[EM] Multipile Transferable Votes

[Stephen H. Sosnick]

>>>In a message about STV that appeared in 
>>>Election-Methods Digest on 29-Aug-2009, I said 
>>>the following:  "1 vote [per ballot] is a 
>>>special case.  And, unfortunately, mentioning 
>>>only that special case gives opponents of 
>>>transferable-vote systems a politically 
>>>effective counter-argument. Instead, what 
>>>people favoring transferable-vote elections 
>>>should say is that (1) every valid ballot will 
>>>cast the same number of votes, and (2) the 
>>>same outcome will emerge regardless of how 
>>>many votes each valid ballot casts, whether 
>>>that number is (a) 1 (which simplifies 
>>>calculation), (b) the number of open seats 
>>>(which might be, say, 3), or even (c) merely a 
>>>letter, say, v."
>>>
>>>Two people posted comments.
>>>
>>>One wrote, "This statement is wrong Š.  In 
>>>STV-PR each voter has one vote and one vote 
>>>only throughout the entire process.  Š  It is 
>>>extremely important to refer to STV as the 
>>>SINGLE Transferable Vote, because each voter 
>>>must have only one vote to ensure PR. Š PR 
>>>cannot be obtained (except by chance) if that 
>>>single vote is not transferable Š."
>>>
>>>The other comment was, "Ahh I see what you are 
>>>aiming for.  Effectively, each voter gets to 
>>>submit v ranked ballots.  This could make 
>>>counting pretty hard."
>>>
>>>Both writers missed the point.
>>>
>>>On the other hand, our leading election 
>>>theorist, that is, Nicolaus Tideman, 
>>>understood. He saw that, as I had asserted, 
>>>changing the number of votes cast by each 
>>>ballot in a transferable-vote election might 
>>>not affect the outcome of the election.  And, 
>>>remarkably, he tested and illustrated my 
>>>assertion.
>>>
>>>To do so, Professor Tideman referred to a file 
>>>he had containing 460 rankings actually 
>>>submitted in a ranked-voting election with 4 
>>>winners and 10 candidates.  He selected three 
>>>different numbers of votes that each ballot 
>>>might have cast in that election, namely, 1 
>>>(the number actually cast per ballot), 2, and 
>>>4 (the number of seats being filled).  For 
>>>each of these three possibilities, he 
>>>determined which candidates would have won the 
>>>election.
>>>
>>>Professor Tideman used three different 
>>>versions of STV.  He used the two 
>>>most-sophisticated versions currently 
>>>available, namely, Meek's method and Warren's 
>>>method, and also the Newland-Britton method. 
>>>The latter is much advanced over early 
>>>versions of STV (for example, over the 
>>>primitive version still used in Massachusetts) 
>>>but, like them, does not transfer a vote from 
>>>a candidate being eliminated to the voter's 
>>>next choice if that next choice was elected 
>>>earlier in the calculations (as a result, that 
>>>voter's following choice receives a larger 
>>>transfer directly instead of a smaller 
>>>transfer indirectly).
>>>
>>>Professor Tideman recently sent me his 
>>>results, along with a copy of voters' 
>>>rankings.  Appendix 1, below, reproduces the 
>>>calculations with Meek's method, and the 
>>>results with Warren and Newland-Britton are 
>>>available on request.
>>>
>>>To understand the calculations, you need to 
>>>notice which variables changed and which 
>>>variables did not change when the number of 
>>>votes cast by each ballot changed.
>>>
>>>With each of the three versions of STV, three 
>>>variables changed as the number of votes per 
>>>ballot changed from 1 to 2 to 4.  The 
>>>variables that changed were the values at each 
>>>stage of (a) the quota (that is, the number of 
>>>votes that a candidate currently needs to be 
>>>elected), (b) the candidates' tallies (that 
>>>is, the number of votes that, after transfers 
>>>of votes that occurred at previous stages, 
>>>each candidate currently has), and (c) the 
>>>"excess" (that is, the number of votes that, 
>>>because some ballots have incomplete rankings, 
>>>cannot be transferred from elected or 
>>>eliminated candidates to active candidates).
