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The de Borda Institute aims to promote the use of inclusive voting procedures on all contentious questions of social choice.

This applies specifically to decision-making, be it for the electorate in regional/national polls, for their elected representatives in councils and parliaments, for members of a local community group, for members of a company board, for members of a co-operative, and so on.

 

 

 

The director alongside the statue of Jean-Charles de Borda, capitaine et savant, in l’École Navale in Brest, 24.9.2010. Photo by Gwenaelle Bichelot. 

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Welcome to the home page of the de Borda Institute, a Northern Ireland-based international organisation (an NGO) which aims to promote the use of inclusive voting procedures on all contentious questions of social choice. For more information use the menu options above or feel free to contact the organisation's headquarters. If you want to check the meaning of any of the terms used, then by all means have a look at this glossary.

As shown in these attachments, there are many voting procedures for use in decision-making and even more electoral systems.  This is because, in decision-making, there is usually only one outcome; but with some electoral systems, as in any proportional ones, there can be several winners.  Sometimes, for any one voters' profile - that is, the set of all their preferences - the outcome of any count may well depend on the voting procedure used.  In this very simple example of a few voters voting on just four options, and in these two hypothetical examples on five, (word document) or (power-point) in which a few cast their preferences on five options, the profiles are analysed according to different methodologies, and the winner could be any one of all the options.  Yet all of these methodologies are called democratic!  Extraordinary!

« 2019-13 A Brexit Solution - a 'preferendum' | Main | 2019-11 The Commons & 'no' to compromise »
Thursday
Apr042019

2019-12 Indicative Votes - Preferential

The full paper can also be found here.

INDICATIVE VOTES, ROUND TWO – MONDAY 1ST APRIL

The aim of this paper is to hazard a guess at how the indicative votes cast in the House of Commons on Monday 1st April might have produced a different outcome if the voting methodology had been different.  As it was, the MPs used a form of approval voting, each having the opportunity to vote ‘for’ or ‘against’ – that or abstain – on four separate options.  As happened in the first round of indicative voting on 27th March, (and as was the case in 2003 when the House voted on five different options on the question of reforming the House of Lords), none of the options received a majority. 

Table 1 records what actually happened, with the additional data of the result of the so-called ‘meaningful vote 3’ on May’s withdrawal deal.

Table 1          The Indicative Votes

Motions:

 

E

C

D

G

Options:

May’s

Deal

May’s Deal + People’s Vote

Customs Union

Common Market 2.0

no deal v revoke

Sponsor:

Theresa May

Peter Kyle

Phil Wilson

Ken Clarke

Nick Boles

Joanna Cherry

MPs

‘for’

‘against’

‘for’

‘against’

‘for’

‘against’

‘for’

‘against’

‘for’

‘against’

Tory

313

277

34

15

253

37

236

33

228

10

260

Lab

245

5

234

203

24

230

10

185

25

121

18

SNP

35

 

34

31

 

 

 

32

 

32

 

L-D

11

 

11

11

 

1

5

2

4

10

 

TIG

11

 

11

11

 

 

11

 

11

11

 

DUP

10

 

10

 

10

 

10

 

10

 

10

Ind

11

4

5

4

5

5

3

5

3

2

4

PC

4

 

4

4

 

 

 

4

 

4

 

GP

1

 

1

1

 

 

1

 

1

1

 

Totals*

641

286

344

280

292

273

276

261

282

191

292

Total voting

630

572

549

543

483

 

*          7 SF members are absent; the Speaker doesn’t vote; and there’s one vacant seat; so to make the total of 650 MPs altogether. 

         More MPs voted on May’s deal, in part because the 28 cabinet members abstained from the indicative votes.

________________

In Table 2, the author estimates how these MPs might have voted if they had been able to cast their preferences, bearing in mind a number of factors:

a)         Some parties sometimes use whips although within parties, some MPs sometimes rebel.

b)         The Tory party is split, and the ERG, of which there are/were about 95 members, is itself split.  The Labour Party also has its various factions.

c)         Sometimes, and especially in majority voting, MPs voting ‘for’ do not necessarily vote in favour of the option concerned or even ‘against’ the perceived opposite, but because of other considerations, even the more remote possibilities of a general election, the resignation of the premier, or whatever.

So now, the analysis.  In Table 2, the right-hand side of each column – E, C, D and G – are running totals which culminate in the bold italicised figures, which coincide with all the various totals given in Table 1.  The author has had to speculate as to how different MPs might have cast their preferences, and given that the data available – the MPs’ indicative votes – showed only their approvals, the data below must be taken as his best estimates.

