Difference between revisions of "Proportionality for Solid Coalitions"
(Adding {{wikipedia}} template, since wikipedia:Proportionality for Solid Coalitions exists) 

(15 intermediate revisions by one other user not shown)  
Line 1:  Line 1:  
+  {{wikipedia}} 

−  '''Proportionality for Solid Coalitions''' ('''PSC''') is a criterion for proportional methods requiring that sufficientlysized groups of voters (solid coalitions) always elect a proportional number of candidates from their set of mutually mostpreferred candidates. In general, any time any group of voters prefers any set of candidates over all others, a certain minimum number of candidates from that set must win to pass the criterion, and the same must hold if the preferred set of candidates for a group can be shrunk or enlargened. It is the main conceptualization of Proportional Representation generally used throughout the world ([[Party List]] and [[STV]] pass versions of it.) 

+  
+  '''Proportionality for Solid Coalitions''' ('''PSC''') is a criterion for proportional methods requiring that sufficientlysized groups of voters (solid coalitions) always elect a proportional number of candidates from their set of mutually mostpreferred candidates. In general, any time any group of voters prefers any set of candidates over all others (i.e. they are solidly committed to/solidly support these candidates), a certain minimum number of candidates from that set must win to pass the criterion, and the same must hold if the preferred set of candidates for a group can be shrunk or enlargened. It is the main conceptualization of Proportional Representation generally used throughout the world ([[Party List]] and [[STV]] pass versions of it.) 

Any voting method that collects enough information to distinguish solid coalitions (generally scored or ranked methods, since preferences can be inferred from their ballots) can be forced to be PSCcompliant by first electing the proportionally correct number of candidates from each solid coalition before doing anything else. 
Any voting method that collects enough information to distinguish solid coalitions (generally scored or ranked methods, since preferences can be inferred from their ballots) can be forced to be PSCcompliant by first electing the proportionally correct number of candidates from each solid coalition before doing anything else. 

== Types of PSC == 
== Types of PSC == 

−  For the purposes of PSC, maximal support to a candidate or set of candidates is generally determined as: 

−  
−   with ranked ballots, all of the candidates are ranked above all candidates not in the set. 

−  
−   with rated, approval, and chooseone ballots, all of the candidates in the set are given the maximum score or are marked. 

−  
=== HarePSC === 
=== HarePSC === 

−  kPSC or HarePSC is a condition requiring a solid coalition of c candidates supported by k Hare quotas to be always elect at least <math>\min(c, k)</math> mostpreferred candidates (i.e. k candidates whenever c is not less than k.) 
+  '''kPSC''' or '''HarePSC''' is a condition requiring a solid coalition of c candidates supported by k [[Hare quotaHare quotas]] to be always elect at least <math>\min(c, k)</math> mostpreferred candidates (i.e. k candidates whenever c is not less than k.) 
=== DroopPSC === 
=== DroopPSC === 

−  k+1PSC or DroopPSC 
+  '''k+1PSC''' or '''DroopPSC''' is the same as HarePSC but holds for [[HagenbachBischoff quotaHagenbachBischoff quotas]] instead (many authors call the [[HagenbachBischoff quota]]/Hb quota a Droop quota), and requires the solid coalition's preferred candidates to be supported by ''more'' than k HB quotas, rather than at least that amount. The DroopPSC criterion is also called the '''Droop proportionality criterion'''. Note that Droop proportionality implies the [[mutual majority criterion]], and more generally guarantees that a majority will always win at least half of the seats. 
The main difference between HarePSC and DroopPSC can be seen with an example: Suppose you can buy two boxes of pizza, with over 2/3rds of voters wanting Cheese pizza, and under 1/3rds of the voters wanting Pepperoni pizza. HarePSC would say that you should buy at least one box of Cheese pizza, but has no opinion on what you should buy for the second box, whereas DroopPSC would say that you should buy two boxes of Cheese pizza. This can be explained as happening partially because if the 2/3rds group of cheesepreferring voters split themselves into two equally sized groups of over 1/3rd of voters each, then these "two" groups that want Cheese would each outnumber the group of under 1/3rds of voters that want Pepperoni. 
The main difference between HarePSC and DroopPSC can be seen with an example: Suppose you can buy two boxes of pizza, with over 2/3rds of voters wanting Cheese pizza, and under 1/3rds of the voters wanting Pepperoni pizza. HarePSC would say that you should buy at least one box of Cheese pizza, but has no opinion on what you should buy for the second box, whereas DroopPSC would say that you should buy two boxes of Cheese pizza. This can be explained as happening partially because if the 2/3rds group of cheesepreferring voters split themselves into two equally sized groups of over 1/3rd of voters each, then these "two" groups that want Cheese would each outnumber the group of under 1/3rds of voters that want Pepperoni. 

