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Tom Monto

The Borda count can be used for multiple winners

Wikipedia says this :

The Borda count is intended to elect broadly acceptable options or candidates, rather than those preferred by a majority, and so is often described as a consensus-based voting system rather than a majoritarian one.[

But I think better to say

The Borda count is intended to elect broadly acceptable options or candidates, rather than those preferred by a single group, perhaps just a minority, as are elected under SMP, It is a consensus-based voting system rather than a majoritarian one.[




here's a report of a Borda election held to choose a naming for a bridge in Dublin City:


it seems a good example of Borda being used to create multiple winners.


the creation of the short list of names on page 2 is like PR in that it produces multiple winners.

but the top half of table on page 2 does not add up to the lower half

looking a the date in the top half, the lower half of the table should have Bermingham as 11, Duff 21, Hackett 10, etc.


and the analysis bewilders me

it says that in PR-STV, the five choices that were first choices of at least one voter would have been winners (which I don't think is true) and it should be six, not five, anyway.


It goes on to say that Block Voting [Multiple plurality] would have derived the same winners as PR-STV, FPTP [Multiple plurality] would have have yielded the winners that way but not PR-STV, I think.


the process did not use Quota Borda but instead the five choices with the most points (although possibly recorded wrongly) were put on the short list.


This is an example of the Borda Count .

The short list was used to find the winning name fohr bridge.


(it is probably best to ignore the Quota Borda method. Here is my problem with that



the Wikipedia article "quota Borda count" shows how Borda works proportionally to elect multi winners in a district. But as you point out, it is not clear in how it would work. and I tried but don't understand it as the following will show...


say 59 voters electing four eight candidates A to H

2 cast a preferential vote that looks like this: A-B-C-D-E-F-G-H

3 cast a preferential vote that looks like this: A-B-F-G-H-A-E

4 cast a preferential vote that looks like this: E-C-G-H-A

5 cast a preferential vote that looks like this: G-B-D-E-C-G-H

5 cast a preferential vote that looks like this: F-B-D-G-H

6 cast a preferential vote that looks like this: C-A-D-E--G-H

7 cast a preferential vote that looks like this: D-H-A-B-E-G-H

8 cast a preferential vote that looks like this: H-D-A-G-H-D-E-F

9 cast a preferential vote that looks like this: B-F-C-G-H-A-E

10 cast a preferential vote that looks like this: F-B-C-E-D-G

total 59


Quota is 12


Stage 1

any single candidate getting quota on first count is declared elected

A 2+3 = 5 B 9 12 C 6 D 7 E 4 F (5+10) 15 G 5 H 8 total 59

F elected with 15 votes


stage 2

any pair of candidates with 2 quotas are declared elected.

But here is where I have trouble.

taking vote as is (not using Borda multiplier) any pair of candidates that has more than two quota will have at least one getting more than quota = he or she will have been elected in stage 1.


But perhaps you just look at combos and stage 2 is just to elect the lesser of the pair (the stronger of the two already being elected in Stage 1):

A-B 5

B-F 5+9+10 = 24

G-B 5

C-A 6

D-H 7+8 = 15

E-C 4

total 59

so if that is the case, B is elected as part of a pair with 2 quota. F already being elected in Stage 1.


if you use Borda multiplier, then quota is way too low -

just looking at first preferences we have:

A 16 (2x8) + 24 (3x8) = 40

B 63 (9x7)

C 36 (6x6)

D 49 (7x7)

E 50 (10x5)

F 24 (4x6) + 25 (5x5) = 49

G 35 (5x7)

H 64 (8x8)

so all have quota if quota is 12.


The other question is - when do we look at back-up preferences? And when do we disregard back-up preferences?

As F was elected in stage 1, are all votes bearing F as first preference taken right out and we don't look at the backup preferences marked thereon?


if we leave aside F because it won in the stage 1, there are no pairs that have two quota (two quota = 24).

A-B or B-A (3+2) 5

B-G or G-B 5

[F-B B-F 24 (10+5+9)]

C-A or A-C 6

H-D or D-H (7+8) 15

E-C 4

total 59

unless we change quota now to 1/4 of 35 votes (59 minus B-F's 24), making a new quota of 9?

if the quota is now 9, then H-D/D-H has quota! so then H and D should be declared elected, in that case.


so the Quota Borda system is an unknown quantity...


