Certificates of optimality for minimum norm biproportional apportionments

Computing a biproportional apportionment that satisfies some given properties may require a high degree of mathematical expertise, that very few voters can share. It seems therefore that the voters have to accept the electoral outcome without any possibility of checking the validity of the stated properties. 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 Serafini and Simeone (Soc Choice Welf 38:247–268, <CitationRef CitationID="CR18">2012</CitationRef>. 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$L_1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>L</mi> <mn>1</mn> </msub> </math> </EquationSource> </InlineEquation>- or an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$L_2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>L</mi> <mn>2</mn> </msub> </math> </EquationSource> </InlineEquation>-norm deviation from given quotas. Copyright Springer-Verlag Berlin Heidelberg 2015