Quota-preferential by Quotient

Quota-preferential by Quotient or QPQ is a proportional multi-winner system for ranked ballots devised by Olli Salmi and studied by Douglas Woodall. The method modifies Phragmén's Sequential Method for ranked ballots by adding a minimum quota required to be elected. If no candidate reaches the quota, the least supported candidates are eliminated and votes are transferred to their next choice. In this sense, QPQ generalizes the D'Hondt method of apportionment in the same way that the single transferable vote generalizes the highest averages method. In the single winner case, the method is equivalent to Instant-runoff voting. It passes the Droop proportionality criterion.

Explanation

Each ballot has a so-called "load", initially set to 0. This can be thought of as the number of winning candidates that ballot has helped elect. So if 10 ballots together elect one candidate, each ballot has elected 1/10 of a candidate and thus has a load of 1/10. In order to elect a candidate, the total load of all ballots supporting that candidate must increase by 1. One at a time, the candidate that would increase the maximum load on a ballot by the least amount is elected, and the loads of their supporters is increased. In Phragmen's method, this process continues until all candidates have been elected. The inverse of the load required to elect a candidate is called the candidate's priority, and so candidates are elected from highest to lowest priority.

If the vote is highly split, this process can cause some candidates to win with very few votes. To prevent this, QPQ requires that an elected candidate could not be stopped by the other voters even if they voted tactically. That is, the votes not supporting the elected candidate could not be modified so that the remaining seats were filled by other candidates with a higher priority. This requirement leads to a minimum priority to get elected called the quota. The quota begins as the Droop quota but lowers as ballots exhaust. If no candidate reaches the quota, the lowest priority candidate is eliminated. Voters supporting that candidate will transfer to their next choice, increasing their priority, or exhaust, decreasing the quota. This minimum quota is what ensures that the method passes proportionality for solid coalitions.

A quirk of the system is that, after an elimination, a candidate getting elected can cause a ballot's load to decrease. To prevent this, Woodall recommends restarting the count after each elimination. He proves that any candidate that was elected during the count will be reelected after the count is restarted.

Method

Every candidate is marked as elected, hopeful, or eliminated. Every ballot is said to support the highest ranked hopeful candidate. If the ballot has no hopeful candidates ranked, the ballot is said to be exhausted. Every ballot has a load that starts at 0.

Compute each helpful candidate's priority as ${\displaystyle p_{c}={\frac {v_{c}}{1+l_{c}}}}$ where ${\textstyle v_{c}}$ is the number of voters supporting the candidate and ${\textstyle l_{c}}$ is the sum of the loads of the supporters. Compute the quota as ${\displaystyle Q={\frac {v-v_{e}}{N+1-l_{e}}}}$ where ${\textstyle v}$ is the total number of voters, ${\textstyle v_{e}}$ is the number of exhausted ballots, ${\textstyle N}$ is the number of candidates that will be elected, and ${\textstyle l_{e}}$ is the sum of the loads of exhausted ballots. If the candidate with the highest priority has a priority above the quota, mark that candidate as elected and set the load of all supporting ballots to ${\displaystyle {\frac {1}{p_{c}}}}$, then transfer those ballots to their next hopeful choice and repeat this step. If no candidate has a priority above the quota, eliminate all candidates with 0 priority and the candidate with the lowest non-zero priority. Restart the election, but all eliminated candidates stay eliminated. Stop when there are no more hopeful candidates.

Related Methods

Phragmén's Method is equivalent to this method except there is no quota. STV produces similar results but is not based on the divisor methods. Ross Hyman has written about a method that generalizes other divisor methods like the Sainte-Laguë method and the Huntington-Hill method.

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