=== Weak forms of PSC === 
=== Weak forms of PSC === 

+  Most methods that pass weak forms of PSC (i.e. any method that passes a weak form of DroopPSC) allow a [[majority]] to strategically vote to get at least half of the seats. The more candidates there are that are preferred by a solid coalition, however, the less likely this may be; for example, [[Approval voting]] will likely pass the [[majority criterion]] in practice more often than the [[mutual majority criterion]], because things like the [[chicken dilemma]] may encourage voters to not approve all solidly supported candidates. 

−  PSC is a requirement that holds for honest voters. Many voting methods pass weaker requirements that hold only for strategic voters, with the difficulty of the strategy depending on the method. In general, any method that passes such weaker versions of PSC is considered to be at least semiproportional. Note that PSC implies all of these weaker forms of PSC. Here are some of these weaker requirements (note that the requirements vary slightly depending on whether you're using the Hare quota, HB quota, or other quota):<blockquote>If a solid coalition of k quotas evenly distributes its support among k of their preferred candidates such that each of the k candidates receives maximal support from at least a quota of voters,then at least k of their preferred candidates must win.</blockquote>[[SNTV]] passes this with Droop quotas.<blockquote>If a solid coalition of k quotas gives maximal support to k of their preferred candidates, and no support to all other candidates, then at least k of their preferred candidates must win.</blockquote>Most cardinal PR methods pass this for Hare quotas.<blockquote>If a solid coalition of k quotas gives maximal support to at least k of their preferred candidates, and lessthanmaximal support to all other candidates, at least k of those preferred candidates must win.</blockquote>[[Sequential Monroe voting]] passes this, making it the best cardinal PR method from the perspective of PSC. This is probably the strongest PSClike requirement that a natural voting method can pass without actually passing PSC. 

+  
+  PSC is a requirement that holds for honest voters. Many voting methods pass weaker requirements that hold only for strategic voters, with the difficulty of the strategy depending on the method. In general, any method that passes such weaker versions of PSC is considered to be at least semiproportional. Note that PSC implies all of these weaker forms of PSC. Here are some of these weaker requirements (note that the requirements vary slightly depending on whether you're using the Hare quota, HB quota, or other quota): 

+  
+  (For the purposes of PSC, maximal support to a candidate or set of candidates is generally determined as: 

+  
+  * with [[Ranked ballotranked ballots]], all of the candidates in the set are ranked above all candidates not in the set. 

+  * with rated, approval, and chooseone ballots, all of the candidates in the set are given the maximum score or are marked.) 

+  <blockquote>If a solid coalition of k quotas evenly distributes its support among k of their preferred candidates such that each of the k candidates receives maximal support from a quota of voters, with each quota of voters giving no support to any other candidates, and the solid coalition as a whole giving no support to any candidates other one of the k candidates, then at least k of their preferred candidates must win.</blockquote>[[SNTV]] and [[Cumulative votingcumulative voting]] pass this with Droop quotas.<blockquote>If a solid coalition of k quotas gives maximal support to k of their preferred candidates, and no support to all other candidates, then at least k of their preferred candidates must win.</blockquote>Most cardinal PR methods pass this for Hare quotas.<blockquote>If a solid coalition of k quotas gives maximal support to at least k of their preferred candidates, and lessthanmaximal support to all other candidates, at least k of those preferred candidates must win.</blockquote>[[Sequential Monroe voting]] passes this, making it the best cardinal PR method from the perspective of PSC. This is probably the strongest PSClike requirement that a natural voting method can pass without actually passing PSC. 

