread index of inequality, and it does satisfy Dalton’s principle [of transfers]. The Gini index is useful for many purposes other than electoral studies (where it has been little used)’ (p.263).”

The calculation above has been straightforward. It must be mentioned that Kedar et al. have a more complex analysis with districts though.

Anish Tailor and Nicolas Veron (2014) look at inequality in the European Parliament. Their problem is that Germany has 700,000 votes per seat while Malta has 70,000 votes per seat. They find a Gini of the UK of 6.3%, but this thus concerns another research question. If one would look at representation by parties then the EU Parliament might be less disproportional.

Kestelman (2005) also considers measures of apportionment and proportionality, and also refers to Taagepera and Shugart for the Gini (p14). He states that the Gini would be complicated to explain and calculate, while it is rather simple, see above Excel sheet. Thus, curiously:

“Fortunately highly correlated with LHI [Loosmore-Hanby Index], the Gini Disproportionality Index (GnI) is rather complicated to explain and calculate (virtually necessitating computerisation).” (p16)

Kestelman also suggests that STV would be a proportional method, but then he might neglect that an application to districts causes disproportionality over the whole nation (see this example).

Alexander Karpov (2008) gives a more analytical overview of the various measures.  He sums over the ratio of shares (%seats / %votes) but I do not see the rationale for this yet. The calculation above uses the ratio only for ordering, and uses levels with their intuitive interpretation.

Karpov’s article got a comment from Michela Chessa (2012) who points to indices that look at power. Indeed, if a party has 49.9% of the vote and the mere technique of apportionment would generate a majority of 50.1% in the seats, then one might wonder whether this is merely technique, and not a major decision on content.

Update June 23 2017: (1) Originally I sorted the parties on the difference between the % seats and the % votes, but the ratio is better, and indeed gives slightly higher Gini coefficients. One can easily check this by sorting differently in the excel sheet. (2) Alan Renwick repeats the useful distinction between the measure of proportionality and the measure on impact (like power). (3) If seats are allocated using one particular criterion (like the Sainte Laguë criterion), then it doesn’t seem so much useful to see what it means in terms of another criterion, for, if the other criterion really is better, then use this to assign the seats. Thus the issue is intellectually rather dead in continental Europe, that already applies proportionality. The issue only comes up here because of the situation in the UK.

Update June 30 2017: (1) Northern Ireland included, see this discussion. (2) The excel sheet for a country/year now contains an unsorted section and a section that sorts automatically. (3) Uniform chart sizes. (4) With a small correction, the Dutch PR Gini is 3.6 instead of 3.5.

Apparently I cannot find a picture of Max Lorenz (1876-1959) but there are some of Corrado Gini (1884-1965).

Corrado Gini (1884-1965) (Source: SIS, Instituto Centrale di Statistica)

If you don’t have proportional representation (PR) then some voters get representatives they did not vote for. Thus it isn’t very democratic not to have PR.

The last weblog criticised the UK Electoral Reform Society (ERS) for erroneously claiming that Single Transferable Vote (STV) was PR.

ERS namely adopts districts, which causes STV to lose the limited PR properties that it has.

A persons affiliated with ERS answered to this criticism:

“We are well aware of the tension between the desire for (overall) proportionality and the desire for guaranteed local representation.  This tension is apparent among British electors when opinion polls have asked relevant questions about the outcomes of voting systems.  British electors want both overall proportionality (of parties) AND the local representation provided by exclusively single-member districts.  That is just not possible, so we aim for a compromise between local representation and overall proportionality through appropriately-sized multi-member districts. (…).” (Personal communication)

This is an unsatisfactory answer since there simply is no such “compromise”. When one must choose between a square and a circle then the answer is not some other graphic with some measure of deviation. If there is no PR then there is no PR, and then ERS should not claim that they have PR. To express their “compromise”, ERS speaks about “STV-PR” but this is like speaking about square-circles, and comes with the grating sound from nails across a blackboard.

