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Electoral systems 


Sequential proportional approval voting (SPAV) or reweighted approval voting (RAV)^{[1]} is an electoral system that extends the concept of approval voting to a multiple winner election. It is a simplified version of proportional approval voting. Proposed by Danish statistician Thorvald N. Thiele in the early 1900s,^{[2]} it was used (with adaptations for party lists) in Sweden for a short period from 19091921, and was replaced by a cruder "partylist" style system as it was easier to calculate.^{[3]}^{[4]}
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Voting Theory: Approval Voting

Finding a Winner Using Approval Voting

voting methods

TickBorne Disease Working Group (TBDWG) Meeting  May 2018  Day 1

Instant Runoff Voting
Transcription
 WELCOME TO A LESSON ON APPROVAL VOTING. IN THIS LESSON WE WILL DEFINE APPROVAL VOTING, AND ALSO DETERMINE AN ELECTION WINNER USING APPROVAL VOTING. NORMALLY WE THINK OF DEMOCRATIC VOTING AS ONE PERSON, ONE VOTE. HOWEVER, SINCE NO VOTING SYSTEM IS PERFECT, VARIOUS VOTING METHODS USE DIFFERENT TECHNIQUES TO TRY TO FAIRLY SELECT A WINNER. MOST VOTING SYSTEMS REQUIRE RANKING OF CANDIDATES, BUT NOT ALL VOTING METHODS. FOR EXAMPLE, APPROVAL VOTING DOES NOT ASK VOTERS TO RANK THE CANDIDATES. VOTERS APPROVE OR DISAPPROVE OF EACH CANDIDATE. THE CANDIDATE WITH THE MOST APPROVAL IS THE WINNER. LET'S BEGIN BY LOOKING AT AN EXAMPLE. THE TABLE ABOVE SUMMARIZES THE RESULTS OF AN APPROVAL VOTE AMONG CANDIDATES "A", B, C, AND D. EACH COLUMN SHOWS THE NUMBER OF PEOPLE WITH A CERTAIN APPROVAL VOTE. APPROVALS ARE MARKED WITH AN X. WE WANT TO FIND THE WINNER UNDER THE APPROVAL VOTING METHOD. SO IF WE START WITH CANDIDATE "A", NOTICE HOW CANDIDATE "A" WAS APPROVED BY 24 + 21 + 25, OR 70 VOTERS. LOOKING AT CANDIDATE B, CANDIDATE B WAS APPROVED BY 24 + 23 + 22, OR 69 VOTERS, CANDIDATE C WAS APPROVED BY 24 + 21 + 22, OR 67 VOTERS, AND FINALLY, CANDIDATE D WAS APPROVED BY 24 + 22 + 25, OR 71 VOTERS. AND BECAUSE CANDIDATE D RECEIVED THE MOST APPROVAL, CANDIDATE D WINS. LET'S TAKE A LOOK AT A SECOND EXAMPLE. CONSIDER THREE CANDIDATES RUNNING FOR MAYOR. LET'S ASSUME CANDIDATE "A" AND CANDIDATE B HAVE SIMILAR POLITICAL VIEWS. 35% APPROVE ONLY OF C, 32% APPROVE "A" FIRST AND B SECOND, 32% APPROVE B FIRST AND "A" SECOND, AND 1% APPROVE ONLY "A". SO NOTICE CANDIDATE "A" IS APPROVED BY THIS 32%, THIS 32%, AND THIS 1%. SO 32% + 32% + 1% = 65% APPROVAL RATING. CANDIDATE B WOULD BE APPROVED BY THIS 32% AND THIS 32%. SO 32% + 32% = 64% APPROVAL RATING. AND THEN CANDIDATE C IS APPROVED BY 35%. AND THEREFORE, UNDER THE APPROVAL VOTING METHOD, NOTICE HOW CANDIDATE "A" WOULD WIN WITH 65% APPROVAL. NOW, LET'S TALK ABOUT WHAT'S WRONG WITH THE APPROVAL VOTING METHOD. SOMETIMES APPROVAL VOTING TENDS TO ELECT THE LEAST DISLIKED CANDIDATE. TO ILLUSTRATE THIS, LET'S LOOK AT THIS EXAMPLE HERE. USING THIS PREFERENCE SCHEDULE, IF WE ASSUME THE VOTERS FIRST TWO CHOICES ARE CONSIDERED APPROVED, LET'S FIND THE WINNER USING THE APPROVAL VOTING METHOD. SO AGAIN, WE'RE ONLY FOCUSING ON THESE FIRST TWO CHOICES. NOTICE THAT CANDIDATE "A" IS ONLY APPROVED HERE BY 39 VOTERS. IF WE LOOK AT CANDIDATE B, AGAIN, WE'RE COUNTING APPROVAL AS THE FIRST OR SECOND CHOICE, AND THEREFORE CANDIDATE B IS APPROVED BY 39 + 8 + 3, OR ALL OF THE VOTERS. AND THEREFORE CANDIDATE B IS APPROVED BY 50 VOTERS, AGAIN, OR ALL OF THE VOTERS. IF WE LOOK AT CANDIDATE C, NOTICE HOW CANDIDATE C IS APPROVED BY 8 + 3, OR 11 VOTERS. SO UNDER APPROVAL VOTING CANDIDATE B WOULD WIN, WHICH MAY NOT SEEM RIGHT. BECAUSE NOTICE HOW ONLY 8 VOTERS SELECTED B AS THEIR FIRST CHOICE, WHILE 39 VOTERS SELECTED "A" AS THEIR FIRST CHOICE. SO "A" ACTUALLY HAS THE MAJORITY OF THE FIRST CHOICE VOTES, AND THEREFORE WOULD HAVE A MAJORITY WIN. BUT UNDER THE APPROVAL VOTING METHOD, CANDIDATE B WINS. SO YOU CAN SEE WHY SOMETIMES THE APPROVAL VOTING METHOD SELECTS THE LEAST DISLIKED CANDIDATE AS THE WINNER. NOBODY REALLY DISLIKES B, BUT ONLY 8 VOTERS ACTUALLY REALLY WANT B