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.
Candidate
Rank
Ivanov
2
Smith
1
Chen
3
Lee
-
• q=[number of votes/(number of seats+1)]+1
Candidate
Ivanov
First
preference
2000
Smith
2500
Chen
6000
Lee
4500
Example
Preferences:
3200
800
1000
1000
2000
1999
A
A
B
C
D
E
D
B
B
C
C
B
E
Number of votes
A
B
C
D
E
Total
4000
1000
1000
2000
1999
9999
Q  9999 /(3  1)  1  2500
Gregory method (1)
• Candidate A is elected.
Transfer of A’s surplus 4000-2500=1500.
TV=1500/4000=0,375
3200
800
A
A
B
C
E
A - elected
B
C
D
E
NonTotal
transferable
2500
1000+
3200*0,375
=2200
1000
2000
1999
800*0,375=
300
9999
All candidates have less than 2500 votes. C has the
smallest number of votes and should be excluded.
Gregory method (2)
C’ exclusion
B receives 1000 votes.
1000
C
B
A - elected
B
C - excluded D
E
NonTotal
transferable
2500
2200+
1000=3200
0
1999
300
B is elected.
2000
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).
A - elected
B - elected
C - excluded D
E
NonTotal
transferable
2500
2500
0
1999
300+700=
1000
2000
9999
D has more votes than E. E excluded. Elections outcome
– A, B, D.
“Bonner syndrome”
• 1974 case in Australian Senate elections
Bonner was third in Liberal ticket
Large proportion o fist preference votes for Bonner had subsequent
preference for Labor candidates
Bonner 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 seat
Problem of random sampling
Problem 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%
A - elected
B - elected
C - excluded D
E
NonTotal
transferable
2500
2500
0
1999+3200
*0,1346=
2429,7
300+1000*
0,1346=
434,6
2000+1000
*0,1346=
2134,6
• Elections outcome – A, B, E.
9999
2001 election
• In 234 count (!!!) under Inclusive Gregory
method shows anomalous situation
• Inclusive Gregory method inflated value of
vote
Weighted Inclusive Gregory method
B 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.
TV 
Surplus * incom.ing value
candidate' number of votes
Surplus
 0,21875
candidate' number of votes
A - elected B - elected C -excluded D
E
Nontransferable
Total
2500
1999+3200*
0,21875*
0,375=2261,5
300+1000*
0,21875*1=
518,75
9999
2500
0
2000+1000*
0,21875*1=
2218,75
Example
Q=2500
First count: 1000
B’s votes (first
preferences)
Second count:
3200 votes from A
Third count:
1000 votes from C
Gregory method
Incoming value
1
0,375
1
Outgoing value
0
0
0,7
Contribution to surplus (%)
0
0
100,0
Incoming value
1
0,375
1
Outgoing value
0,1346
0,1346
0,1346
19,2
61,5
19,2
Incoming value
1
0,375
1
Outgoing value
0,219
0,082
0,219
Contribution to surplus (%)
31,325
37,5
31,325
Inclusive Gregory method
Contribution to surplus (%)
Weighted inclusive Gregory
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 example
Ballot A  B  C
KV=1 non-elected
0<KV<1 elected
KV=0 excluded
Meek method (iteration 1)
Candidates
A
B
C
D
E
Non-transferable votes
Total
KV
1,000000000
1,000000000
1,000000000
1,000000000
1,000000000
Votes
4000,000000000
1000,000000000
1000,000000000
2000,000000000
1999,000000000
0
9999,000000000
votes
Q
 2499,750000001
seats  1
A is elected. Total surplus = 4000 - 2499,750000001 = 1500,249999999
Difference between two candidates with minimal number of votes
1000-1000=0,000000000 < Total Surplus. Therefore, Total Surplus
should be transferred.
Meek method (iteration 2)

 

 current KV    current Q 
 
  1* 2499,750000001/4000 = 0,624937501
KVA  


 current number of votes


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, 10,624937501)=0,375062499 became non-transferable.
Candidates
KV
Votes
A
B
C
D
E
0,624937501
1,000000000
1,000000000
1,000000000
1,000000000
2499,750004000 =4000*0,624937501
2200,199996800 =1000+3200*0,375062499
1000,000000000
2000,000000000
1999,000000000
300,049999200 =800*0,375062499
Non-transferable
Total
9999,000000000
Meek method (iteration 2)
Q
votes  non.transferable votes
 (9999 - 300,049999200)/4 = 2424,737500201
seats  1
• 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.
Meek method (iteration 3)
KVA  0,624937501* 2424,737500201/2499,750004000 = 0,606184376
KVC  0
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.
Candidates
KV
Votes
A
B
C
D
E
Non-transferable
Total
0,606184376
1,000000000
0,000000000
1,000000000
1,000000000
2424,737504000 =4000*0,606184376
3260,209996800 =1000+3200*0,393815624
0,000000000
=1000*0
2000,000000000
1999,000000000
315,052499200 =800*0,393815624
9999,000000000
Meek method (iteration 3)
Q  (9999 - 315,052499200) / 4 = 2420,986875201
• 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.
Meek method (iteration 4)
KVA  0,606184376 * 2420,986875201/2424,737504000 = 0,605246719
KVB  1* 2420,986875201/3260,209996800 = 0,742586177
KVC  0
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
Meek method (iteration 4)
Candidates
A
B
C
D
E
Nontransferable
Total
KV
Votes
0,605246719 2420,986876000 =4000*0,605246719
0,742586177 2423,215009347 =1000*0,742586177+3200*0,394753281*
0,742586177 +1000*0,742586177
0,000000000 0,000000000
1,000000000 2257,413823000 =2000+1000*0,257413823
1,000000000 2324,167843853 =1999+3200*0,394753281*0,257413823
573,216447800 =800*0,394753281+1000*0,257413823
9999,000000000
• After iteration 5 E will be elected. Elections
outcome – A, B, E.
Local Electoral Amendment Act 2002 No
85, Public Act. New Zealand
“1A Algorithm and article
The 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.”
Alternatives
Other ordinal methods:
• Warren Method
• The Wright system
• The Iterative by comparison method
• Sequential STV
• CPO-STV
• STV(EES)
• Borda-Type methods
Thanks for
your attention
Algorithm 123