If n is the number of voters, s the number of seats, then q = n / s is the threshold or quota, of voters per member. A candidate can be elected when he or she meets the quota. When the district size is 2q, then the district representative must get 50%+1 of the vote to attain the quota. At best s / 2 seats can be filled in this manner, since s / 2 * 2q = n again. All unfilled seats can be allocated using overall PR. This shows that districts are not a key design feature, while PR is. (These formulas can be adjusted for turnout, when district size is defined in terms of the electorate and not actual voters. See here.)

By focusing on districts, ERS loses track of the key design feature, and it lets its logic be occluded by a less relevant issue.

Wikipedia follows ERS

Apparently the editors at wikipedia follow ERS rather uncritically. The wikipedia statement in red is what ERS claims falsely and what is adopted by wikipedia too. The statement in green is true. Since the statement in green is true, the statement in red can only be true by chance.

Proportional representation (PR) characterizes electoral systems by which divisions in an electorate are reflected proportionately in the elected body. If n% of the electorate support a particular political party, then roughly n% of seats will be won by that party. The essence of such systems is that all votes contribute to the result: not just a plurality, or a bare majority, of them. Proportional representation requires the use of multiple-member voting districts (also called super-districts); it is not possible using single-member districts alone.[1][2][3] In fact, the most proportional representation is achieved when just one super-district is used.

The two most widely used families of PR electoral systems are party list PR and single transferable vote (STV).[4][5] Mixed member proportional representation (MMP), also known as the Additional Member System, is a hybrid Mixed Electoral System that uses party list PR as its proportional component. MMP has the potential to be proportional or semi-proportional depending on a number of factors such as the ratio of first past the post (FPTP) seats to PR seats, the existence or nonexistence of compensatory seats to make up for overhang seats, and election thresholds.[6][7][8][9]   (Source: Wikipedia on PR)

ERS thus is confusing the world including wikipedia. My advice for the editors of wikipedia (and the ERS) is:

  • Maintain conceptual integrity.
  • Restrict PR to the notion that p% of the votes translates into p% of the seats.
  • For PR the first preferences are relevant and not what is done with the subsequent preferences. Thus do not label STV as a PR-system but as “potentially PR”, or as STV-PPR.
  • For PR it suffices when the electorate selects parties. A single candidate is a party with a single candidate.
  • The professionals in parliament can use more complex systems like STV. The use of STV (there) must be compared to other systems, like Borda Fixed Point.
  • Get rid of the hangup on district representation.

Unfortunately, the person affiliated with ERS writes to me, with an unrelenting hangup about districts, and neglecting that PR should hold nation-wide:

“Neither the ERS nor I would be prepared to label STV-PT as “potentially PR” or anything similar.  I have seen some academics describe STV-PR as “a semi-proportional” system.  That is just nonsense.  For the same district magnitude, STV-PR and party-list PR both deliver the same degree of proportionality.  The fact that some electorates are prepared to accept electoral districts that cover the whole country for party-PR but don’t like the idea of “large” electoral districts for STV-PR is completely irrelevant.  It is the district magnitude that is the determining factor, not the voting system.”  (Personal communication)

Scotland is an example

Scotland has four electoral systems, and I copy from Wikipedia:

Does this mean that Scotland comprehends democracy or that they don’t ?

The Party List System as used for the EU Parliament generates proportional representation (PR), and this would be the criterion for representative democracy.

(Obviously, for the election of a local council, the norm for PR are the local votes, and not nationwide PR. Once the issue here is reduced to apportionment, then STV is one of the options and a choice depends upon one’s criteria.)

Let us look at the Scottish implementation Additional Member System (AMS) a.k.a. Supplementary Member System a.k.a. Mixed Member System (MMS). I would prefer the latter term, since there is nothing “additional” about an elected MP. Sometimes the term “Mixed Member Proportional” (MMP) is used but this is only warranted when there really is overall PR.

The current Scottish system

Scotland has 73 constituencies, in which the candidate is selected by FPTP. There are 8 regions with 7 seats per region, to a total of 56 regional seats. These “additional seats” are used to make the outcome more proportional. Brief explanations of the current Scottish system are by the Parliament itself and The Scotsman. The Scottish Parliament elections of May 5 2016 have these full data. The turnout was 55.6%.

