Certificates of optimality: the third way to biproportional apportionment

Abstract

The biproportional apportionment problem (BAP) must be faced in many proportional electoral systems where seats must be allocated to parties within regions. BAP is a non trivial optimization problem, and only sophisticated algorithms are currently available for solving it. The issue is: are they "writable" as an actual law? Citizens rightly demand simple, easy to understand, voting systems. The alternative, though, seems to have simple, but unsound electoral laws. We propose the following way out of this dilemma: leave to a mathematically sophisticated algorithm the task of producing an optimal apportionment, but attach to it a "certificate of optimality", that is, describe a simple procedure whereby anybody can check, through some elementary operations, that the seat allocation output by the algorithm is indeed an optimal apportionment. We discuss one such certificate, based on the Max Flow-Min Cut Theorem, relative to a parametric max flow method of ours for BAP.

Certificates of optimality for minimum norm biproportional apportionments

“…In this paper we continue the type of investigation started in [18] and show that a similar certificate can be indeed attached also to apportionments satisfying different measures of deviation from given quotas. In particular we consider the Controlled rounding procedure [4], i.e., an L 2 -norm minimization within quotas, and the L 1 -norm minimization (without restrictions).…”

“…However, it is possible in some cases to attach to the computed apportionment a certificate which can guarantee the voters of the validity of the apportionment. This type of investigation has been first proposed in [18]. In this paper we pursue the same line of approach and show that a certificate can be produced and easily checked by a layman for apportionments that minimize either an L1-or an L2-norm deviation from given quotas.…”

“…In a recent paper [18] the question was posed of how to 'guarantee' the voters that the seats assigned by the competent authority to parties and constituencies are consistent with certain properties the apportionment is claimed to satisfy.…”

“…In [18] the following dilemma was stated: 'Which is better? To have simple, but unsound electoral laws, or sound, but complex ones?'…”

“…The specific electoral system addressed by [18] was the biproportional apportionment problem that minimizes an L ∞ -norm deviation from given quotas (this type of apportionment has been fully investigated in [17]). In that paper it was shown that it is possible to attach to an apportionment a certificate of guarantee that anybody can check with only an elementary knowledge of mathematics.…”