>>>
>>>When the number of votes cast by each ballot 
>>>changed, each of those three variables changed 
>>>in proportion.  Specifically, when each ballot 
>>>cast 2 votes, the quota, the tallies, and the 
>>>excess at every stage were 2 times the level 
>>>they had when each ballot cast 1 vote. 
>>>Similarly, when each ballot cast 4 votes, the 
>>>values at every stage were 4 times the level 
>>>that those variables had when each ballot cast 
>>>1 vote.
>>>
>>>To see the re-scaling, compare three columns 
>>>in Appendix 1, namely, (a) the third column 
>>>from the left (that is, the column with the 
>>>heading "1"), which contains the quota, 
>>>tallies, and excess that emerged at each stage 
>>>when each ballot cast one vote, (b) the fourth 
>>>column from the left (that is, the column 
>>>headed "2"), which contains the quota, 
>>>tallies, and excess that emerged at each stage 
>>>when each ballot cast 2 votes, and (c) the 
>>>fifth column from the left (that is, the 
>>>column headed "4"), which contains the quota, 
>>>tallies, and excess that emerged at each stage 
>>>when each ballot cast 4 votes.
>>>
>>>Equally important is what did NOT change as 
>>>the number of votes cast by each ballot 
>>>changed from 1 to 2 to 4.  There was no change 
>>>at any stage in the "retention proportion" of 
>>>any candidate. (If a candidate's current tally 
>>>exceeds the current quota, a 
>>>portion--specifically, 1 minus the retention 
>>>proportion--of each whole or fractional vote 
>>>in that tally will transfer to the next choice 
>>>on the ballot where that whole or fractional 
>>>vote originated, provided that next-choice 
>>>candidate is still "active").
>>>
>>>The retention proportions did not change 
>>>because they were determined by the algorithm 
>>>being used, that is, by Meek's, Warren's, or 
>>>N-B's method. Hence, Appendix 1 shows the 
>>>retention proportions, not in three columns, 
>>>but rather in one, namely, the second column, 
>>>which is headed "retain."
>>>
>>>Because changing the number of votes cast by 
>>>each ballot caused a proportionate change in 
>>>the quota, the tallies, and the excess at 
>>>every stage, but did not change the retention 
>>>proportion of any candidate at any stage, the 
>>>outcome was--as predicted--the same whether a 
>>>ballot cast 1, 2, or 4 votes.  In particular, 
>>>there was no change in (a) the candidates who 
>>>won and who lost the election, (b) the number 
>>>of stages (namely, nine) needed to determine 
>>>which candidates won and lost, or (c) the 
>>>stage at which each candidate was elected or 
>>>eliminated. The column in Appendix 1 labeled 
>>>"Status" shows the latter.
>>>
>>>On the other hand, because the three versions 
>>>of STV use different retention proportions, 
>>>they yield different outcomes.  The winners 
>>>are (a) candidates #10, #1, #7, and #2 with 
>>>N-B, (b) #10, #1, #7, and #4 (instead of #2) 
>>>with Meek, and (c) #10, #1, #7, and #6 
>>>(instead of #2 or #4) with Warren (which 
>>>Professor Tideman prefers).  Starting in stage 
>>>4, Meek's retention proportions differ from 
>>>Warren's.
>>>
>>>A question probably has come to mind:  If 
>>>changing the number of votes cast by each 
>>>ballot would not change the outcome, then why 
>>>not continue to have each ballot cast one vote 
>>>and continue to refer to SINGLE transferable 
>>>vote? The answer is the second point made in 
>>>my message of 29-Aug-2009, namely, that making 
>>>each ballot cast as many votes as there are 
>>>open seats would help win public support for a 
>>>transferable-vote system.