+  <br /> 

== Examples == 
== Examples == 

Line 61:  Line 66:  
Looking at the top 5 lines, 50 voters, a Hare quota, mutually most prefer the set of candidates (A15) over all other candidates, so HarePSC requires at least one of (A15) must win. (Note that Sequential Monroe voting fails HarePSC in this example. However, one could forcibly make SMV do so by declaring that the candidate with the highest Monroe score within the set (A15) must win the first seat, for example.) <ref name="The Center for Election Science 2020">{{cite web  title=An example of maximal divergence between SMV and HarePSC  website=The Center for Election Science  date=20200131  url=https://forum.electionscience.org/t/anexampleofmaximaldivergencebetweensmvandharepsc/586  accessdate=20200219}}</ref> 
Looking at the top 5 lines, 50 voters, a Hare quota, mutually most prefer the set of candidates (A15) over all other candidates, so HarePSC requires at least one of (A15) must win. (Note that Sequential Monroe voting fails HarePSC in this example. However, one could forcibly make SMV do so by declaring that the candidate with the highest Monroe score within the set (A15) must win the first seat, for example.) <ref name="The Center for Election Science 2020">{{cite web  title=An example of maximal divergence between SMV and HarePSC  website=The Center for Election Science  date=20200131  url=https://forum.electionscience.org/t/anexampleofmaximaldivergencebetweensmvandharepsc/586  accessdate=20200219}}</ref> 

−  +  The reason DroopPSC guarantees a majority wins at least half of the seats is that majority solid coalitions always constitute more voters than the number of HagenbachBischoff quotas corresponding to half of the seats, while with Hare quotas they can only guarantee they will win just under half the seats and have over half a Hare quota left to win the additional seat required to get at least half the seats. 5winner example using STV with Hare quotas: 

{ class="wikitable" 
{ class="wikitable" 

Line 83:  Line 88:  
} 
} 

−  Note that 51 voters, a majority, prefer (A13) over all other candidates, and thus electing all 3 of them would mean 3 out of 5 seats, a majority, would belong to the majority. Also, the 
+  Note that 51 voters, a majority, prefer (A13) over all other candidates, and thus electing all 3 of them would mean 3 out of 5 seats, a majority, would belong to the majority. Also, the HB quota here is ~16.667, and thus by DroopPSC the majority has (51/16.667 rounded down to the nearest number)=3 seat or PR guarantees, which would give them a majority of seats. However, by HarePSC, they only have (51/20 rounded down) = 2 PR guarantees, and thus STV using Hare quotas, which passes HarePSC, yields: 
<blockquote>So, the Hare quota here is 20. A1 and A2 are immediately elected, but posttransfer A3 only has 11 votes, and is thus eliminated first. B1, B2, B3 take the remaining 3 seats.<ref name="reddit 2011">{{cite web  title=Can RankedChoice Voting Save American Democracy? : EndFPTP  website=reddit  date=20110126  url=https://www.reddit.com/r/EndFPTP/comments/ermb1s/comment/ff7a7f8  accessdate=20200219}}</ref></blockquote> 
<blockquote>So, the Hare quota here is 20. A1 and A2 are immediately elected, but posttransfer A3 only has 11 votes, and is thus eliminated first. B1, B2, B3 take the remaining 3 seats.<ref name="reddit 2011">{{cite web  title=Can RankedChoice Voting Save American Democracy? : EndFPTP  website=reddit  date=20110126  url=https://www.reddit.com/r/EndFPTP/comments/ermb1s/comment/ff7a7f8  accessdate=20200219}}</ref></blockquote> 

+  === Solid coalition overlaps === 

There can be quota overlaps when assigning PSC claims; suppose a group constituting 80% of a quota of voters vote A>B>C=D, another group of 80% of a quota vote B>A>C=D, and another group of 50% of a quota vote C>A=B=D. Then, 2 candidates must be elected from the set (A, B, C, D), since in total 2.1 quotas mutually most prefer that set, but a further constraint is that 1 candidate must win from within (A, B), since 1.6 quotas mutually most prefer them. It would not satisfy PSC if the final winner set had neither A or B in it in other words, even if it had C and D. 
There can be quota overlaps when assigning PSC claims; suppose a group constituting 80% of a quota of voters vote A>B>C=D, another group of 80% of a quota vote B>A>C=D, and another group of 50% of a quota vote C>A=B=D. Then, 2 candidates must be elected from the set (A, B, C, D), since in total 2.1 quotas mutually most prefer that set, but a further constraint is that 1 candidate must win from within (A, B), since 1.6 quotas mutually most prefer them. It would not satisfy PSC if the final winner set had neither A or B in it in other words, even if it had C and D. 

