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. 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 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 2000 1999 9999
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 Non. Total 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 Non. Total 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 Non. Total transferable 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 Non. Total transferable 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. 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 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, 00000 Votes 4000, 000000000 1000, 00000 2000, 00000 1999, 00000 0 9999, 00000 A is elected. Total surplus = 4000 - 2499, 750000001 = 1500, 249999999 Difference between two candidates with minimal number of votes 1000 -1000=0, 00000 < Total Surplus. Therefore, Total Surplus should be transferred.
Meek method (iteration 2) 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 2499, 750004000 =4000*0, 624937501 2200, 199996800 =1000+3200*0, 375062499 1000, 00000 2000, 00000 1999, 00000 300, 049999200 =800*0, 375062499 Non-transferable Total 9999, 00000
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.
Meek method (iteration 3) 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, 00000 0, 000000000 1, 00000 2424, 737504000 3260, 209996800 0, 00000 2000, 00000 1999, 00000 315, 052499200 9999, 00000 =4000*0, 606184376 =1000+3200*0, 393815624 =1000*0 =800*0, 393815624
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.
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 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 1, 00000 2257, 413823000 =2000+1000*0, 257413823 1, 00000 2324, 167843853 =1999+3200*0, 394753281*0, 257413823 573, 216447800 =800*0, 394753281+1000*0, 257413823 9999, 00000 • After iteration 5 E will be elected. Elections outcome – A, B, E.
Local Electoral Amendment Act 2002 No 85, Public Act. New Zealand “ 1 A 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
- Slides: 28