>>>
>>>I say that after having a bad experience.  My 
>>>city, like many others, elects either 2 or 3 
>>>members of a 5-member city council every 2 
>>>years, using the multi-vote plurality system. 
>>>With that system, a voter may vote for as many 
>>>candidates are there are open seats, and the 
>>>leading vote-getters win those seats. 
>>>Recently, there was a popular vote on whether 
>>>to substitute STV.
>>>
>>>STV lost, and I think a major reason was that, 
>>>over and over, opponents asserted that 
>>>introducing STV would deprive voters of their 
>>>2nd and 3rd votes.  Meanwhile, our side 
>>>implicitly conceded that point whenever we 
>>>mentioned SINGLE transferable vote or tried to 
>>>explain how a single vote could support more 
>>>than one candidate.
>>>
>>>Conversely, I do not see that anything would 
>>>be lost by making a ranking ballot cast as 
>>>many votes as there are openings.  In 
>>>particular:
>>>
>>>(1) As with 1 vote per ballot, every vote cast 
>>>would automatically be apportioned among the 
>>>candidates in a way that reflects both the 
>>>voter's preferences and how others have voted.
>>>
>>>(2) As with 1 vote per ballot, voters would no 
>>>longer need to choose between (a) helping 
>>>their 1st choice beat their 2nd choice, and 
>>>(b) helping their 2nd choice beat candidates 
>>>liked even less (of the ballots submitted in 
>>>my city's last seven 2-seat elections, 29% 
>>>cast only one vote, thereby giving priority to 
>>>helping the 1st choice).
>>>
>>>(3) As with 1 vote per ballot, whether 
>>>spoilers are nominated would become less 
>>>important.
>>>
>>>(4) As with 1 vote per ballot, the proportion 
>>>of open seats filled by a faction's favorite 
>>>candidates would, at times, become more like 
>>>the proportion of votes cast by that faction.
>>>
>>>Moreover, with multiple transferable votes, it 
>>>probably would be easier for the public to 
>>>understand and appreciate these advantages.
>>>
>>>Stressing benefit (4), some advocates of STV 
>>>call the system "proportional representation," 
>>>not STV.  For them, especially, it should be 
>>>interesting to compare the STV outcomes 
>>>described above with the outcome that 
>>>multi-vote plurality would have produced in 
>>>the same election.
>>>
>>>If multi-vote plurality had been used in that 
>>>election, then (a) each voter would have been 
>>>invited to vote for up to 4 candidates, (b) a 
>>>candidate would have received one vote for 
>>>each voter who had voted for that candidate, 
>>>and (c) the winners would have been the 4 
>>>candidates receiving the most votes.
>>>
>>>How the 460 voters would have voted is not 
>>>obvious.  Because of the dilemma mentioned 
>>>above (see benefit (2)), a voter might have 
>>>voted for 4 candidates or for 3, 2, or 1. 
>>>Accordingly, I learned which candidates would 
>>>have won if every voter voted only for his or 
>>>her (a) 1st choice, (b) 1st and 2nd choices, 
>>>(c) top 3 choices, and (d) top 4 choices. 
>>>Appendix 2 contains the figures.
>>>
>>>In all four cases, the outcome with multi-vote 
>>>plurality was, to my surprise, reasonable and, 
>>>indeed, arguably better than with N-B. 
>>>Specifically, the winners in cases (a), (b), 
>>>and (c) were, as with Meek's method, 
>>>candidates #1, #4, #7, and #10 and, in case 
>>>(d), were the same as with Warren's method, 
>>>that is, #1, #6, #7, and #10.  Hence, changing 
>>>from multi-vote plurality to STV--at least to 
>>>either of those versions of STV--would not 
>>>have made the proportion of open seats filled 
>>>by a faction's favorite candidates more like 
>>>the proportion of votes cast by that faction.
