PSC-CLE

PSC-CLE (Proportionality of Solid Coalitions by Condorcet Loser Elimination) is a preference voting system for multi-winner elections.

Voting
Each voter ranks all candidates in order of preference. For example:


 * 1) Andrea
 * 2) Carter
 * 3) Brad
 * 4) Delilah

Procedure
Step 1: Compute the elimination order.

The elimination order is determined using a Condorcet ranking method. The ranking is inverted so that the Condorcet Winner, if one exists, will be last in the elimination order.

Step 2: Compute the proportionality rules.

Each possible set of candidates is entitled to a number of seats equal to either the greatest integer less than (proportion of voters solidly committed to that set) × (seats + 1), or the number of candidates in that set, whichever is lower. ("Solidly committed" is defined as in Descending Solid Coalitions.)

Step 3: Using the elimination order found in Step 1, eliminate candidates one-by-one until the number of candidates remaining is equal to the number of seats. But if the elimination of a candidate would cause a proportionality rule to be violated, then do not eliminate that candidate.

Example
2 seats to be filled, 4 candidates: Andrea (A), Brad (B), Carter (C), and Delilah (D). The ballots are:


 * 15: A>B>C>D
 * 51: A>C>B>D
 * 4: D>A>B>C
 * 4: D>A>C>B
 * 4: D>B>A>C
 * 4: D>B>C>A
 * 4: D>C>A>B
 * 4: D>C>B>A

The pairwise victories are:


 * 78-12: A>B, A>C
 * 66-24: A>D, B>D, C>D
 * 63-27: C>B

The Condorcet ranking is A>C>B>D, so the elimination order is D,B,C,A.

The sets of candidates with solidly-committed voters are:

Thus, the proportionality rules are:


 * Elect A.
 * Elect at least 1 candidate from {A, C}.
 * Elect at least 2 candidates from {A, B, C}.
 * Elect at least 2 candidates from {A, B, C, D}.

or, equivalently:


 * Elect A.
 * Elect either B or C.

The first 2 candidates in the elimination order, D and B, can be removed without violating the proportionality rules. We're now left with the required 2 candidates, and so are finished.

The winners are Andrea and Carter.