===================


After assembling the short list of names for the bridge in Dublin, Borda was used to find the winner. the 51 councillors cast preferential ballots to choose. The preferences are not used as back-up preferences but instead to weight the ranking of the names.

If five preferences were marked, a first preference is worth five points, four points for the second choice, etc.


the election of the one name from the short list is analysed on page 3.

there is only one winner. but on page 3 writer uses term PRSTV when he or she must mean Alternative Voting.


The data seems clear

most voters allocated 15 points (giving 5-4-3-2-1 points for the five preferences).

the one with the most points wins.


it is not AV nor multi-round voting but the result seems fair ans seems to be the same as if AV or multi-round voting had been used. (as noted on p. 3)


Interestingly on page 5, much space is devoted to examination of how the Modified Borda Count inhibits strategic voting. we in Canada don't like strategic voting but we are happy just to have people voting - in Ireland apparently they want people to vote and to vote authentically. Good for them


Here's their definition of strategic voting:

‘strategic voting’, i.e. voting differently from one’s true preferences for the sake of achieving a more favoured outcome (Dummett 1984) So the Borda works, through weighted vote totals and simple comparison of vote tallies at the end. It is nice to see a practical example. and good to see a city council devoting such energy to an exotic form of voting for a name for a bridge. I wish the Edmonton city council was that cool...


-------------------------------------------------


(it is probably best to ignore the Quota Borda method.


the Wikipedia article "Quota Borda count" shows how Borda works proportionally to elect multi winners in a district. But it is not clear in how it would work. and I tried but don't understand it as the following will show...


say 59 voters electing four eight candidates A to H

2 cast a preferential vote that looks like this: A-B-C-D-E-F-G-H

3 cast a preferential vote that looks like this: A-B-F-G-H-A-E

4 cast a preferential vote that looks like this: E-C-G-H-A

5 cast a preferential vote that looks like this: G-B-D-E-C-G-H

5 cast a preferential vote that looks like this: F-B-D-G-H

6 cast a preferential vote that looks like this: C-A-D-E--G-H

7 cast a preferential vote that looks like this: D-H-A-B-E-G-H

8 cast a preferential vote that looks like this: H-D-A-G-H-D-E-F

9 cast a preferential vote that looks like this: B-F-C-G-H-A-E

10 cast a preferential vote that looks like this: F-B-C-E-D-G

total 59


Quota is 12


Stage 1

any single candidate getting quota on first count is declared elected

A 2+3 = 5 B 9 12 C 6 D 7 E 4 F (5+10) 15 G 5 H 8 total 59

F elected with 15 votes


stage 2

any pair of candidates with 2 quotas are declared elected.

But here is where I have trouble.

taking vote as is (not using Borda multiplier) any pair of candidates that has more than two quota will have at least one getting more than quota = he or she will have been elected in stage 1.


But perhaps you just look at combos and stage 2 is just to elect the lesser of the pair (the stronger of the two already being elected in Stage 1):

A-B 5

B-F 5+9+10 = 24

G-B 5

C-A 6

D-H 7+8 = 15

E-C 4

total 59

so if that is the case, B is elected as part of a pair with 2 quota. F already being elected in Stage 1.


if you use Borda multiplier, then quota is way too low -

just looking at first preferences we have:

A 16 (2x8) + 24 (3x8) = 40

B 63 (9x7)

C 36 (6x6)

D 49 (7x7)

E 50 (10x5)

F 24 (4x6) + 25 (5x5) = 49

G 35 (5x7)

H 64 (8x8)

so all have quota if quota is 12.


The other question is - when do we look at back-up preferences? And when do we disregard back-up preferences?

As F was elected in stage 1, are all votes bearing F as first preference taken right out and we don't look at the backup preferences marked thereon?


if we leave aside F because it won in the stage 1, there are no pairs that have two quota (two quota = 24).

A-B or B-A (3+2) 5

B-G or G-B 5

[F-B B-F 24 (10+5+9)]

C-A or A-C 6

H-D or D-H (7+8) 15

E-C 4

total 59

unless we change quota now to 1/4 of 35 votes (59 minus B-F's 24), making a new quota of 9?

if the quota is now 9, then H-D/D-H has quota! so then H and D should be declared elected, in that case.


so the Quota Borda system is an unknown quantity...

-------------------


Thanks for reading.

===================================

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