Characteristics of different versions of Single Transferable Vote

Characteristics of different versions of Single Transferable Vote

Karpov A.V. (Higher School of Economics)Volsky V.I. (Institute of Control Science RAS)

The paper was partially supported by the Scientific Foundation of the State University-Higher School of Economics under grant №10-04-0030 and Laboratory of Analysis and Decision Making.

Single Transferable Vote

• STV (Hare-Clark Proportional method in Australia) is the Family of vote counting rules

• Classic form of Gregory method (in ACT and Tasmania), Northern Ireland (UK)

• Inclusive Gregory method (Australian Senate, South Australia and Western Australia)

• Weighted Inclusive Gregory method (Scotland, 2007)• Meek method (New Zealand)

Single Transferable Vote• Each voter ranks candidates according his/her preference.

• q=[number of votes/(number of seats+1)]+1

Candidate Rank

Ivanov 2

Smith 1

Chen 3

Lee -

Candidate First preference

Ivanov 2000

Smith 2500

Chen 6000

Lee 4500

ExamplePreferences:

Number of votes

3200 800 1000 1000 2000 1999

A A B C D E

B D B C

C B

E

A B C D E Total

4000 1000 1000 2000 1999 9999

25001)13/(9999 Q

Gregory method (1)• Candidate A is elected.Transfer of A’s surplus 4000-2500=1500.TV=1500/4000=0,375

All candidates have less than 2500 votes. C has the smallest number of votes and should be excluded.

A - elected B C D E Non-transferable

Total

2500 1000+3200*0,375=2200

1000 2000 1999 800*0,375=300

9999

3200 800

A A

B

C

E

Gregory method (2)C’ exclusionB receives 1000 votes.

B is elected.

1000

C

B

A - elected B C - excluded D E Non-transferable

Total

2500 2200+1000=3200

0 2000 1999 300 9999

Gregory method (3)• B has 1000 own first preference votes,

3200*0,375=1200 votes transferred from A, 1000 from C.

• Surplus=3200-2500=700 votes• In this case Gregory method transfers votes from the

last parcel (C’s votes transfer).

D has more votes than E. E excluded. Elections outcome – A, B, D.

A - elected B - elected C - excluded D E Non-transferable

Total

2500 2500 0 2000 1999 300+700= 1000

9999

“Bonner syndrome”• 1974 case in Australian Senate electionsBonner was third in Liberal ticketLarge proportion o fist preference votes for Bonner had subsequent

preference for Labor candidatesBonner was elected after transferring votes from another candidate. None of the second preferences from Bonner’s first preferences were

transferred

• Labor Party candidate, Colston, failed to win a seatProblem of random samplingProblem of taking in account only of the last parcel received

• Senate electoral reform in 1983

Inclusive Gregory method• In our example the first two steps of counting process

are the same (as in Gregory method).• Distinction in B’s surplus transfer (700 votes).B has 1000 own first preference votes, 3200*0,375=1200 votes -

from A, 1000 - from C

• IGM takes into account all votes TV=700/(5200)=13,46%

• Elections outcome – A, B, E.

A - elected B - elected C - excluded D E Non-transferable

Total

2500 2500 0 2000+1000*0,1346=2134,6

1999+3200*0,1346= 2429,7

300+1000*0,1346=434,6

9999

2001 election• In 234 count (!!!) under Inclusive Gregory

method shows anomalous situation• Inclusive Gregory method inflated value of

vote

Weighted Inclusive Gregory methodB has 1000 own first preference votes with incoming value 1, 3200 votes from A with incoming value 0,375, 1000 - from C with incoming value 1.

A - elected B - elected C -excluded D E Non-transferable

Total

2500 2500 0 2000+1000*0,21875*1= 2218,75

1999+3200* 0,21875* 0,375=2261,5

300+1000* 0,21875*1=518,75

9999

votesofnumbercandidate

valueingincomSurplusTV

'

.*

0,21875'

votesofnumbercandidate

Surplus

ExampleQ=2500 First count: 1000

B’s votes (first preferences)

Second count:3200 votes from A

Third count:1000 votes from C

Gregory methodIncoming value 1 0,375 1

Outgoing value 0 0 0,7

Contribution to surplus (%) 0 0 100,0

Inclusive Gregory methodIncoming value 1 0,375 1

Outgoing value 0,1346 0,1346 0,1346

Contribution to surplus (%) 19,2 61,5 19,2

Weighted inclusive Gregory

Incoming value 1 0,375 1

Outgoing value 0,219 0,082 0,219

Contribution to surplus (%) 31,325 37,5 31,325

Note: Calculations are subject to rounding errors

Meek method• On every iteration each candidate has “keep

value”. The portion candidate obtains from the ballot

• For exampleBallot A B C

KV=1 non-elected0<KV<1 electedKV=0 excluded

Meek method (iteration 1)Candidates KV VotesA 1,000000000 4000,000000000B 1,000000000 1000,000000000C 1,000000000 1000,000000000D 1,000000000 2000,000000000E 1,000000000 1999,000000000

Non-transferable votes 0Total 9999,000000000

00012499,750001

seats

votesQ

A is elected. Total surplus = 4000 - 2499,750000001 = 1500,249999999Difference between two candidates with minimal number of votes 1000-1000=0,000000000 < Total Surplus. Therefore, Total Surplus should be transferred.

