Condorcet PR
Condorcet PR is a term that currently encompasses (at most) all voting methods that reduce to a Condorcet method in the single-winner case. There is no general agreement on how to further specify the category, though most would likely agree that some multi-winner voting methods are not proportional in any meaningful sense (such as Bloc Ranked Pairs), and thus do not qualify to be Condorcet PR methods.[1] Further, many would likely argue that PSC, probably Droop-PSC, is a must. Generally, most discussions of "Condorcet PR" methods are about CPO-STV and Schulze STV.
Discussion
Examples can be used to test certain properties that may or may not be desirable for different specifications of Condorcet PR to have. A general idea behind Condorcet PR that can be used to distinguish it from other PR methods is that Droop quotas of voters found in pairwise comparisons are guaranteed representation, rather than only Droop quota solid coalitions. Consider the following 2-winner example:
11 A>B>C
5 B>A=C
9 C>B>A
(space added to emphasize the distinct solid coalitions)
11 D>E>F
5 E>D=F
9 F>E>D
One possible specification of Condorcet PR is that (B, E) should win; this is because there are two disjoint Hare solid coalitions for (A, B, C) and (D, E, F) respectively, with B and E each being Condorcet winners in their own solid coalitions (B/E pairwise beats A/D 14 to 11, and pairwise beats C/F 16 to 9). So essentially, though B and E aren't 1st choice candidates for a Droop quota, they do show up as prominently supported by Droop quotas in every comparison. A 2-winner example used to demonstrate one possible generalization of the Smith set for the multi-winner case:
1 A>B>C (space for clarity) >I=J
1 B>C>A (space for clarity) >I=J
1 C>A>B (space for clarity) >I=J
1 D (space for clarity) >I=J
1 E>F>G (space for clarity) >I=J
1 F>G>E (space for clarity) >I=J
1 G>E>F (space for clarity) >I=J
1 H (space for clarity) >I=J
One possible 2-winner Smith set here could be considered to be all winner sets with 1 candidate from (A, B, C) and 1 candidate from (E, F, G) (note that these can be visualized with a combinatorics calculator such as https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html); this is because 2 disjoint Droop solid coalitions support each of those sets of candidates. Note that this particular generalization would disqualify candidates I and J from being in the Smith set winner sets, even though they are the top 2 candidates in the Condorcet ranking. A 2-winner example with discussion on how a Hare-PSC compliant Smith set can be shrunk even further using the concept of "strategic imposition":
8 D>A>B>C
7 B>C>A>D
5 C>B>A>D
8 D2>A2>B2>C2
7 B2>C2>A2>D2
5 C2>B2>A2>D2
There are 2 disjoint Hare solid coalitions for A through D and A2 through D2 respectively, so a Hare-PSC compliant Smith set of winner sets must have at most 1 winner from each solid coalition set of candidates in its winner sets. But note that 12 voters in each Hare solid coalition prefer the set (A, B, C) > D and (A2, B2, C2) > D2 to 8 voters who prefer it the other way around respectively; those voters would have the power to force their preferred candidates to win in many even semi-proportional methods. Based on this, we could say that the Smith set can be even further shrunk to only those winner sets which have 1 candidate from each strategically imposed set, meaning D and D2 are eliminated from all Smith winner sets.
There are various ways to extend the Independence of Smith-dominated Alternatives criterion to the multi-winner case. 3-winner example with discussion for Droop-PSC compliant Condorcet PR:
1 c1>c2>c3>e1>e2>e3>e4>d1
1 c2>c3>c1>e1>e2>e3>e4>d1
1 c3>c1>d1>c2>e1>e2>e3>e4
6 e1>e2>e3>e4>c1>c2>c3>d1
3 Droop quotas of voters prefer anyone other than d1 to win, so d1 can't be in a Droop-compliant Smith set winner set. One way to apply ISDA then would be to say that the election result should be the same whether or not d1 had run. This can sometimes allow elimination-based logic to shrink the number of possible winner sets; in this case, eliminating d1 yields:
3 (Party C candidates)>all others
6 (Party E candidates)>all others
And so the Droop proportionality criterion would require one Party C candidate and 2 E candidates to win. This example would disqualify the Quota Borda method from being Condorcet PR, since it would elect 3 E candidates (see the Expanding Approvals Rule article).
