Proportional approval voting
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Proportional approval voting (PAV) is an electoral system which is an extension of approval voting to multiplewinner elections. It applies proportional representation principles with a ballot which is no more complicated than ballots for plurality voting. It allows each voter to vote for as many or as few candidates as they choose. The system was invented by Thorvald N. Thiele.^{[1]} ^{[2]} ^{[3]} It was rediscovered by Forest Simmons in 2001,^{[4]} who coined the name "proportional approval voting".
Description
PAV works by looking at how "satisfied" each voter is with each potential result or outcome of the election. The calculated satisfaction with any particular result for an individual voter is a function of how many of the elected candidates the individual originally voted for.^{[5]} Under PAV, to calculate the satisfaction of an individual, only the elected candidates that the individual voted for are counted  the unsuccessful candidates that they voted for, as well as the elected candidates that they did not vote for are not taken into account. Assuming that an individual voted for n candidates that were successful, their satisfaction would be calculated using the formula^{[4]}
Summing the satisfaction across all voters with any potential result gives the population's total satisfaction with that result. The total satisfaction is calculated for every possible set of candidates, and the set of candidates with the highest total satisfaction is deemed to be the winning set.
In an election with only one winner, PAV operates in exactly the same way as normal approval voting. If on the other hand, each voter voted exclusively for all of the candidates within a single party, PAV would function in the same way as the D'Hondt method of partylist proportional representation.
Counting the votes in PAV is NPhard, making it a very computationally demanding voting method as the number of candidates and seats increase.^{[6]} If there were c candidates and s seats, then there would be
combinations of candidates to compare with each election,^{[7]} for example if there were 24 candidates for 4 seats, there would be 10,626 combinations to calculate total satisfaction for. An election requiring this many calculations would necessitate vote counting by computer.
Example
2 seats to be filled, four candidates: Andrea (A), Brad (B), Carter (C), and Delilah (D), and 30 voters. The ballots are:
 5: AB
 17: AC
 8: D
There are 6 possible results: AB, AC, AD, BC, BD, and CD.
AB  AC  AD  BC  BD  CD  

Voters approving at least 1 successful candidate (satisfaction of 1 for 1st approved candidate)  22  22  30  22  13  25 
Voters approving at least 2 successful candidates (satisfaction of 1/2 for 2nd approved candidate)  5  17  0  0  0  0 
Total satisfaction  24.5  30.5  30  22  13  25 
Andrea and Carter are elected.
See also
 Approval voting
 Block approval voting
 Sequential proportional approval voting
 Satisfaction approval voting
 Pluralityatlarge voting
Advantages and disadvantages
Most systems of proportional representation use party lists. The system was designed to have both proportional representation and personal votes (voters vote for candidates, not for a party list). It deserves to be called a "proportional" system because if votes turn out to follow a partisan scheme (each voter votes for all candidates from a party and no other) then the system elects a number of candidates in each party that is proportional to the number of voters who chose this party^{[8]}
References
 ^ Thiele, Thorvald N. (1895). "Om Flerfoldsvalg". Oversigt over det Kongelige Danske Videnskabernes Selskabs Forhandlinger: 415–441.
 ^ http://www2.math.uu.se/~svante/papers/sjV9.pdf
 ^ https://rangevoting.org/QualityMulti.html#acknow
 ^ ^{a} ^{b} 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.
 ^ Aziz, Haris; Brill, Markus; Conitzer, Vincent; Elkind, Edith; Freeman, Rupert; Walsh, Toby (2014). "Justified Representation in ApprovalBased Committee Voting". arXiv:1407.8269 [cs.MA].
 ^ Aziz, Haris; Serge Gaspers, Joachim Gudmundsson, Simon Mackenzie, Nicholas Mattei, Toby Walsh. "Computational Aspects of MultiWinner Approval Voting". Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems. pp. 107–115. arXiv:1407.3247. ISBN 9781450334136.CS1 maint: multiple names: authors list (link)
 ^ Enric Plaza: "Technologies for political representation and accountability": p9 [1]
 ^ Brill, Markus; Laslier, JeanFrançois; Skowron, Piotr (2016). "Multiwinner Approval Rules as Apportionment Methods". arXiv:1611.08691 [cs.GT].