Sequential Ebert

From electowiki

Sequential Ebert is a system of proportional representation that uses approval voting, where candidates are elected one at a time, rather than each potential winning set considered as a whole. It is a variant of the optimal Ebert's Method.

Each elected candidate has a “load” of 1 that is spread equally among their approvers (every elected candidate must be approved by at least one voter), and each newly elected candidate is the one which minimises the sum of the squared voter loads given the candidates already elected.

The candidate with the most approvals always minimises the sum of the squared loads for a single winner, so is elected first. In the following example, there are three voters, and A, B and C are candidates. Two candidates are to be elected. The approval ballots are as follows:

2 voters: A, B

1 voter: C

A and B are tied for the most approvals, so one of these would be elected at first.

The sum of the squared voter loads for electing either A or B would be 2*(1/2)^2 + 1*(0/1)^2 = 0.5.

For candidate C, it would be 2*(0/2)^2 + 1*(1/1)^2 = 1.

Since lower is better, we can see that A or B would be a better choice than C by this measure. Let's say that by a tie-break procedure, A is elected. To see which candidate fills the second seat, we would calculate the sum of the squared voter loads for each of B and C in the second seat, and see which gives the lower figure. As A is already elected, the loads the voters received from A are taken into account in this calculation.

For candidates A and B, the sum of the squared voter loads is 2*(2/2)^2 + 1*(0/1)^2 = 2.

For candidates A and C, the sum of the squared voter loads is 2*(1/2)^2 + 1*(1/1)^2 = 1.5.

Although candidate B has more total approvals than candidate C, candidate C is elected second, because electing C along with the already elected A gives a lower total sum of squared voter loads than electing B along with A.