2-winner example:
18 A>B
16 B
18 C>D
16 D
32 E
It may be desirable for a Droop-PSC compliant method to elect one candidate from (A, B) and one from (C, D). This is because if A and C drop out, B and D become a Droop quota's 1st choice, and therefore it'd be a spoiler effect if their presence were to help E win. Defeat-dropping Condorcet methods appear to be the best Condorcet methods for this in the single-winner case.
2-winner example:
10 A>C>D
10 B>C>D
9 E
2 F
There is arguably some type of coalition for candidates A through D, since if A and B don't win, 20 voters will prefer C and D to win. 20 voters/2 seats=10 voters per seat, which is more voters per seat than E or F can muster to win, so a D-Hondt based Condorcet PR method could eliminate everyone other than A through D, at which point A and B are a Hare quota's 1st choice (when computing the Hare quota to account for the fact that only 20 voters have preferences among the uneliminated candidates) and would thus be considered winners by Hare-PSC.
It is also worth considering how various specifications of Condorcet PR behave in the variants of the Party List case (where voters rank their party's candidates 1st and all other parties' candidates last), regarding whether they reduce to various divisor or highest-remainders methods and which ones.
Because of the inherent connection between Condorcet and Score voting (see Score voting#Connection to Condorcet methods), it may be worth modeling or testing a Condorcet PR method to see whether it looks like any particular cardinal PR method when strength of preference is used in the pairwise matchups (see Pairwise counting#Cardinal methods). In addition, concepts from cardinal PR methods can be used to construct Condorcet PR methods; for example, cardinal methods often elect candidates with the most points. That can abstractly be thought of as "elect the Condorcet winner" since the Score winner is a Self-referential Smith-efficient Condorcet method type of winner.
And when cardinal PR methods talk about reweighting voters' ballots in proportion to their support for a winner, this can be thought of as "reweight voters in proportion to how much they helped a candidate win." One way to transfer this concept to Condorcet, when there is a Condorcet winner, is to reweight voters' ballots based on how much they contributed support to the CW in their weakest pairwise victories (according to margins); this naturally takes more weight from those who ranked the CW 1st (since such voters helped the CW in all of their victories, not just some), and for example, if the CW has a victory of a 2% margin and another victory of a 30% margin, with some voters only contributing to the 2% victory and other voters only to the 30% victory, would take much more weight from the 2%-contributing voters, as they were much more pivotal in helping to make the CW pairwise beat all others.
One consideration for a Condorcet PR method is its "approval case", which is the situation where all voters rank every candidate either 1st or last. In these situations, it may be worth considering what kind of Approval PR method the method reduces to. If the method is sequential and reduces to SPAV, for example, then it may be necessary to make it take the most weight from voters who ranked the winner above or equal to every other candidate; this is because in SPAV, a voter who bullet voted the winner has as much weight taken from them as a voter who approved the winner and every other candidate.
Here is a
2-winner example that documents some of the difficulties with creating a simple sequential Condorcet PR method:
33 A1>A2
32 A2>A1
35 B
Arguably A1 should win first, but the trouble is deciding how to reweight the ballots. The 33 A1-top ballots clearly fully support A1, yet the A2>A1 ballots don't. So if you take full weight from the A2-top ballots here to allow B to win the next seat, then it may even be justifiable to take full weight from a voter who only ranked the round's winner above one other candidate out of, say, 100.
See Asset voting for further discussion. Arguably, Condorcet can be thought of as a model of negotiations where each voter maximally aims for their own benefit, and this can be extended to the PR case by allowing each voter to only support up to one candidate at a time. This means that in a situation like
26 A>B
25 B
49 C
B must win, since C always has 49 votes to win the negotiations, and thus the only way for A-top voters to beat C is to make sure B has 51 votes; this can be extended to consider situations with multiple of these "semi-solid coalitions".
Here is an example illustrating the difficulty of creating a Condorcet multiwinner method along the lines of RRV:
34 A
35 B>C
31 C
B is the CW, so they'd win the first seat. If their supporters' ballots are reweighted by half, then C pairwise beats A 48.5 to 34 and wins, despite A being bullet voted by a Droop quota. One complicated way of possibly fixing this is to, after electing B, say that if B hadn't been in the election, C would have been the winner, and therefore both B and C voters' ballots should be reweighted by half since they both rank C above all candidates other than B (A), thus allowing A to beat C 34 to 33.