+  
+  == Complying methods == 

+  [[STV]], [[Party list]], and [[Expanding Approvals Rule]] pass forms of PSC. Specifically, the choice of quota in those methods determines which type of PSC they pass. 

+  
+  [[D'Hondt]] passes DroopPSC, so many methods that reduce to it in the [[Party list caseparty list case]] (such as [[Schulze STV]]), with the notable exception of most [[:Category:Cardinal PR methodscardinal PR methods]], pass it too. 

== Generalised solid coalitions == 
== Generalised solid coalitions == 

Line 113:  Line 124:  
By analogy to [[Descending Acquiescing Coalitions]], the generalized PSC could also be called proportionality for acquiescing coalitions. 
By analogy to [[Descending Acquiescing Coalitions]], the generalized PSC could also be called proportionality for acquiescing coalitions. 

+  
+  == Factions == 

+  Inherent to the discussion of PSC are factions and subfactions, which are defined here as solid coalitions (i.e. they prefer some candidates above all others). PSC not only guarantees a certain number of seats to a political faction or party, but also guarantees that the wings or subfactions of that group are also proportionally represented i.e. if 40% of the voters are Democrats, and 50% of the Democrats are more liberal Democrats, then 40% of the seats will go to the Dems and 20% of all seats will go to liberal Dems. There is some discussion over how valid the PSC guarantees for subfactions are, however; for example, in [[Party list]], the ordering of the candidates on the list may not accurately represent the subfactions (i.e. if there are 5 spots on the list, with the party expected to get 4 seats, and there is a subfaction that constitutes 25% of the party, then that subfaction proportionally deserves 1 of the 4 seats, but the party leaders might put other subfactions' candidates in the top 4 spots of the list, effectively preventing that subfaction from winning a seat).<ref>{{Cite weburl=https://forum.electionscience.org/u/Marylandertitle=Marylanderwebsite=The Center for Election Sciencelanguage=enUSaccessdate=20200509}}</ref> 

== Notes == 
== Notes == 

+  
+  === Rated ballot adaptations === 

+  With rated ballots, it is possible for a voter to express lessthanfull support for any candidate. Because of this, one way to apply PSC to rated ballots and thus cardinal PR methods would be to require that every voter in each solid coalition give at least one candidate in their coalition the maximum score. An even stricter requirement could be to require them to give every candidate in the coalition a perfect score, though this could instead be thought of as a genuinely slightly weaker form of PSC; it is already satisfied by [[SMV]]. This is modeled off of the [[Majority criterion for rated ballotsmajority criterion for rated ballots]]. 

+  
+  === Singlewinner case === 

DroopPSC implies HarePSC, since a Hare quota is simply a large Droop quota, but the same doesn't hold the other way around. HarePSC is equivalent to the unanimity criterion and DroopPSC to the mutual majority criterion in the singlewinner case. Note that this means cardinal PR methods can only pass HarePSC and not DroopPSC in order to reduce to cardinal methods that fail the mutual majority criterion in the singlewinner case, which is most of them. 
DroopPSC implies HarePSC, since a Hare quota is simply a large Droop quota, but the same doesn't hold the other way around. HarePSC is equivalent to the unanimity criterion and DroopPSC to the mutual majority criterion in the singlewinner case. Note that this means cardinal PR methods can only pass HarePSC and not DroopPSC in order to reduce to cardinal methods that fail the mutual majority criterion in the singlewinner case, which is most of them. 

+  === Freeriding === 

+  Though DroopPSC guarantees a majority half of the seats, it doesn't guarantee a plurality group half the seats when it could take them using [[vote management]]. 5winner example: 

+  
+  51 A>C 

+  49 B 

+  10 C 

+  
+  The HB quota is (110/(5)+1)=~18.333, so A is guaranteed only (51/18.333 rounded down) = 2 out of the 5 seats, less than half. With vote management, Party A could split into 3 groups of 17 votes each, and most PR methods would then give them 3 seats. So it may be worth considering a stronger type of DroopPSC based on giving solid coalitions at least as many seats as they'd get in [[D'Hondt]]. [[Schulze STV]] is an example of a method that does so. 

