Researchers in the field of electoral systems have
recently turned their attention to network flow
techniques, with the intention to resolve certain
practically relevant problems of contemporary
electoral systems. Here we review some of this
work, with a focus on biproportional apportionment
methods and on the give-up problem.
In the biproportional apportionment problem, the
whole electoral region is subdivided into several
electoral districts. The input data consists of a
matrix of the vote counts a party receives in a
district. The task is to convert the vote matrix
into a (integer) seat matrix, maintaining
proportionality ``as much as possible''. Moreover
each district must be allocated its pre-specified
number of seats, and each party must receive the
number of seats it is entitled to on the basis of
the aggregate, national vote counts.
The give-up problem arises in the current
electoral law for the Italian Parliament. For
each region each party submits a blocked, ordered
lists of candidates. Candidates may run on more
than one list (that is, in more than one region),
and many of them do in order to advertise their
national standing. Candidates winning a seat in
more than one region must give-up all of them but
one. The give-up problem finds a schedule of
give-ups, for all lists of a party, that results
in a globally best, in some sense, choice of
deputies for that party.
