They allow voters to vote for individual candidates rather than parties, but still guarantee proportional representation.
They were published by Thorvald Thiele in Danish in 1895,[1] and translated to English by Svante Janson in 2016.
The goal of Thiele's methods is to find a committee W that maximizes the total satisfaction (following the utilitarian rule).
Optimization methods: find the committee that maximizes the total satisfaction.
It was proposed in the Swedish parliament in 1912 and rejected; but was later adopted for elections inside city and county councils, and is still used for that purpose.
[2]: Sec.10 For each possible ballot b, let vb be the number of voters who voted exactly b (for example: approved exactly the same set of candidates).
So if the numbers of votes are all multiplied by the same constant, the method returns the same outcome.
In fact, Thiele's optimization method satisfies house-monotonicity only for the (normalized) satisfaction function f(r)=r.
Moreover, If the satisfaction-score of the i-th approved candidate is (1/p)i, for various values of p, we get the entire spectrum between CC and AV.