Meek method (iteration 2)

Candidates KV VotesA 0,624937501 2499,750004000 =4000*0,624937501B 1,000000000 2200,199996800 =1000+3200*0,375062499C 1,000000000 1000,000000000D 1,000000000 2000,000000000E 1,000000000 1999,000000000Non-transferable 300,049999200 =800*0,375062499

Total 9999,000000000

10,62493750=0001/40002499,75000*1

votesofnumbercurrent

QcurrentKVcurrentKVA

For 3200 votes A B C E 0,624937501 of every vote keeps candidate A, (1-0, 624937501)=0,375062499 transfers to candidate B.For 800 votes A 0,624937501 keeps candidate A, 1-0,624937501)=0,375062499 became non-transferable.

Meek method (iteration 2)

• Total surplus = 2499,750004000 - 2424,737500201 = 75,012503799

• Difference between two candidates with minimal number of votes 1999-1000=999 > Total Surplus. Therefore, Candidate with minimal number of votes should be excluded.

• C is excluded.

02012424,73750=200)/4300,049999-(99991

.

seats

votesletransferabnonvotesQ

Meek method (iteration 3)

Candidates KV VotesA 0,606184376 2424,737504000 =4000*0,606184376B 1,000000000 3260,209996800 =1000+3200*0,393815624C 0,000000000 0,000000000 =1000*0D 1,000000000 2000,000000000E 1,000000000 1999,000000000Non-transferable 315,052499200 =800*0,393815624Total 9999,000000000

60,60618437=7500040000201/2499,2424,73750*10,62493750AKV

For 1000 votes C B 0 has C, 1 has B.For 3200 votes A B C E 0,606184376 of every vote keeps candidate A, (1-0,606184376)= 0,393815624 transfers to candidate B.For 800 votes A 0,606184376 keeps candidate A 1- 0,606184376)= 0,393815624 became non-transferable.

0CKV

Meek method (iteration 3)

• B is elected• Total surplus = (2424,737504000 - 2420,986875201)

+ (3260,209996800 - 2420,986875201) = 842,973750398

• Difference between two candidates with minimal number of votes 2000-1999=1 < Total Surplus. Therefore, Total Surplus should be transferred.

52012420,98687 = 4 / 200)315,052499 - (9999Q

Meek method (iteration 4)90,60524671=7375040005201/2424,2420,98687*60,60618437AKV

For 1000 votes C B 0 has C, 1 has B.For 3200 votes A B C E 0,605246719 of every vote keeps candidate A, (1 - 0,605246719) * 0,742586177 = 0,293138330 transfers to candidate B, (1 - 0,605246719) * (1 -0,742586177) * 0 = 0 transfers to C, (1 - 0,605246719) * (1 - 0,742586177) * (1 - 0) = 0,101614951 transfers to E.For 800 votes A 0,605246719 keeps candidate A (1 - 0,605246719)= 0,394753281 became non-transferable.For 1000 votes B D 0,742586177 keeps B, (1 - 0,742586177) transfers to D

70,74258617=2099968005201/3260,2420,98687*1BKV

0CKV

Meek method (iteration 4)

• After iteration 5 E will be elected. Elections outcome – A, B, E.

Candidates KV VotesA 0,605246719 2420,986876000 =4000*0,605246719B 0,742586177 2423,215009347 =1000*0,742586177+3200*0,394753281*

0,742586177 +1000*0,742586177C 0,000000000 0,000000000D 1,000000000 2257,413823000 =2000+1000*0,257413823E 1,000000000 2324,167843853 =1999+3200*0,394753281*0,257413823Non-transferable

573,216447800 =800*0,394753281+1000*0,257413823

Total 9999,000000000

Local Electoral Amendment Act 2002 No 85, Public Act. New Zealand

“1A Algorithm and articleThe New Zealand method of counting single transferable

votes is based on a method of counting votes developed by Brian Meek in 1969 that requires the use of Algorithm 123. That method (with developments) is described in an article in The Computer Journal (UK), Vol 30 No 3, 1987, pp 277-81 (the article). A discussion of the mathematical equations that prove the existence and uniqueness of that method is set out in the article. The New Zealand method of counting single transferable votes includes modifications to Meek's method and incorporates certain rules relevant to the operation of New Zealand local electoral legislation.”

AlternativesOther ordinal methods:• Warren Method• The Wright system• The Iterative by comparison method• Sequential STV• CPO-STV• STV(EES)• Borda-Type methods

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