# 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:

- Andrea
- Carter
- Brad
- 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:

Set | Solidly Committed | Quotas | Int. Quotas | # Candidates | Min. Seats |
---|---|---|---|---|---|

{A} | 66/90 = 11/15 | (11/15)×(2+1) = 2+1/5 | 2 | 1 | 1 |

{D} | 24/90 = 4/15 | (4/15)×(2+1) = 4/5 | 0 | 1 | 0 |

{A, B} | 15/90 = 1/6 | (1/6)×(2+1) = 1/2 | 0 | 2 | 0 |

{A, C} | 51/90 = 17/30 | 17/30×(2+1) = 1+7/10 | 1 | 2 | 1 |

{A, D} | 8/90 = 4/45 | (4/45)×(2+1) = 4/15 | 0 | 2 | 0 |

{B, D} | 8/90 = 4/45 | (4/45)×(2+1) = 4/15 | 0 | 2 | 0 |

{C, D} | 8/90 = 4/45 | (4/45)×(2+1) = 4/15 | 0 | 2 | 0 |

{A, B, C} | 66/90 = 11/15 | (11/15)×(2+1) = 2+1/5 | 2 | 3 | 2 |

{A, B, D} | 8/90 = 4/45 | (4/45)×(2+1) = 4/15 | 0 | 3 | 0 |

{A, C, D} | 8/90 = 4/45 | (4/45)×(2+1) = 4/15 | 0 | 3 | 0 |

{B, C, D} | 8/90 = 4/45 | (4/45)×(2+1) = 4/15 | 0 | 3 | 0 |

{A, B, C, D} | 90/90 = 1 | 1×(2+1) = 3 | 2 | 4 | 2 |

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.