Biproportionality

A proportional representation method is biproportional if it's close to proportional in more than one way. For instance, a biproportional party list method might give each party close to its national support in parliament, while retaining approximate within each district as well.

These methods work by accepting some error in both types of proportionality in exchange for reasonably good results along both. A biproportional party list method of the above type might give parties with fewer votes more representatives in a few districts to ensure reasonable national proportionality, similar to the top-up seats in MMP and Scandinavian party list PR w:leveling seats.

The most commonly known biproportional apportionment method is Pukelsheim's method, which is used in Switzerland. Fair majority voting, a biproportional method with single-member constituencies, has been proposed by Michel Balinski as a solution to gerrymandering problems in the United States.

Biproportionality is an instance of multidimensional proportionality (in this case, with two dimensions). Higher order proportionality (e.g. three dimensions: party support, relative population, and gender) is possible, but at the cost of exact proportionality: fully general solutions are no longer guaranteed to exist.