+  
+  === Semisolid coalitions === 

Note that PSC doesn't hold if some voters in a coalition back outofcoalition candidates i.e. 1winner example with Droop quota of 51: 
Note that PSC doesn't hold if some voters in a coalition back outofcoalition candidates i.e. 1winner example with Droop quota of 51: 

Line 144:  Line 173:  
Arguably there is some kind of coalition of 45 voters backing candidates A through J here, and since the largest opposing coalition is 8 voters, D'Hondt would say that the 45voter coalition ought to win all 5 seats. At that point, one could eliminate all candidates outside the 45voter coalition (K and L) at which point A through E all are a Hare quota's 1st choice and must all win. This sort of thinking is generally what Condorcet PR methods such as Schulze STV do. 
Arguably there is some kind of coalition of 45 voters backing candidates A through J here, and since the largest opposing coalition is 8 voters, D'Hondt would say that the 45voter coalition ought to win all 5 seats. At that point, one could eliminate all candidates outside the 45voter coalition (K and L) at which point A through E all are a Hare quota's 1st choice and must all win. This sort of thinking is generally what Condorcet PR methods such as Schulze STV do. 

+  
+  === Other multiwinner properties analogous to PSC === 

+  Properties analagous to PSC can be considered for multiwinner voting methods that aren't proportional. For example, [[Bloc votingBloc Score voting]] guarantees that a majority solid coalition can elect all of its preferred candidates if they set their [[approval threshold]] between their preferred candidates and all other candidates. 

+  
+  === Finding solid coalitions from ballots === 

+  Because a solid coalition is defined as a group of voters who all agree on their k highestranked candidates, it can be found by looking at the first choice(s) of a ballot, then the second choices, etc. until some group of voters are found to be in agreement on the candidates examined so far. Note that if some voters equally rank candidates, then it is necessary to avoid counting more candidates on their ballots than the number of candidates that have already been examined on other ballots. 

== See Also == 
== See Also == 

Line 156:  Line 191:  
[[Category:Voting theory]] 
[[Category:Voting theory]] 

+  [[Category:Proportionalityrelated concepts]] 
Latest revision as of 03:35, 7 April 2021
Proportionality for Solid Coalitions (PSC) is a criterion for proportional methods requiring that sufficientlysized groups of voters (solid coalitions) always elect a proportional number of candidates from their set of mutually mostpreferred candidates. In general, any time any group of voters prefers any set of candidates over all others (i.e. they are solidly committed to/solidly support these candidates), a certain minimum number of candidates from that set must win to pass the criterion, and the same must hold if the preferred set of candidates for a group can be shrunk or enlargened. It is the main conceptualization of Proportional Representation generally used throughout the world (Party List and STV pass versions of it.)
Any voting method that collects enough information to distinguish solid coalitions (generally scored or ranked methods, since preferences can be inferred from their ballots) can be forced to be PSCcompliant by first electing the proportionally correct number of candidates from each solid coalition before doing anything else.
Types of PSC[edit  edit source]
HarePSC[edit  edit source]
kPSC or HarePSC is a condition requiring a solid coalition of c candidates supported by k Hare quotas to be always elect at least mostpreferred candidates (i.e. k candidates whenever c is not less than k.)
DroopPSC[edit  edit source]
k+1PSC or DroopPSC is the same as HarePSC but holds for HagenbachBischoff quotas instead (many authors call the HagenbachBischoff quota/Hb quota a Droop quota), and requires the solid coalition's preferred candidates to be supported by more than k HB quotas, rather than at least that amount. The DroopPSC criterion is also called the Droop proportionality criterion. Note that Droop proportionality implies the mutual majority criterion, and more generally guarantees that a majority will always win at least half of the seats.
The main difference between HarePSC and DroopPSC can be seen with an example: Suppose you can buy two boxes of pizza, with over 2/3rds of voters wanting Cheese pizza, and under 1/3rds of the voters wanting Pepperoni pizza. HarePSC would say that you should buy at least one box of Cheese pizza, but has no opinion on what you should buy for the second box, whereas DroopPSC would say that you should buy two boxes of Cheese pizza. This can be explained as happening partially because if the 2/3rds group of cheesepreferring voters split themselves into two equally sized groups of over 1/3rd of voters each, then these "two" groups that want Cheese would each outnumber the group of under 1/3rds of voters that want Pepperoni.
Weak forms of PSC[edit  edit source]
Most methods that pass weak forms of PSC (i.e. any method that passes a weak form of DroopPSC) allow a majority to strategically vote to get at least half of the seats. The more candidates there are that are preferred by a solid coalition, however, the less likely this may be; for example, Approval voting will likely pass the majority criterion in practice more often than the mutual majority criterion, because things like the chicken dilemma may encourage voters to not approve all solidly supported candidates.
PSC is a requirement that holds for honest voters. Many voting methods pass weaker requirements that hold only for strategic voters, with the difficulty of the strategy depending on the method. In general, any method that passes such weaker versions of PSC is considered to be at least semiproportional. Note that PSC implies all of these weaker forms of PSC. Here are some of these weaker requirements (note that the requirements vary slightly depending on whether you're using the Hare quota, HB quota, or other quota):
(For the purposes of PSC, maximal support to a candidate or set of candidates is generally determined as:
 with ranked ballots, all of the candidates in the set are ranked above all candidates not in the set.
 with rated, approval, and chooseone ballots, all of the candidates in the set are given the maximum score or are marked.)
If a solid coalition of k quotas evenly distributes its support among k of their preferred candidates such that each of the k candidates receives maximal support from a quota of voters, with each quota of voters giving no support to any other candidates, and the solid coalition as a whole giving no support to any candidates other one of the k candidates, then at least k of their preferred candidates must win.SNTV and cumulative voting pass this with Droop quotas.
If a solid coalition of k quotas gives maximal support to k of their preferred candidates, and no support to all other candidates, then at least k of their preferred candidates must win.Most cardinal PR methods pass this for Hare quotas.
If a solid coalition of k quotas gives maximal support to at least k of their preferred candidates, and lessthanmaximal support to all other candidates, at least k of those preferred candidates must win.Sequential Monroe voting passes this, making it the best cardinal PR method from the perspective of PSC. This is probably the strongest PSClike requirement that a natural voting method can pass without actually passing PSC.
Examples[edit  edit source]
5winner example, Hare quota 50 (example done using scored ballots):
Number  Ballots 