The "May's deal" column is not included in the analysis of the indicative votes shown in Table 3.  It would have been at least interesting to have included the withdrawal deal in the indicative vote, but it wasn’t; so it is not included in this analysis either except, as for example in the four Tories in the third row, where it is then assumed that the four voted for option E as a 2nd preference.

Table 2          The Indicative Votes

Motions:

 

E

C

D

G

Options:

May’s

Deal

May’s Deal + People’s Vote

Customs Union

Common Market 2.0

no deal v revoke

Sponsor:

Theresa May

Peter Kyle

Phil Wilson

Ken Clarke

Nick Boles

Joanna Cherry

Tory

313

258

1st

258

 

 

 

 

 

 

 

 

10

1st

268

 

 

 

 

 

 

2nd

10

4

1st

272

2nd

4

 

 

 

 

 

 

5

1st

277

 

 

2nd

5

 

 

 

 

14

 

 

 

 

1st

19

2nd

14

 

 

2

 

 

 

 

1st

21

 

 

 

 

8

 

 

 

 

 

 

1st

22

 

 

5

 

 

 

 

2nd

26

1st

27

 

 

5

 

 

3rd

9

1st

31

2nd

32

 

 

5

 

 

2nd

14

1st

36

 

 

 

 

1

 

 

3rd

15

2nd

37

1st

33

 

 

Labour

245

25

 

 

 1st

25

2nd

25

3rd

25

 

 

25

 

 

 1st

50

2nd

50

 

 

3rd

25

60

 

 

 1st

110

2nd

110

3rd

85

 

 

30

 

 

 

 

1st

140

2nd

115

 

 

17

 

 

2nd 

127

1st

157

 

 

3rd

42

13

 

 

 

 

1st

170

2nd

128

3rd 

55

45

 

 

3rd

172

2nd 

215

 

 

1st 

100

15

 

 

2nd

187

1st

230

3rd

143

 

 

21

 

 

 

 

 

 

1st

164

2nd 

121

16

 

 

2nd

203

 

 

1st 

180

 

 

5

1st

5

 

 

 

 

2nd

185

 

 

SNP

35

27

 

 

1st

27

 

 

2nd

27

3rd

27

4

 

 

2nd

31

 

 

1st

31

3rd

31

1

 

 

 

 

 

 

2nd

32

1st

32

L-D

11

9

 

 

1st

9

 

 

 

 

2nd

9

1

 

 

3rd

10

 

 

2nd

1

1st

10

1

 

 

2nd

11

1st

1

3rd

2

 

 

TIG 11

 

 

 

1st

11

 

 

 

 

2nd

11

IND

11

4

1st

4

 

 

 

 

 

 

 

 

3

 

 

2nd

3

1st

3

3rd

3

 

 

2

 

 

 

 

2nd

5

1st

5

3rd

2

1

 

 

1st

4

 

 

 

 

 

 

PC 4

 

 

 

1st

4

 

 

2nd

4

3rd

4

GP 1

 

 

 

1st

1

 

 

 

 

2nd

1

TOTALS

 

286

 

280

 

273

 

261

 

191

 

Table 3 is an extract from Table 2, to show the totals of MPs’ cast preferences.  The figures shown in bold italics confirm the mathematics.  The important data, the numbers at the bottom of this table, will be used in Table 4, in the final analysis.

Table 3          The Estimated Preferences Cast

Motions:

E

 C

D

G

Options:

May’s Deal + People’s Vote

Customs Union

Common Market 2.0

no deal v revoke

Sponsor:

Peter Kyle

Phil Wilson

Ken Clarke

Nick Boles

Joanna Cherry

Tory

1st

 

26

14

 

2nd

9

11

19

10

3rd

6

 

 

 

Totals

15

37

33

10

Lab

1st

110

75

37

45

2nd

48

155

48

21

3rd

45

 

100

55

Totals

203

230

185

121

SNP

1st

27

 

4

1

2nd

4

 

28

-

3rd

 

 

 

31

Totals

31

 

32

32

L-D

1st

9

1

 

1

2nd

1

 

1

9

3rd

1

 

1

 

Totals

11

1

2

10

TIG

1st

11

 

 

 

2nd

 

 

 

11

3rd

 

 

 

 

Totals

11

 

 