TO WIN. APPROVAL VOTING CAN ALSO BE SUSCEPTIBLE TO STRATEGIC OR INSINCERE VOTING, WHERE INSINCERE VOTING IS WHEN A VOTER DOES NOT VOTE THEIR TRUE PREFERENCE, TO TRY TO INCREASE THE CHANCE OF A PARTICULAR CANDIDATE OF WINNING. LET'S GO BACK AND TAKE A LOOK AT ONE OF THE PREVIOUS EXAMPLES. AS WE SAW EARLIER USING APPROVAL VOTING, D WINS WITH THE MOST APPROVAL VOTES. HOWEVER, IF THREE VOTERS WHO APPROVE OF "A" AND D, OR THREE VOTERS FROM THIS COLUMN HERE, REMOVE THEIR APPROVAL FOR D, LET'S SEE HOW THIS WOULD AFFECT THE RESULTS OF THIS ELECTION. SO AGAIN, THREE OF THE VOTERS HERE, THAT APPROVE OF BOTH "A" AND D, ARE GOING TO REMOVE THEIR APPROVAL FOR D, AND ONLY APPROVE CANDIDATE "A". WHICH WILL LEAVE 22 VOTES THAT APPROVE OF "A" AND D, AND 3 VOTES THAT APPROVE FOR ONLY "A". NOTICE HOW THIS SUM HERE IS STILL 25 FROM THE PREVIOUS PREFERENCE SCHEDULE. SO NOW USING THIS TABLE TO FIND THE WINNER, NOTICE HOW CANDIDATE "A" STILL RECEIVES 24 + 21 + 25, OR 70 APPROVAL VOTES. CANDIDATE B STILL HAS 24 + 23 + 22, OR 69 APPROVAL VOTES, CANDIDATE C STILL HAS 24 + 21 + 22, OR 67 APPROVAL VOTES, BUT BECAUSE 3 VOTERS REMOVED THEIR APPROVAL FOR CANDIDATE D, NOTICE NOW CANDIDATE D, THE PREVIOUS WINNER, HAS 24 + 22 + 22, OR ONLY 68 APPROVAL VOTES. SO BECAUSE OF THE INSINCERE VOTING, NOTICE HOW CANDIDATE D IS NO LONGER THE WINNER. NOW CANDIDATE "A" IS THE WINNER UNDER APPROVAL VOTING. SO AS YOU CAN SEE FROM THIS EXAMPLE, IN SOME CASES APPROVAL VOTING CAN BE SUSCEPTIBLE TO STRATEGIC OR INSINCERE VOTING. I HOPE YOU FOUND THIS HELPFUL.
Description
Sequential Proportional Approval Voting (SPAV) uses Approval Voting ballots to elect multiple winners equitably^{[5]} by selecting a candidate in each round and then reweighing the approvals for the subsequent rounds.
Each ballot is assigned a value equal to the reciprocal of one more than the number of candidates approved on that ballot who have been designated as elected. Each ballot is counted at its current value as a vote for all continuing candidates approved on that ballot. The candidate with the most votes in the round is elected. The process continues until the number of elected candidates is equal to the number of seats to be filled.^{[6]}
At each stage, the unelected candidate with the highest approval score is elected. Then the value of each voter’s ballot is set at where s is the number of candidates approved on that ballot who were already elected, until the required number of candidates is elected. This reweighting is based on the D'Hondt method (Jefferson method). Other weighting formulas such as SainteLague method may be used while still being referred to as SPAV.
There is an incentive towards tactical voting where a voter may withhold approval from candidates who are likely to be elected in any case, as with cumulative voting and the single nontransferable vote.
It is a much computationally simpler algorithm than harmonic proportional approval voting, permitting votes to be counted either by hand or by computer, rather than requiring a computer to determine the outcome of all but the simplest elections.^{[7]}
When comparing Sequential Proportional Approval Voting to Single Transferable Vote, SPAV is better at selecting more central candidates, that represent all the voters, where STV is better at mimicking the distribution of the voters.^{[8]}
Example
For this example, there is an election for a committee with 3 winners. There are six candidates from two main parties: A, B, and C from one party, and X, Y, and Z from another party. About 2/3 of the voters support the first party, and the other roughly 1/3 of the voters support the second party. Each voter casts their vote by selecting the candidates they support. The following table shows the results of the votes. Each row starts by saying how many voters voted in that way and marks each candidate that group of voters supported. The bottom row shows the number of votes each candidate received.
# of votes  Candidate A  Candidate B  Candidate C  Candidate X  Candidate Y  Candidate Z 