Scotland like the UK has a hangup on the distinction between the local candidate and the party. It is claimed: “In the second vote the voter votes for a party rather than a candidate.” Indeed, when the first vote has a FPTP selection, then voters may be forced to vote strategically for a candidate of reduced preference, in trying to prevent that a worst candidate is selected. Thus the explanation about local representation may be a misrepresentation about what might really motivate voters.

When we compare the votes for the constituencies (districts) and the regions, then we don’t see much of a difference, except for the Greens and Others. (This are totals though, and there might be differences over districts.)



























We take the summed region vote as determining what the proportions for the parties should be. The additional 56 seats and their restriction to regions are not enough for correction of the error in the local vote. The SNP got 7% more seats than warranted under PR.


























































An alternative for Scotland

Let us consider a rough alternative for Scotland:

  • A local winner must get at least 50% of the vote of a district (constituency).
  • All 129 seats are allocated in proportion to the summed region vote.

The data file allows us to determine which candidates are elected now. This generates a quite different result. In the local vote, only 29 candidates manage to get at least 50% of their district (constituency). 95 candidates are selected via the Party List, which puts the ERS argument for locality into perspective. In this rough alternative, there are 5 seats that cannot be allocated due to rounding errors. But having 4% empty seats is not unfair given that 4.5% of the votes are wasted on the small parties.













































Others  0  0  0










A general observation

The quota is q = 2,285,752 / 129 = 17720. Above criterion of 50% of the local vote might be too lax. With 73 districts, the number or electors per district might be too small. If the number of districts is 129 / 2 ≈ 65, then the average district has size 2q, and the criterion of at least 50% of the votes would fit the overall condition of winning a seat via satisfying q.

A google showed this page by Andrew Ducker who also wondered about PR in Scotland. He mentions: (1) The region votes must be summed for nationwide PR indeed. (2) A 50%:50% distinction between local and national seats would be helpful indeed. In reply to this: why still allow FPTP when it may cause that a minority winner would become the “representative” ? It is better to require at least q and/or at least 50% of the district.

The UK Electoral Reform Society (ERS)

The UK ERS falsely claims that STV applied to districts would be PR while it is not. The ERS also criticises the Scottish system, but perhaps for the wrong reasons.

In 2011, the current Scottish system was already in place, and the ERS advised a change. See the Guardian or the BCC:

One of the authors of the report, Prof John Curtice of Strathclyde University, said: “The widespread expectation that the Scottish Parliament would be a multi-party parliament, in which no party would ever have an overall majority, has been dashed. “In truth, although the electoral system bequeathed to the Scottish Parliament by Labour was far more proportional than first-past-the-post, it was never one that was best fitted to the realisation of that original expectation. “It still favours larger parties over smaller ones, who, indeed, are actually being discouraged from standing in the constituency contests.” (BBC 2011-11-15)

A similar critique is given w.r.t. the 2016 outcome. Some changes like an “open party” list (i.e. the ability to vote for individual party candidates on the list) and the allocation of Sainte-Laguë may indeed be mentioned.

But this is small beer compared to the major critique on the Scottish system, that it still allows for the hangup on district representation.

While the ERS should warn voters and legislators about this hangup, the ERS suffers from this hangup itself too, and propounds STV for districts, which destroys PR.

The person affiliated with ERS writes to me:

I am not in favour of electing MPs (or other representatives) in two different ways. In Scotland we have experience of the Scottish Parliament where MSPs are elected by the Additional Member System (AMS = a regionalised version of MMP). Some of the worst problems of electing MSPs in two different ways (Constituency and Region) have abated over the years, but the tension remains and surfaces from time to time. It would have been much better if all the MSPs had been elected by STV-PR, but AMS was a political compromise as one the major parties (Labour) would just not accept STV-PR at any price. (Personal communication)

Again, this person at ERS suggests that STV would be PR, even calling it STV-PR, while the very application of STV to districts destroys the PR.

