characteristics of different versions of single transferable vote
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Characteristics of different versions of Single Transferable Vote. Karpov A.V. (Higher School of Economics ) Volsky V.I. ( Institute of Control Science RAS ). - PowerPoint PPT PresentationTRANSCRIPT
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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