112  ✓  ✓  ✓  
6  ✓  ✓  
4  ✓  ✓  ✓  ✓  
73  ✓  ✓  ✓  
4  ✓  ✓  ✓  ✓  
1  ✓  ✓  
Total Votes  116  122  126  82  78  77 
Because Candidate C has the most support, they are the first winner, w_{1}, and their vote is not counted in later rounds. For the second round, anyone who voted for Candidate C has their vote counted as only 1/2. Below is the chart for round 2. A second column on the left has been added to indicate the weight of each ballot.
# of votes  Weight of Vote  Candidate A  Candidate B  Candidate C  Candidate X  Candidate Y  Candidate Z 

112  1/2  ✓  ✓  ✓  
6  1/2  ✓  ✓  
4  1/2  ✓  ✓  ✓  ✓  
73  1  ✓  ✓  ✓  
4  1/2  ✓  ✓  ✓  ✓  
1  1  ✓  ✓  
Weighted Votes  58  61  78  76  75 
Despite Candidates A and B having so many votes in the first round, Candidate X is the second winner, w_{2}, because not as many of the votes for Candidate X were halved. In round 3, anyone who voted for either Candidates C or X has their vote count 1/2, and anyone who voted for both has their vote count 1/3. If anyone had voted for neither, their vote would remain at 1. Below is that table.
# of votes  Weight of Vote  Candidate A  Candidate B  Candidate C  Candidate X  Candidate Y  Candidate Z 