Missing Scottish voters

ERS Scotland director Willie Sullivan wrote a book about the structurally low turnout for Scottish elections: “The Missing Scotland: Why over a million Scots choose not to vote and what it means for our democracy” (publisher).

In an article, Sullivan summarises:

“If the working people wanted democracy, why do so many now not vote? Surely these are the people that should be most eager to flex their democratic muscle? In research for my book, Missing Scotland, I tried to find out why more than a million Scots choose not to vote. What I found is worrying. Most important of all, people don’t think voting will make anything better. They have tried voting, and they have tried not voting, and there is no difference. They think politicians are all the same, don’t understand their lives and they make promises they never keep. This is not a question of not caring. The people I spoke to care a lot about their families and communities. They are worried about losing their homes or their jobs. They even like the idea of democracy, they just don’t think we have it. Not voting is often a deliberate act.”

I haven’t read this book, but only find it relevant enough to mention its existence. My guess is that Sullivan hasn’t mentioned two elements:

  1. When the Scottish electoral system was changed, they didn’t adopt the PR system like in Holland, but kept their hangup on districts, and ERS itself was an agent in this. Potentially Sullivan might feel ashamed that he doesn’t quite comprehend what democracy is.
  2. Scotland isn’t aware of the failure of the Trias Politica model and the need for an Economic Supreme Court.


My finding is that major political distortions in the UK, France, USA and India arise because of lack of PR.  A lobby for STV for districts doesn’t resolve this, and it is falsely claimed to be PR. Thus I would tend to advice electoral reform in this order of priority:

  • first PR, like the system in Holland or the EU Parliament (Open Party List)
  • if this is up and running, secondly allow for an element of locality for half of the seats (s / 2, district size 2q, and the district representative is elected with at least 50% of the district vote, potentially corrected for turnout)
  • if this is up and running, improve the system by allowing voters freedom on how they vote
  • compare STV and Borda Fixed Point and other methods for the selection of the local representative.

The UK Electoral Reform Society hinders clarity on electoral reform since they show a hangup on districts. They better focus on establishing Proportional Representation (PR), while regarding the issue of districts as of secondary importance.

Given overall PR, one might even let voters determine on the ballot how to deal with the district representation, for the s / 2 seats available for district representation.

  1. Some voters might vote for a party, and be done with that. Seats are allocated to the party in proportion to the total number of votes. (Closed Party List) Some voters might wish to select a party but also a particular person in that party, so that the party order takes over if the person would not be elected. (Open Party List) These approaches can be combined (as in Holland) when the Closed List voters vote for the party leader.
  2. Some voters might indicate where their vote would go, if their party of choice isn’t elected. (Remember that a single candidate is a party with a single candidate.)
  3. Other voters might wish to vote for particular candidates across parties, and then might want to indicate how votes would have to be transferred if the candidate doesn’t get elected. (Otherwise it is apportioned automatically.) There is still the comparison between STV and e.g. (repeated) application of the Borda Fixed Point method. STV runs the risk of eliminating a compromis candidate, who receives few votes in the initial stage, but who can collect support because of secondary preferences. This might not be relevant for the party proportion but be quite relevant for voters and the candidates themselves. This would not be an issue of PR but of Quality Representation (QR).

PS. Dan Hodges (Telegraph June 1 2015) has a very entertaining article “No, Britain does not want proportional representation“. The weak spot in his argument is that the 2011 referendum on AV was misrepresented as a referendum on PR while it actually was a referendum on AV. The strong point is that ERS cannot be convincing if its arguments are confused. There still is a case for sound arguments and good education.

PPS. The subtle relation between proportional representation (PR) and district representation (DR) shows also in the existence of a Senate or House of Lords, in which districts / States might be represented by 2 senators per State like in the USA. For a Senate the DR might be acceptable since the Senate has the role of guardian for the nation itself. The House would be sensitive to the preferences of the electorate, and in that case PR would be logical.