10  A1:10 A2:7 A3:7 A4:7 A5:7 B1:1 C1:0 D1:0 E1:0 F1:0 
10  A1:7 A2:10 A3:7 A4:7 A5:7 B1:0 C1:1 D1:0 E1:0 F1:0 
10  A1:7 A2:7 A3:10 A4:7 A5:7 B1:0 C1:0 D1:1 E1:0 F1:0 
10  A1:7 A2:7 A3:7 A4:10 A5:7 B1:0 C1:0 D1:0 E1:1 F1:0 
10  A1:7 A2:7 A3:7 A4:7 A5:10 B1:0 C1:0 D1:0 E1:0 F1:0 
40  A1:2 A2:0 A3:0 A4:0 A5:1 B1:10 C1:0 D1:0 E1:0 F1:0 
40  A1:0 A2:2 A3:0 A4:0 A5:1 B1:0 C1:10 D1:0 E1:0 F1:0 
40  A1:0 A2:0 A3:2 A4:0 A5:1 B1:0 C1:0 D1:10 E1:0 F1:0 
40  A1:0 A2:0 A3:0 A4:2 A5:1 B1:0 C1:0 D1:0 E1:10 F1:0 
40  A1:0 A2:0 A3:0 A4:0 A5:0 B1:0 C1:0 D1:0 E1:0 F1:10 
Looking at the top 5 lines, 50 voters, a Hare quota, mutually most prefer the set of candidates (A15) over all other candidates, so HarePSC requires at least one of (A15) must win. (Note that Sequential Monroe voting fails HarePSC in this example. However, one could forcibly make SMV do so by declaring that the candidate with the highest Monroe score within the set (A15) must win the first seat, for example.) ^{[1]}
The reason DroopPSC guarantees a majority wins at least half of the seats is that majority solid coalitions always constitute more voters than the number of HagenbachBischoff quotas corresponding to half of the seats, while with Hare quotas they can only guarantee they will win just under half the seats and have over half a Hare quota left to win the additional seat required to get at least half the seats. 5winner example using STV with Hare quotas:
Number  Ballots 