>>>
>>>On the other hand, factions were either absent 
>>>or invisible.  While 4 seats were open, STV 
>>>elected only 1 candidate, namely, #10, before 
>>>votes were transferred from an eliminated 
>>>candidate.  The next-most-frequent 1st choice, 
>>>namely, candidate #1, was top-ranked by merely 
>>>81 of the 460 voters, and surplus transferred 
>>>from #10 was not large enough to carry #1 (or 
>>>any other candidate) over the threshold.  As a 
>>>result, no outcome could have made the 
>>>proportion of open seats filled by factions' 
>>>favorite candidates resemble the proportion of 
>>>votes cast by those factions.
>>>
>>>But there are cases, at least hypothetical 
>>>cases, where STV probably would--and 
>>>multi-vote plurality probably would 
>>>not--produce proportional representation, in 
>>>the sense that STV would make the proportion 
>>>of open seats filled by each faction's 
>>>favorite candidates as close to the proportion 
>>>of votes cast by the faction as is possible 
>>>when the proportion of open seats filled by a 
>>>faction's favorites must be a multiple of 
>>>(1/number of open seats).
>>>
>>>For example, suppose that (a) 2 seats are 
>>>open, (b) 6 candidates, namely, A, B, C, D, E, 
>>>and F, are running; (c) 100 people vote; (d) 
>>>35 voters rank the candidates A > B > C > D > 
>>>E > F, 34 voters think C > D > E > F > B > A, 
>>>and 31 voters think F > E > D > C > B > A; (e) 
>>>with multi-vote plurality, voters will vote 
>>>for both their 1st choice and their 2nd 
>>>choice; and (f) with a transferable-vote 
>>>system, voters will report both their 1st 
>>>choice and their 2nd choice.
>>>
>>>With multi-vote plurality, candidates (A, B, 
>>>C, D, E, F) would receive, respectively, (35, 
>>>35, 34, 34, 31, 31) votes.  Hence, candidates 
>>>A and B would win.  As a result, candidates 
>>>favored by merely 35% of the voters (and 
>>>disfavored by the other 65%) would fill 100% 
>>>of the open seats.
>>>
>>>With a transferable-vote system, in contrast, 
>>>candidate A and candidate C would immediately 
>>>receive more than fraction 1/(2+1) of the 
>>>votes cast and therefore (in that order) would 
>>>quickly be elected.  As a result, a candidate 
>>>who is the 1st choice of 35% of the voters 
>>>would fill 50% of the open seats, and a 
>>>candidate who is the 1st choice of 34% of the 
>>>voters would fill the other 50%. 
>>>Proportionality!
>>>
>>>I conclude that (a) a transferable-vote system 
>>>will produce the same outcome regardless of 
>>>the number of votes cast by each ballot; (b) 
>>>to win greater public support for a 
>>>transferable-vote system, each ballot should 
>>>cast as many votes as there are open 
>>>positions; (c) when the electorate is not 
>>>polarized, multi-vote plurality may produce 
>>>the same outcome as a transferable-vote 
>>>system; and (d) even when a transferable-vote 
>>>system does not increase proportionality, it 
>>>will have other benefits.
