The give-up problem for blocked regional lists with multi-winners

The current electoral law for the Italian Parliament prescribes blocked, linearly ordered lists of candidates for each party within each constituency. The peculiarity of the Italian electoral system is that a party can present the same candidate in different constituencies. There are several seats at stake in each constituency; these seats are allocated to the parties proportionally to the total number of votes they get. If the blocked list mechanism-which assigns the seats obtained by a party in a constituency to the first candidates of the corresponding ordered list-causes some candidates to win in more than one constituency, they may retain only one of the seats, giving up all the remaining ones. Thus, the problem arises for a party to find a suitable "schedule of give-ups" that produces the final set of winners for that party. In order to do this, we assume that such decision is centralized and based on some models of global (inter-regional) preferences over the set of candidates. In this paper, we introduce two classes of models to formulate the "give-up problem", i.e., utility and ordinal models, and we show that for both of them some natural formulations of the problem can be efficiently solved by network flows techniques.