26  A2>A1>A3>B1>B2>B3 
25  A1>A3>A2>B1>B2>B3 
17  B1>B2>B3>A1>A3>A2 
16  B2>B1>B3>A1>A3>A2 
16  B3>B2>B1>A1>A3>A2 
Note that 51 voters, a majority, prefer (A13) over all other candidates, and thus electing all 3 of them would mean 3 out of 5 seats, a majority, would belong to the majority. Also, the HB quota here is ~16.667, and thus by DroopPSC the majority has (51/16.667 rounded down to the nearest number)=3 seat or PR guarantees, which would give them a majority of seats. However, by HarePSC, they only have (51/20 rounded down) = 2 PR guarantees, and thus STV using Hare quotas, which passes HarePSC, yields:
So, the Hare quota here is 20. A1 and A2 are immediately elected, but posttransfer A3 only has 11 votes, and is thus eliminated first. B1, B2, B3 take the remaining 3 seats.^{[2]}
Solid coalition overlaps[edit  edit source]
There can be quota overlaps when assigning PSC claims; suppose a group constituting 80% of a quota of voters vote A>B>C=D, another group of 80% of a quota vote B>A>C=D, and another group of 50% of a quota vote C>A=B=D. Then, 2 candidates must be elected from the set (A, B, C, D), since in total 2.1 quotas mutually most prefer that set, but a further constraint is that 1 candidate must win from within (A, B), since 1.6 quotas mutually most prefer them. It would not satisfy PSC if the final winner set had neither A or B in it in other words, even if it had C and D.
Complying methods[edit  edit source]
STV, Party list, and Expanding Approvals Rule pass forms of PSC. Specifically, the choice of quota in those methods determines which type of PSC they pass.
D'Hondt passes DroopPSC, so many methods that reduce to it in the party list case (such as Schulze STV), with the notable exception of most cardinal PR methods, pass it too.
Generalised solid coalitions[edit  edit source]
The Expanding Approvals Rule passes a stricter PR axiom than PSC:
Definition 5 (Generalised solid coalition) A set of voters N′ is a generalised solid coalition for a set of candidates C′ if every voter in N′ weakly prefers every candidate in C′ at least as high as every candidate in C\C′. That is, for all i ∈ N′ and for any c′ ∈ C′
 .
We note that under strict preferences, a generalised solid coalition is equivalent to solid coalition. Let c(i, j) denotes voter i’s jth most preferred candidate. In case the voter’s preference has indifferences, we use lexicographic tiebreaking to identify the candidate in the jth position.
Definition 6 (Generalised qPSC) Let q ∈ (n/(k + 1), n/k]. A committee W satisfies generalised qPSC if for all generalised solid coalitions N′ supporting candidate subset C′ with size N′ ≥ ℓq, there exists a set C′′ ⊆ W with size at least min{ℓ, C′} such that for all c′′ ∈ C′′
 ∃i ∈ N′ : c′′ i c(i,C′ ).
The idea behind generalised qPSC is identical to that of qPSC and in fact generalised qPSC is equivalent to qPSC under linear preferences. Note that in the definition above, a voter i in the solid coalition of voters N′ does not demand membership of candidates from the solidly supported subset C′ but of any candidate that is at least as preferred as a least preferred candidate in C′. Generalised weak qPSC is a natural weakening of generalised qPSC in which we require that C′ is of size at most ℓ.
Definition 7 (Generalised weak qPSC) Let q ∈ (n/(k + 1), n/k]. A committee W satisfies weak generalised qPSC if for every positive integer ℓ, and every generalised solid coalition N′ supporting a candidate subset C′
 C′ ≤ ℓ with size N′ ≥ ℓq, there
exists a set C′′ ⊆ W with size at least min{ℓ, C′} such that for all c′′ ∈ C′′
 ∃i ∈ N′ : c′′ i c(i,C′ ).^{[3]}
By analogy to Descending Acquiescing Coalitions, the generalized PSC could also be called proportionality for acquiescing coalitions.
Factions[edit  edit source]
Inherent to the discussion of PSC are factions and subfactions, which are defined here as solid coalitions (i.e. they prefer some candidates above all others). PSC not only guarantees a certain number of seats to a political faction or party, but also guarantees that the wings or subfactions of that group are also proportionally represented i.e. if 40% of the voters are Democrats, and 50% of the Democrats are more liberal Democrats, then 40% of the seats will go to the Dems and 20% of all seats will go to liberal Dems. There is some discussion over how valid the PSC guarantees for subfactions are, however; for example, in Party list, the ordering of the candidates on the list may not accurately represent the subfactions (i.e. if there are 5 spots on the list, with the party expected to get 4 seats, and there is a subfaction that constitutes 25% of the party, then that subfaction proportionally deserves 1 of the 4 seats, but the party leaders might put other subfactions' candidates in the top 4 spots of the list, effectively preventing that subfaction from winning a seat).^{[4]}