112  1/2  ✓  ✓  ✓  
6  1/2  ✓  ✓  
4  1/3  ✓  ✓  ✓  ✓  
73  1/2  ✓  ✓  ✓  
4  1/3  ✓  ✓  ✓  ✓  
1  1/2  ✓  ✓  
Weighted Votes  57 1/3  60 1/3  38 1/3  37 5/6 
Candidate B is the third and final winner, w_{3}. The final result has 2/3 winners from the party that had about 2/3 of the votes, and 1/3 winner from the party that had about 1/3 of the votes. If approval voting had been used instead, the final committee would be all three candidates from the first party, as they had the highest three vote totals without scaling.
Properties
SequentialPAV satisfies the fairness property called justified representation whenever the committee size is at most 5, but might violate it when the committee size is at least 6.^{[9]}^{[10]}
SPAV is not precinct summable, and requires the ballot information to be centralized before a complete winner set can be determined.
Pareto efficiency  Committee monotonicity  Support monotonicity with additional voters  Support monotonicity without additional voters  Consistency  inclusion strategyproofness  Computational complexity  

Approval voting  strong  ✓  ✓  ✓  ✓  ✓  P 
Proportional approval voting  strong  ×  ✓  cand  ✓  ×  NPhard 
Sequential Proportional Approval Voting  ×  ✓  cand  cand  ×  ×  P 
See also
 Proportional approval voting
 Satisfaction approval voting
 Reweighted range voting
 Approval voting
 Single transferable vote
References
 ^ Brams, Steven; Brill, Markus (2018). "The Excess Method: A Multiwinner Approval Voting Procedure to Allocate Wasted Votes". SSRN Electronic Journal. doi:10.2139/ssrn.3274796. ISSN 15565068. S2CID 53600917.
 ^ E. Phragmén (1899): "Till frågan om en proportionell valmetod." Statsvetenskaplig tidskrifts Vol. 2, No. 2: pp 8795 [1]
 ^ Lewis, Edward G. (1950). "Review of Modern Foreign Governments". The American Political Science Review. 44 (1): 209–211. doi:10.2307/1950372. ISSN 00030554. JSTOR 1950372. S2CID 152254976.
 ^ Humphreys, John H. (20060101). Proportional Representation: A Study in Methods of Election.
 ^ Kilgour, D. Marc (2010). "Approval Balloting for Multiwinner Elections". In JeanFrançois Laslier; M. Remzi Sanver (eds.). Handbook on Approval Voting. Springer. pp. 105–124. ISBN 9783642028397.
 ^ Steven J. Brams, D. Marc Kilgour (2009): "Satisfaction Approval Voting": p4 [2]
 ^ Aziz, Haris; Serge Gaspers, Joachim Gudmundsson, Simon Mackenzie, Nicholas Mattei, Toby Walsh (2014). "Computational Aspects of MultiWinner Approval Voting". Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems. pp. 107–115. arXiv:1407.3247v1. ISBN 9781450334136.
{{cite book}}
: CS1 maint: multiple names: authors list (link)  ^ Faliszewski, Piotr; Skowron, Piotr; Szufa, Stanisław; Talmon, Nimrod (20190508). "Proportional Representation in Elections: STV vs PAV". Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems. AAMAS '19. Richland, SC: International Foundation for Autonomous Agents and Multiagent Systems: 1946–1948. ISBN 9781450363099.
 ^ SánchezFernández, Luis; Elkind, Edith; Lackner, Martin; Fernández, Norberto; Fisteus, Jesús; Val, Pablo Basanta; Skowron, Piotr (20170210). "Proportional Justified Representation". Proceedings of the AAAI Conference on Artificial Intelligence. 31 (1). doi:10.1609/aaai.v31i1.10611. ISSN 23743468. S2CID 17538641.
 ^ Aziz, H., Brill, M., Conitzer, V., et al. (2014): "Justified Representation in ApprovalBased Committee Voting", arXiv:1407.8269 p5 [3]