Proportional Ordering

Proportional Ordering is a ranking method proposed by James Green-Armytage in December 2004. The ordering is intended so that a semi-proportional set of n candidates can be obtained by taking the first n candidates from the ranking.

Definition
Start by choosing the Condorcet winner. To choose the kth candidate, perform a CPO-STV tally considering only the outcomes that include the previously chosen k-1 candidates; this will add one new candidate to the ranking. Repeat the process until all candidates have been ranked.

Example
Suppose there are four candidates: Andrea (A), Brad (B), Carter (C), and Delilah (D). The ballots are:


 * 5: A>B>C>D
 * 17: A>C>B>D
 * 8: D

First Candidate
Andrea is the Condorcet winner, so she is the first candidate in the ordering.

Second Candidate
To choose the second-place candidate, we consider the sets of two candidates including Andrea: {A, B}, {A, C}, and {A, D}.

In the CPO-STV comparison for {A, B} versus {A, C}, D is eliminated, leaving


 * 5: A>B>C
 * 17: A>C>B

After transferring A's excess votes, {A, C} beats {A, B}.

The CPO-STV comparison for {A, B} versus {A, D} eliminates C, leaving


 * 22: A>B>D
 * 8: D

With the two-seat Droop quota of 11 votes, A meets the quota with 11 excess votes, all transferring to B, giving a vote count of:


 * A: 11
 * B: 11
 * D: 8
 * A+B: 22
 * A+D: 19

So {A, B} beats {A, D}.

Finally, the CPO-STV comparison for {A, C} versus {A, D} gives a vote count of:


 * A: 11
 * C: 11
 * D: 8
 * A+C: 22
 * A+D: 19

So {A, C} beats {A, D}.

The set {A, C} is the Condorcet winner (beating both {A, B} and {A, D}). Therefore, Carter is the second candidate in the proportional ordering.

Third candidate
We now consider the sets of three candidates containing both Andrea and Carter, i.e., {A, C, B} and {A, C, D}. The inital first-choice vote counts are:


 * A: 22
 * B: 8

Andrea is in both outcomes, so her votes are transferred. Using the three-seat Droop quota of 8 votes with fractional transfer gives her a retention fraction of 1-8/22 = 7/11. Applying this to the A>B>C>D and A>C>B>D ballots gives.


 * 3+2/11: (A)>B>C>D
 * 10+9/11: (A)>C>B>D
 * 8: D

Now, Carter has a surplus as well, all of which transfers to Brad, giving the ballots:


 * 3+2/11: (A)>B>(C)>D
 * 2+9/11: (A>C)>B>D
 * 8: D

Vote transfers are now complete. The vote totals for each candidate are:


 * A: 8 (quota)
 * B: 6
 * C: 8 (quota)
 * D: 8
 * A+C+B: 22
 * A+C+D: 24

Thus, {A, C, D} beats {A, C, B}, so Delilah is the next candidate chosen.

Final ordering
Brad is now the only candidate left and is added to the end of the ranking. The final proportional ordering is:


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