Abstract
A new voting rule for electing committees is described. Specifically, we use approval balloting and propose a new voting procedure that guarantees that if there is a committee that represents (with a given proportion of representatives) all of the existing voters, then the selected committee has to represent all of voters in at least the same proportion. This property is a way of selecting a committee that represents completely all of voters when such a committee exists. The usual voting rules in this context do not satisfy this condition.
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Notes
For the elections to the Senate of Spain (The Upper House of the Spanish Parliament) each province elects four senators regardless of their population. In order to obtain a more representative committee, under current legislation, each voter is allowed to mark three or fewer candidates’ names.
The lexicographical order is defined as \(( x_1,x_2,\ldots ,x_m ) \succ _L ( y_1,y_2,\ldots ,y_m ) \) if and only if there is \(p \in \lbrace 1,2,\ldots ,m \rbrace \) such that \(x_q=y_q\) for all \(q<p\) and \(x_p>y_p\). The weak relation \(( x_1,x_2,\ldots ,x_m ) \succcurlyeq _L ( y_1,y_2,\ldots ,y_m ) \) means \(( x_1,x_2,\ldots ,x_m ) \succ _L ( y_1, y_2,\ldots ,y_m )\) or \(( x_1,x_2,\ldots ,x_m ) = ( y_1,y_2,\ldots ,y_m )\). Note that being the set of vectors \(\left\{ U^{{\alpha _s}}\left( V,C\right) : C \in \xi _s \right\} \) finite, the lexicographical order always reaches a unique maximum (although several committees may provide this maximum).
We are grateful to H. Aziz for pointing out this fact in a private communication.
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Acknowledgements
Financial support from Spanish Ministry of Economy and Competitiveness under Projects ECO2013-43119-P and ECO2016-77200-P is gratefully acknowledged. We thank Roger Laing and the anonymous reviewers for their careful reading of our paper and their insightful comments.
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Subiza, B., Peris, J.E. A Representative Committee by Approval Balloting. Group Decis Negot 26, 1029–1040 (2017). https://doi.org/10.1007/s10726-017-9527-9
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DOI: https://doi.org/10.1007/s10726-017-9527-9