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Fair Proportional Representation Problems with Mixture Operators

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Algorithmic Decision Theory (ADT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10576))

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Abstract

This paper deals with proportional representation problems in which a set of winning candidates must be selected according to the ballots of the voters. We investigate the use of a new class of optimization criteria to determine the set of winning candidates, namely mixture operators. In a nutshell, mixture operators are similar to weighted means where the numerical weights are replaced by weighting functions. In this paper: (1) we give the mathematical condition for which a mixture operator is fair and provide several instances of this operator satisfying this condition; (2) we show that when using a mixture operator as optimization criterion, one recovers the same complexity results as in the utilitarian case (i.e., maximizing the sum of agent’s utilities) under a light condition; (3) we present solution methods to find an optimal set of winners w.r.t. a mixture operator under both Monroe and Chamberlin-Courant multi-winner voting rules and test their computational efficiency.

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Notes

  1. 1.

    Considering an open interval simplifies the writing of Proposition 1 in Sect. 3.

  2. 2.

    Note that we follow the convention to use bold letters to represent vectors.

  3. 3.

    if, \(u(v_{ij}) - \lambda ^l w(v_{ij}) > u(v_{kl}) - \lambda ^l w(v_{kl})\), just reverse inequalities 10 and 11.

  4. 4.

    All methods were implemented in C++ using Gurobi version 5.6.3 to solve the LPs. Times are wall-clock times on a 2.4 GHz Intel Core i5 machine with 8GB of RAM.

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Acknowledgements

This work is supported by the ANR project CoCoRICo-CoDec ANR-14-CE24-0007-01.

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Authors and Affiliations

  1. Sorbonne Universités, UPMC Univ Paris 06, CNRS, LIP6 UMR 7606, 4 place Jussieu, 75005, Paris, France

    Hugo Gilbert

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  1. Hugo Gilbert
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Correspondence to Hugo Gilbert .

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  1. Heinrich Heine University, Düsseldorf, Germany

    Jörg Rothe

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Gilbert, H. (2017). Fair Proportional Representation Problems with Mixture Operators. In: Rothe, J. (eds) Algorithmic Decision Theory. ADT 2017. Lecture Notes in Computer Science(), vol 10576. Springer, Cham. https://doi.org/10.1007/978-3-319-67504-6_8

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  • DOI: https://doi.org/10.1007/978-3-319-67504-6_8

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