Biproportional allocation of parliamentary seats via vector apportionment and minimum paths

The Biproportional Apportionment Problem arises in countries which allocate electoral seats at two different territorial levels by a proportional rule. Mathematically, given the constituency/party vote matrix V , vectors c of national party seats and r of constituency seats, the problem is to compute an integer matrix A śś proportional to V whose column- and row- totals correspond to c and r , respectively. To pursue proportionality, one can first find a fractional matrix Q satisfying the above requirements, and then search for an integer matrix A that minimizes the distance from Q . Starting from the Italian electoral law for the Chamber of Deputies, in this paper, we develop a two-phase procedure for BAP which exploits Hare Quota plus Largest Remainders for the seat allocation in the first phase, while, in the second phase, it performs seat transfers by shortest paths on a suitable network. We test our method on real elections of different countries and compare our results with the corresponding institutional allocations. The quality of an apportionment is measured by its distance to the fractional Fair-Share matrix, a benchmark for biproportionality introduced by Balinski and Demange in 1989. The empirical results show that, even if the proposed methods are heuristics, they are able to output high quality solutions. The application to elections in Italy and in other European countries confirms both the effectiveness of the methods and their general applicability.