# Vacancy re-tabulations with Meek's method

When there are one or more vacancies in an elected office, re-tabulating voter's preferences from a previous RCV / STV election can be an effective way to decide which of the unelected candidates in that election will fill the vacancies. This allows voters to still decide who best represents them, but without the expense or delay of holding a special election.

The vacancy re-tabulation procedures implemented in this project are an extension of the PRF reference rule. The extended procedures still follow the principles and general methods of Meek's method while honoring the goal of proportional representation within the constraints of filling a vacancy.

Vacancy re-tabulations are performed by following three simple rules:

- Designate as protected candidates any incumbents who will continue to serve.
- Designate as excluded candidates any candidates who are ineligible to serve or who are unwilling to serve.
- Set the number of winners (a.k.a. the number of seats to fill) equal to the sum of the number of protected candidates and the number of vacancies to be filled by the re-tabulation.

An excluded candidate is treated as if the candidate were already defeated before the tabulation begins. In contrast, a protected candidate is protected from being defeated and as a result is ensured of being elected.

The vacancy re-tabulation ensures that all of the protected candidates and the correct number of unprotected candidates are elected. This is accomplished in part by using different quotas for protected candidates versus unprotected candidates.

For those who are interested, the remainder of this page provides more background and explanation about the details of how a vacancy re-tabulation is calculated. You do not have to be familiar with this material in order to effectively use the tabulation web page to perform a vacancy re-tabulation.

During a vacancy re-tabulation, there are two groups of candidates competing for votes. One consists of the protected candidates, the other consists of all of the other candidates who are not protected and not excluded. The second group are the candidates who are competing to be chosen to fill the one or more vacancies, so we'll call them the vacancy candidates.

Candidates from both groups are awarded votes in the same way as in a regular Meek's method tabulation. However to ensure that the correct number of candidates are elected from each group, each group has its own quota (a.k.a threshold) for being elected and for distributing surplus votes.

The quota for the protected candidates is calculated in the same way as for a regular election. The calculation of the quota for the vacancy candidates follows the same principles, but applied according to the special circumstances of the vacancy candidates. The vacancy quota will always be at least as big as the quota for protected candidates. Typically the gap will narrow as the tabulation progresses.

Here are two PDF files that document how the PRF reference rule is extended to support vacancy re-tabulations.

- The first PDF shows the reference rule with the extensions.
- The second PDF shows a comparison of the reference rule to the extended rule, which makes it easier to see what has changed.

Ignoring for the moment the relatively minor issues of rounding and the precision of arithmetic calculations, the quota for vacancy candidates is still calculated by the traditional formula:

- Q = V / (1 + S)

However V is the total number of votes that could still count for vacancy candidates, and S is the number of vacancy candidates that are allowed to be elected. Since protected candidates are never defeated, only their surplus votes can potentially count for vacancy candidates. So any votes for a protected candidate that are below the protected candidate quota are not counted as part of V for the vacancy candidate quota.

One might wonder why the above interpretation is not also applied to the protected candidates. If it were, that would mean that the V for protected candidates would be all votes counting for any candidate and S would be the number of protected candidates that can be elected.

The problem with that approach is that the votes counting for vacancy candidates would be double counted in the numerators of the respective quotas, while the denominators are strictly partitioned between the the two groups.

The way to avoid that problem is to rethink how the candidates are grouped. Instead of partitioning the candidates into two separate groups, the candidates are put into two overlapping groups. The first group is the group of all non-excluded candidates, both protected candidates and vacancy candidates. The second group is just the vacancy candidates.

Now the generalized formula for a group of candidates can be applied to both groups. When a candidate is in more than one group, the rule is that the highest quota of any such group applies to that candidate. In other words, in order to be elected, a candidate has to satisfy all of the quotas for the groups the candidate is a member of.

So while there is still some double counting of votes between the two groups, as applied to the numerators of the quotas, there is now a commensurate double counting of electable positions in the denominators of the quotas.

This provides a uniform basis for the approach that is reflected in the extended rule for vacancy re-tabulations.

In practice, it means that protected candidates only have to satisfy the regularly calculated quota that applies to everyone, while vacancy candidates also have to satisfy the typically higher vacancy quota that only applies to vacancy candidates.