11

IND

1st

1

3

2

 

2nd

3

2

 

 

3rd

 

 

3

2

Totals

4

5

5

2

PC

1st

4

 

 

 

2nd

 

 

4

 

3rd

 

 

 

4

Totals

4

 

4

4

GP

1st

1

 

 

 

2nd

 

 

 

1

 

Totals

1

 

 

1

 

1st

163

105

55

47

 

2nd

65

168

102

51

 

3rd

52

-

104

93

Grand Totals

280

273

261

191

 

 

 

 

 

 

 

 

 

 

The Analysis

A preferential vote can be analysed in any one of at least four ways, and these include a Borda count, BC; an alternative vote, AV;[i] a modified Borda count, MBC; and the Condorcet rule.  In many analyses, especially in those where most voters have cast all or nearly all of their preferences, so to submit a full or nearly full ballot, the MBC social choice is the same as the Condorcet winner, and even the two social rankings are similar.  Accordingly, this analysis will focus on the first three methodologies: BC, AV and NBC.

It might have been possible to suggest that a vote ‘against’ was a 4th preference, and that an abstention was a 2nd and/or 3rd.  The wiser course of action, however, given that so many assumptions have already been made, was to keep the number of assumptions as small as possible.  So there is no account of 4th preferences; and in any case, a vote of 1st-2nd-3rd-nothing is not too different from one of 1st-2nd-3rd-4th.

It is necessary to pause a little, to explain the difference between a BC and an MBC, which is this: in a BC, the voter is not encouraged to cast more than his 1st preference; in an MBC, she is.  Basically, in a vote on n-options, voters may cast as many preferences as they want – the number cast is referred to as m and, needless to say, n ≥  m ≥  1.  Now in a BC, points are awarded to (1st, 2nd … last) preferences cast according to the rule (n, n-1 … 1).[ii]  So she who casts only one preference gets n points for her favourite, and thus an n point advantage over all the other options.  In an MBC, in contrast, the rule is (m, m-1 … 1), so he who casts only one preference gives his favourite just 1 point, and thus gains only a 1 point advantage over the other options.  But she who casts m preferences, gets m points for her favourite, m-1 points for her 2nd choice, and so on.  Overall, therefore, the last-named voter gains a 1 point advantage for her favourite over her 2nd choice but, if she casts a full ballot, an n-1 point advantage over her least favourite option.

Now of course, MPs casting their preferences in an indicative vote might adjust their voting pattern if they know which methodology is to be used.

It is time, now, to analyse the data from Table 3. 

The BC Analysis

Every 1st preference gets 3 points, every 2nd preference gets 2 points and every 3rd gets 1 point.

Option E gets                163 x 3            +          65 x 2              +          52 x 1§

                        =          489                  +          130                  +          52

                        =          671

 

Option C gets                105 x 3            +          168 x 2            +          0§        

                        =          315                  +          336

                        =          651

 

Option D gets                57 x 3              +          100 x 2            +          104 x 1§

                        =          171                  +          200                  +          104

                        =          475

 

Option G gets             49 x 3              +          52 x 2              +          90 x 1§

                        =          147                  +          104                  +          90

                        =          341

 

So the winner is option E on 671.

 

§          And     163 + 65 + 52   =  280

                      105 + 168 + 0   =  273

                      57 + 100 + 104 =  261

                      49 + 52 + 90     =  191

 

The AV Analysis

In stage (i), the scores are:

            E 163              C 105              D 55                G 49

So G is eliminated, and because it has no transfers, the stage (ii) scores are:

            E 163              C 105              D 55   

Which means D is now eliminated, and from these 55 1st preferences, 8 votes go to C, 20 go to E, and 27 are non-transferable, so          

in stage (iii), the scores are:

            E 183              C 113

 

So the winner is again E on 183.

 

The MBC Analysis

An MBC analysis is best (and usually) done on a computer.  When laid out on paper, it does get a bit complex.  For the sake of clarity, however, that is what will be done here.  The rule, then, is as follows:

The 1st of 3 preferences gets 3 points;

the 2nd of 3, or a 1st of 2 preferences, gets 2 points;

the 3rd  of 3, a 2nd of 2, or a 1st and only of 1 preference, gets 1 point.

