# CPO-STV

CPO-STV (Comparison of Pairs of Outcomes by Single Transferable Vote) is a preference voting system designed to provide proportional representation in multi-seat elections while electing the Condorcet winner in single-winner elections. It is based on STV and pairwise counting between every possible combination of candidates that could win ("winner sets") to determine the winner.

## Voting

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

1. Andrea
2. Carter
4. Delilah

### Setting the Quota

When all the votes have been cast, a winning quota is set. Possible formulas for the quota include the Droop Quota, the Hare Quota, and the Imperiali Quota.

### Comparison of Pairs of Outcomes

In CPO-STV, each possible outcome (set of candidates) is compared with every other possible outcome in a pairwise competition. The pairwise competition is performed as follows:

1. Eliminate all candidates who are not in either outcome.
2. Transfer excess votes from candidates who are in both outcomes.
3. The number of pairwsie votes for an outcome is equal to the sum of votes for the candidates in that outcome.

Example: Compare {Escher, Andre, Gore} versus {Escher, Nader, Gore} given a quota of 100 and the ballots

• 100: Escher
• 6: Gore>Bush
• 45: Bush>Gore

First, eliminate Bush, who is in neither outcome.

• 100: Escher
• 51: Gore

Next, transfer the excess votes for Escher, who is in both outcomes. (Do not transfer the votes for Andre, who is only in one outcome.) Because Echer happens to meet the quota exactly, there is nothing to do here.

The number of first-choice votes for each candidate is now

• 100: Escher
• 110: Andre
• 72: Gore

• Escher Andre Gore = 100 110 72 = 282
• Escher Nader Gore = 100 18 72 = 190

Thus, {Escher, Andre, Gore} pairwise beats {Escher, Nader, Gore}, 282 to 190.

Process A: If any candidate has a quota of top-preference votes, declare them elected. Distribute excess votes (determined by random selection or by fractional transfer) for the winning candidates to the next-highest ranked candidates on the ballots. Repeat this process until there there are no more candidates who meet the quota. (This process is optional, but can greatly simplify Process B.)

Process B: Conduct a pairwise comparison between every possible set of candidates that includes all of the elected candidates from Process A. Choose the winning outcome with a Condorcet method.

## Example

2 seats to be filled, 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

The Droop Quota is floor(22/3) 1 = 11.

Andrea has 22 first-choice votes, and is declared elected. Her 11 excess votes are reallocated to their second preferences. If this is done by fractional transfer, the resulting ballots are:

• 2.5: B>C>D
• 8.5: C>B>D
• 8: D

No more candidates meet the quota, so Process A is completed.

Since Andrea must be elected, there are only 3 possible outcomes to consider: {A, B}, {A, C}, and {A, D}.

To compare {A, B} and {A, C}, first eliminate D:

• 5: A>B>C
• 17: A>C>B
• 8: (blank)

Andrea is elected with 11 excess votes. After transferring these, the ballots become:

• 11: A
• 2.5: B>C
• 8.5: C>B
• 8: (blank)

and so {A, C} beats {A, B}, 19.5 to 13.5.

Similarly, to compare {A, C} and {A, D}, first eliminate B:

• 22: A>C>D
• 8: D

Andrea is elected with 11 excess votes, which all transfer to C, producing:

• 11: A
• 11: C
• 8: D

and so {A, C} beats {A, D}, 22 to 19.

At this point, we know that {A, C} is the Condorcet winner. Therefore, CPO-STV elects Andrea and Carter.

### Notes

CPO-STV can be highly computationally complex and thus difficult to calculate when there are many candidates, as if there are, say, 5 seats to be filled and 60 candidates, then there are 60 choose 5 = 5,461,512 possible outcomes.

It has not been proven whether CPO-STV is proportional for Droop solid coalitions. However, if it can be, then its cycle resolution method likely must choose from the Smith Set of winner sets in order to do so, as Smith-efficiency guarantees Droop proportionality (the mutual majority criterion) in the single-winner case. One type of procedure that requires among the fewest pairwise comparisons to find one of the Smith Set winner sets is Sequential comparison Condorcet methods. Since a winner set in the Smith Set can only be eliminated by another Smith winner set by this procedure, the final remaining winner set will guaranteeably be in the Smith Set. If desired, it is then possible to discover the rest of the Smith Set by checking which winner sets beat or tie the final remaining winner set, which beat or tie these winner sets, etc. One well-known procedure that works along these lines is BTR-IRV.

One suggestion to modify CPO-STV to be guaranteeably proportional for Droop solid coalitions is to first eliminate all outcomes from consideration that fail Droop proportionality. In the above example, if there are 4 solid coalitions of 5 candidates each, then the upper bound of outcomes to consider is ((5^4) * 60) = 37500 outcomes, which is a reduction of outcomes to consider by a factor of about 145.

Several other such modifications are possible to reduce the number of outcomes to consider, some of which can potentially elect some outcome other than what CPO-STV would. Some are:

- As a first guess, calculate the STV outcome and see if it can win against all other outcomes (that are in consideration). (It is estimated that the STV winner set is almost always the same as the CPO-STV winner set.)

- If a set of candidates X is ranked above or equal to a set of candidates Y on all ballots, ignore all outcomes featuring candidates from Y but not X. (Based on unanimity criterion).