Ebert's method

Ebert's Method is a system of proportional representation that uses approval voting as a modification of Phragmén's Method and uses the same concept of loads. It was first knowingly defined by Bjarke Dahl Ebert in 2003.

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). For example, if an elected candidate is approved by 100 voters, each of these voters would have a load of 1/100 from this candidate, which would be added to their loads from the other winning candidates. The winning set of candidates is the one that minimises the sum of the squared voter loads. For example, if a voter approves two elected candidates who each had 100 approvers, this voter would have a squared load of (1/100 + 1/100)^2 = 1/2500. This would be added to the squared loads of every other voter.

Definition
Let:
 * V voters
 * C candidates
 * W winners, 0<W<C
 * Each voter approves or disapproves each candidate.
 * Assume each voter approves at least one candidate.

A "load distribution" is a two-dimensional array $$X_{v,c}$$ with $$v=1\ldots V,\,c=1\ldots C$$ such that:
 * 1) $$ 0 \leq X_{v,c} \leq 1$$
 * 2) $$X_{v,c}=0$$ unless v approves c
 * 3) $$\sum_{v}\sum_{c}\,X_{v,c} = W$$
 * 4) for each candidate c, $$\sum_{v} X_{v,c} = 1$$ if c is a winner, otherwise $$=0$$.

The winner set is the set which minimizes $$\sum_{v}(\sum_{c} X_{v,c} )^2$$.

Variants
There is a Sequential version called Sequential Ebert.

There is also a modified version by Toby Pereira called PAMSAC