Parametric maximum flow methods for minimax approximation of target quotas in biproportional apportionment

Abstract

In this article, we study the biproportional apportionment problem, which deals with the assignment of seats to parties within regions. We consider the minimization of both the maximum absolute error and the maximum relative error of the apportioned seats with respect to target quotas. We show that this can be done polynomially through a reduction to a parametric maximum flow problem. Moreover, the maximum absolute error can be minimized in strongly polynomial time. More generally, our method can be used for computing ∞ projections onto a flow polytope. We also address the issue of uniqueness of the solution, proposing a method based on finding unordered lexicographic minima. Our procedure is compared to other well-known ones available in the literature. Finally we apply our procedures to the data of the 2008 Italian political elections, for which the procedure stated by the law produced an inconsistent assignment of seats.

Certificates of optimality: the third way to biproportional apportionment

“…In the table the quotas have been rounded to three decimal digits. We report in Table 12 the strongly optimal apportionment computed via the algorithm in [16]. Indeed we claim that it is optimal and we provide the following certificate to support our claim: The Verifier can compute the maximum error which is 0.75372 for the lower pair (Veneto 1, UDC) (1 seat against a target quota of 1.75372).…”

“…The target quotas in the 26 regions, already modified to take into account the majority bonus to the winning coalition (PDL, LN, MPA), are reported in Table 11. As already pointed out in [11,12,13,14,16] the final seats assigned in each region after the election are different from the ones assigned by the law before the election. But this is only due to a flaw in the procedure of the Italian law.…”

“…We mention the Discrete Alternate Scaling Method (DAS) [15,9], the convex separable formulation of BAP by Gaffke and Pukelsheim [7,8], the minimax parametric network flow method by Serafini and Simeone [16]. All these procedures share quite a degree of mathematical sophistication and so it is problematic to embody them in a real-life electoral law.…”

“…An optimal solution is in addition strongly optimal if the following requirement holds (see also [16]). Let x * be a feasible apportionment.…”

“…As explained in [16], the problem of finding a unique strongly optimal apportionment can be modeled as a parametric capacitated transportation problem and efficiently computed by solving a sequence of Max Flow problems.…”