>>>
>>>
>>>APPENDIX 1:
>>>OUTCOME WITH MEEK'S METHOD
>>>
>>>Votes per ballot=	1	2	4
>>>
>>>Stage 1	Quota=	92	184	368
>>>Cand.	Retain	Tally	Tally	Tally	Status
>>>#1	1.000	81	162	324
>>>#2	1.000	40	80	160
>>>#3	1.000	15	30	50
>>>#4	1.000	42	84	168
>>>#5	1.000	27	54	108
>>>#6	1.000	41	82	164
>>>#7	1.000	70	140	280
>>>#8	1.000	12	24	48
>>>#9	1.000	14	28	56
>>>#10	1.000	118	236	472	Newly elected
>>>	Subtotal=	460	920	1830
>>>	Excess=	0	0	0
>>>	Total=	460	920	1830
>>>
>>>Stage 2	Quota=	91.87	183.73	367.5
>>>Cand.	Retain	Tally	Tally	Tally	Status
>>>#1	1.000	88.53	177.06	354.1
>>>#2	1.000	41.55	83.10	166.2
>>>#3	1.000	15.89	31.77	63.5
>>>#4	1.000	43.77	87.54	175.1
>>>#5	1.000	28.77	57.54	115.1
>>>#6	1.000	43.21	86.43	172.9
>>>#7	1.000	73.10	146.20	292.4
>>>#8	1.000	15.54	31.09	62.2	To be excluded
>>>#9	1.000	17.10	34.20	68.4
>>>#10	0.779	91.87	183.73	367.5	Elected
>>>	Subtotal=	459.33	918.66	1837.4
>>>	Excess=	0.66	1.33	2.7
>>>	Total=	460.00	920.00	1840.0
>>>
>>>Stage 3	Quota=	91.84	183.69	367.4
>>>Cand.	Retain	Tally	Tally	Tally	Status
>>>#1	1.000	93.63	187.26	374.5	Newly elected
>>>#2	1.000	42.59	85.19	170.4
>>>#3	1.000	18.30	36.59	73.2
>>>#4	1.000	44.59	89.19	178.4
>>>#5	1.000	31.37	62.74	125.5
>>>#6	1.000	44.11	88.22	176.4
>>>#7	1.000	74.89	149.78	299.6
>>>#8	0	0	0	0	Excluded
>>>#9	1.000	17.89	35.78	71.6
>>>#10	0.741	91.84	183.69	367.4	Elected
>>>	Subtotal=	459.21	918.44	1837.0
>>>	Excess=	0.78	1.56	3.1
>>>	Total=	460.00	920.00	1840.0
>>>
>>>Stage 4	Quota=	91.84	183.68	367.4
>>>Cand.	Retain	Tally	Tally	Tally	Status
>>>#1	0.980	91.84	183.68	367.4	Elected
>>>#2	1.000	42.95	85.89	171.8
>>>#3	1.000	18.52	37.04	74.1
>>>#4	1.000	44.94	89.89	179.8
>>>#5	1.000	31.51	63.02	126.0
>>>#6	1.000	44.48	88.98	177.9
>>>#7	1.000	75.15	150.29	300.6
>>>#8	0	0	0	0	Excluded
>>>#9	1.000	17.97	35.95	71.9	To be excluded
>>>#10	0.738	91.84	183.68	367.4	Elected
>>>	Subtotal=	459.20	918.42	1836.9
>>>	Excess=	0.80	1.59	3.2
>>>	Total=	460.00	920.00	1840.0
>>>
>>>Stage 5	Quota=	91.83	183.66	367.3
>>>Cand.	Retain	Tally	Tally	Tally	Status
>>>#1	0.952	91.83	183.66	367.3	Elected
>>>#2	1.000	44.62	89.24	178.5
>>>#3	1.000	21.16	42.32	84.6	To be excluded
>>>#4	1.000	47.12	94.23	188.5
>>>#5	1.000	34.92	69.84	139.7
>>>#6	1.000	48.37	96.74	193.5
>>>#7	1.000	79.30	158.59	317.2
>>>#8	0	0	0	0	Excluded
>>>#9	0	0	0	0	Excluded
>>>#10	0.723	91.83	183.66	367.3	Elected
>>>	Subtotal=	459.15	918.28	1836.6
>>>	Excess=	0.86	1.72	3.4
>>>	Total=	460.00	920.00	1840.0
>>>
>>>Stage 6	Quota=	91.77	183.54	367.1
>>>Cand.	Retain	Tally	Tally	Tally	Status