 

So Option E gets:       (25 + 25 + 60 + 27  + 4) x 3  + 

                                                            (5 + 15 + 16 + 4 + 9 + 1 + 11 + 3 + 1) x 2  + 

                                                            (4 + 5 + 1 + 17 + 45 + 1 + 1) x 1

                        =          141 x 3            +          65 x 2             +          74 x 1§§

                        =          423                  +          130                  +          74

                        =          691

 

Option C gets:               (5 + 17 + 13 + 15 + 1 + 3) x 3 +

                                          (14 + %5 + 5 + 1 + 25 + 25 + 60 + 30 + 45 + 2) x 2

                                          (5 + 2) x 1

                        =          54 x 3              +          212 x 2            +          7 x 1§§

                        =          162                  +          424                  +          7

                        =          593

 

Option D gets             :         (1 + 4 + 2) x 3

                        =                 (5 + 5 + 13 + 21 + 16 + 27 +1 + 1 + 4) x 2  +

                        =                 (14 + 8 + 25 + 60 + 30 + 15 + 5 + 1 + 3) x 1

                        =          7 x 3                +          93 x 2              +          161 x 1§§

                        =          21                    +          186                  +          161

                        =          368

 

Option G gets:                        (45 + 1) x 3  +

                                             (1) x 2  +

                                             (10 + 25 + 17 + 13 + 21 + 27 + 4 + 9 + 11 + 2 + 4 + 1) x 1

                        =          46 x 3              +          1 x 2                +          144 x 1§§

                        =          138                  +          2                      +          144

                        =          284

 

§§         So again,          141 + 65 + 74 = 280

                                    54 + 212 + 7   = 273

                                    7 + 93 + 161   = 261

                        and      46 + 1 + 144   = 191

 

So the winner is yet again option E, now on 691.

 

All of the above results are summarised in Table 4.

 

Table 4                      The Results

 

Motions:

E

C

D

G

 

Social ranking

Options:

May’s Deal + People’s Vote

Customs Union

Common Market 2.0

no deal v revoke

Sponsor:

Peter Kyle

Phil Wilson

Ken Clarke

Nick Boles

Joanna Cherry

 

BC

671

651

475

341

E-C-D-G

AV

183

113

-

-

E-C

MBC

691

593

368

284

E-C-D-G

 

So the social choice in each analysis is option E.

 

Conclusion

As was seen in the House in 2003, if and when there are more than two options ‘on the table’, taking a corresponding number of majority votes is not a suitable methodology.  Indeed, to quote Lord Meghnad Desai[iii] at the time, it was “daft;” instead he recommended (but did not name) a rankings system, the BC.  (Hansard, 22.1 2003.)

With preference voting, an outcome is almost guaranteed; and if that outcome has a sufficiently high score, it may then be declared the winner, just as in majority voting, a result might depend on a similarly pre-determined simple or weighted majority.  Of the three methodologies analysed in this paper, it should be emphasised that the BC is not necessarily consensual; AV can be capricious; while along with Condorcet, whenever “there are more than two” options, the MBC is “the best interpretation of majority rule.”  (Oxford Concise Dictionary of Politics, Iain McLean, 2003, oup, p 139.)

In a nutshell, majority voting cannot best identify the consensus or even the best possible compromise; indeed, with so many votes ‘for’ and so many ‘against’, it measures the very opposite, the degree of dissent.

In contrast, by comparing the totals each option receives with the maximum possible score, the MBC can measure the degree of consensus with a high degree of accuracy.

Postscript

As with any form of voting, additional rules can be applied viz-a-viz minimum turnout and so on.  Suffice here to say that the MBC is a highly accurate and very robust counting procedure and, if used with electronic voting, very practicable.

Recommendation

Whenever there are more than two options ‘on the table’, parliament should use a form of preference voting, and ideally, either the MBC and/or the Condorcet rule.

 

 

Peter Emerson

The de Borda Institute

36 Ballysillan Road

Belfast Bt14 7QQ

 

02890711795

07837717979

 

pemerson@deborda.org

www.deborda.org 

 

 

[i]           Also known as single transferable vote, STV, (the latter usually in its PR format as an electoral system, PR-STV), but also as instant run-off voting, IRV, in North America, and preference vote, PV, in Australasia.

 

[ii]           Some prefer the rule (n-1, n-2 … 0), which makes no difference to the outcome.

 

[iii]          For the current author’s analysis of that “daft” voting procedure, see Reforming the House of Lords: Choosing from the Options.  In Representation, 2005, Vol. 41,  4. 

http://www.tandfonline.com/doi/abs/10.1080/00344890508523322

 

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