Abstract
In some proportional electoral systems with more than one constituency the number of seats allotted to each constituency is pre-specified, as well as, the number of seats that each party has to receive at a national level. “Bidimensional allocation” of seats to parties within constituencies consists of converting the vote matrix V into an integer matrix of seats “as proportional as possible” to V, satisfying constituency and party totals and an additional “zero-vote zero-seat” condition. In the current Italian electoral law this Bidimensional Allocation Problem (or Biproportional Apportionment Problem—BAP) is ruled by an erroneous procedure that may produce an infeasible allocation, actually one that is not able to satisfy all the above conditions simultaneously.
In this paper we focus on the feasibility aspect of BAP and, basing on the theory of (0,1)-matrices with given line sums, we formulate it for the first time as a “Matrix Feasibility Problem”. Starting from some previous results provided by Gale and Ryser in the 60’s, we consider the additional constraint that some cells of the output matrix must be equal to zero and extend the results by Gale and Ryser to this case. For specific configurations of zeros in the vote matrix we show that a modified version of the Ryser procedure works well, and we also state necessary and sufficient conditions for the existence of a feasible solution. Since our analysis concerns only special cases, its application to the electoral problem is still limited. In spite of this, in the paper we provide new results in the area of combinatorial matrix theory for (0,1)-matrices with fixed zeros which have also a practical application in some problems related to graphs.
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Notes
Law n. 270/2005.
In contrast with the erroneous Italian electoral law, the underlying idea of our approach is to guarantee an allocation procedure both correct and conceptually simple.
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Bruno Simeone was really fond of the subject of this paper of which he was an earnest promoter. He left us precious intuitions, a lot of ideas and much enthusiasm that gave us the necessary support to reach our results. Even if we still have not proved all Bruno’s conjectures, we believe to have made a step forward in this area of research and we dedicate this work to Bruno who missed us so much during the preparation of this paper.
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Lari, I., Ricca, F. & Scozzari, A. Bidimensional allocation of seats via zero-one matrices with given line sums. Ann Oper Res 215, 165–181 (2014). https://doi.org/10.1007/s10479-013-1440-2
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DOI: https://doi.org/10.1007/s10479-013-1440-2