Notes[edit  edit source]
Rated ballot adaptations[edit  edit source]
With rated ballots, it is possible for a voter to express lessthanfull support for any candidate. Because of this, one way to apply PSC to rated ballots and thus cardinal PR methods would be to require that every voter in each solid coalition give at least one candidate in their coalition the maximum score. An even stricter requirement could be to require them to give every candidate in the coalition a perfect score, though this could instead be thought of as a genuinely slightly weaker form of PSC; it is already satisfied by SMV. This is modeled off of the majority criterion for rated ballots.
Singlewinner case[edit  edit source]
DroopPSC implies HarePSC, since a Hare quota is simply a large Droop quota, but the same doesn't hold the other way around. HarePSC is equivalent to the unanimity criterion and DroopPSC to the mutual majority criterion in the singlewinner case. Note that this means cardinal PR methods can only pass HarePSC and not DroopPSC in order to reduce to cardinal methods that fail the mutual majority criterion in the singlewinner case, which is most of them.
Freeriding[edit  edit source]
Though DroopPSC guarantees a majority half of the seats, it doesn't guarantee a plurality group half the seats when it could take them using vote management. 5winner example:
51 A>C 49 B 10 C
The HB quota is (110/(5)+1)=~18.333, so A is guaranteed only (51/18.333 rounded down) = 2 out of the 5 seats, less than half. With vote management, Party A could split into 3 groups of 17 votes each, and most PR methods would then give them 3 seats. So it may be worth considering a stronger type of DroopPSC based on giving solid coalitions at least as many seats as they'd get in D'Hondt. Schulze STV is an example of a method that does so.
Semisolid coalitions[edit  edit source]
Note that PSC doesn't hold if some voters in a coalition back outofcoalition candidates i.e. 1winner example with Droop quota of 51:
26 A>B25 B
49 C
STV would elect C here. Yet if A hadn't run, then B would've been 51 voter's 1st choice, so a Droop solid coalition would've been backing them, and since STV passes DroopPSC, B would've guaranteeably won.
One could also consider various extensions of the solid coalition idea, in part to address examples like the above one; 5winner example:
9 A>F>G>H>I>J9 B>F>G>H>I>J
9 C>F>G>H>I>J
9 D>F>G>H>I>J
9 E>F>G>H>I>J
8 K
7 L
Arguably there is some kind of coalition of 45 voters backing candidates A through J here, and since the largest opposing coalition is 8 voters, D'Hondt would say that the 45voter coalition ought to win all 5 seats. At that point, one could eliminate all candidates outside the 45voter coalition (K and L) at which point A through E all are a Hare quota's 1st choice and must all win. This sort of thinking is generally what Condorcet PR methods such as Schulze STV do.
Other multiwinner properties analogous to PSC[edit  edit source]
Properties analagous to PSC can be considered for multiwinner voting methods that aren't proportional. For example, Bloc Score voting guarantees that a majority solid coalition can elect all of its preferred candidates if they set their approval threshold between their preferred candidates and all other candidates.
Finding solid coalitions from ballots[edit  edit source]
Because a solid coalition is defined as a group of voters who all agree on their k highestranked candidates, it can be found by looking at the first choice(s) of a ballot, then the second choices, etc. until some group of voters are found to be in agreement on the candidates examined so far. Note that if some voters equally rank candidates, then it is necessary to avoid counting more candidates on their ballots than the number of candidates that have already been examined on other ballots.
See Also[edit  edit source]
References[edit  edit source]
 ↑ "An example of maximal divergence between SMV and HarePSC". The Center for Election Science. 20200131. Retrieved 20200219.
 ↑ "Can RankedChoice Voting Save American Democracy? : EndFPTP". reddit. 20110126. Retrieved 20200219.
 ↑ Aziz, Haris; Lee, Barton E. (20190809). "The expanding approvals rule: improving proportional representation and monotonicity". Social Choice and Welfare. Springer Science and Business Media LLC. 54 (1): 8. doi:10.1007/s00355019012083. ISSN 01761714.
 ↑ "Marylander". The Center for Election Science. Retrieved 20200509.