Eleccions mitjançant el vot d'aprovació. El mètode de Phragmén i algunes variants

Abstract

We study certain variants of a method for proportional representation that was proposed at the end of the nineteenth century by Edvard Phragmén. Every elector expresses his opinion by means of an approval ballot, where he indicates all the options that he deems suitable. The options can be individual candidates, as it was considered by Phragmén, or party candidatures which can provide several representatives each. Given a set of votes of this kind and a prefixed number of seats, the aim is to distribute these seats among the candidatures so as to achieve as much representativeness as possible. This work pays special attention to a condition of monotonicity which requires that adding approvals in favour of a certain candidature cannot result in its getting a smaller number of seats. In the basic variant, where seats are assigned by means of a sequential procedure, such a monotonicity holds in the case of individual candidates but not in the general case. We briefly explore also some direct variants, where seats are assigned all at once by means of a suitable optimization criterion. These variants seem to be better from the point of view of monotonicity. However, they suffer from a problem of multiplicity of solutions that calls for a supplementary criterion to choose among them.