Characteristics of different versions of Single Transferable Vote

1. 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.

2. 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)

3. Single Transferable Vote • Each voter ranks candidates according his/her preference. • q=[number of votes/(number of seats+1)]+1

4. Example Preferences: Number of votes

5. 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.

6. Gregory method (2) C’ exclusion B receives 1000 votes. B is elected.

7. 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.

8. “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

9. 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.

10. 2001 election • In 234 count (!!!) under Inclusive Gregory method shows anomalous situation • Inclusive Gregory method inflated value of vote

11. 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.

12. Example Note: Calculations are subject to rounding errors

13. 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

14. Meek method (iteration 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.

15. Meek method (iteration 2) For 3200 votes A B C E 0,624937501of every vote keeps candidate A, (1-0, 624937501)=0,375062499 transfers to candidate B. For 800 votes A0,624937501 keeps candidate A, 1-0,624937501)=0,375062499 became non-transferable.

16. 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.

17. Meek method (iteration 3) For 1000 votes C B 0 has C, 1 has B. For 3200 votes A B C E0,606184376 of every vote keeps candidate A, (1-0,606184376)= 0,393815624 transfers to candidate B. For 800 votes A0,606184376 keeps candidate A 1- 0,606184376)= 0,393815624 became non-transferable.

18. 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.

19. Meek method (iteration 4) 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 A0,605246719 keeps candidate A(1 - 0,605246719)= 0,394753281 became non-transferable. For 1000 votes B D0,742586177 keeps B, (1 - 0,742586177) transfers to D

20. Meek method (iteration 4) • After iteration 5 E will be elected. Elections outcome – A, B, E.

21. 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.”

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

23. Thanks for your attention