>>>#1	0.899	91.77	183.54	367.1	Elected
>>>#2	1.000	50.27	100.54	201.1
>>>#3	0	0	0	0	Excluded
>>>#4	1.000	51.78	103.56	207.1
>>>#5	1.000	38.95	77.91	155.8	To be excluded
>>>#6	1.000	49.81	99.63	199.3
>>>#7	1.000	84.49	168.93	338.0
>>>#8	0	0	0	0	Excluded
>>>#9	0	0	0	0	Excluded
>>>#10	0.699	91.77	183.54	367.1	Elected
>>>	Subtotal=	458.84	917.65	1835.5
>>>	Excess=	1.15	2.31	4.6
>>>	Total=	460.00	920.00	1840.0
>>>
>>>Stage 7	Quota=	91.04	182.08	364.2
>>>Cand.	Retain	Tally	Tally	Tally	Status
>>>#1	0.804	91.04	182.08	364.2	Elected
>>>#2	1.000	57.68	115.36	230.7
>>>#3	0	0	0	0	Excluded
>>>#4	1.000	59.62	119.23	238.5
>>>#5	0	0.00	0.00	0.0	Excluded
>>>#6	1.000	58.93	117.86	235.7
>>>#7	1.000	96.90	193.79	387.6	Newly elected
>>>#8	0	0	0	0	Excluded
>>>#9	0	0	0	0	Excluded
>>>#10	0.644	91.04	182.08	364.2	Elected
>>>	Subtotal=	455.21	910.40	1820.9
>>>	Excess=	4.79	9.59	19.2
>>>	Total=	460.00	920.00	1840.0
>>>
>>>Stage 8	Quota=	90.85	181.70	363.4
>>>Cand.	Retain	Tally	Tally	Tally	Status
>>>#1	0.791	90.85	181.70	363.4	Elected
>>>#2	1.000	59.02	118.04	236.1	To be excluded
>>>#3	0	0	0	0	Excluded
>>>#4	1.000	61.92	123.84	247.7
>>>#5	0	0.00	0.00	0.0	Excluded
>>>#6	1.000	60.76	121.51	243.0
>>>#7	0.932	90.85	181.70	363.4	Elected
>>>#8	0	0	0	0	Excluded
>>>#9	0	0	0	0	Excluded
>>>#10	0.635	90.85	181.70	363.4	Elected
>>>	Subtotal=	454.25	908.49	1817.0
>>>	Excess=	5.76	11.52	23.0
>>>	Total=	460.00	920.00	1840.0
>>>
>>>Stage 9	Quota=	87.86	175.71	351.4
>>>Cand.	Retain	Tally	Tally	Tally	Status
>>>#1	0.625	87.86	175.71	351.4	Elected
>>>#2	0	0.00	0.00	0.0	Excluded
>>>#3	0	0	0	0	Excluded
>>>#4	1.000	88.33	176.67	353.3	Newly elected
>>>#5	0	0	0	0	Excluded
>>>#6	1.000	87.38	174.76	349.5	To be excluded
>>>#7	0.726	87.86	175.71	351.4	Elected
>>>#8	0	0	0	0	Excluded
>>>#9	0	0	0	0	Excluded
>>>#10	0.520	87.86	175.71	351.4	Elected
>>>	Subtotal=	439.29	878.56	1757.0
>>>	Excess=	20.72	41.43	82.9
>>>	Total=	460.00	920.00	1840.0
>>>
>>>
>>>
>>>APPPENDIX 2
>>>OUTCOME WITH MULTI-VOTE PLURALITY
>>>
>>>Column (1) shows votes received if voters vote only for their 1st choice.
>>>Column (2) shows votes received if voters vote 
>>>only for their 1st and 2nd choices.
>>>Column (3) shows votes received if voters vote only for their top 3 choices.
>>>Column (4) shows votes received if voters vote only for their top 4 choices.
>>>
>>>Cand.	(1)	(2)	(3)	(4)
>>>#10	118	192	239	289
>>>#1	81	163	215	260
>>>#7	70	126	174	214
>>>#4	42	89	133	166
>>>#6	41	86	129	171
>>>#2	40	68	114	159
>>>#5	27	65	111	146
>>>#8	12	42	79	114
>>>#9	14	37	71	113
>>>#3	15	41	74	102
>>>Total	460	909	1,339	1,734