A multi-criteria approach for assigning weights in voting systems


Following a new analytical orientation, this paper proposes an approach for investigating group versus individual trade-offs in general voting situations. From specific particularizations of a general family of p-norms, three social decision rules have been derived. These rules trigger three different systems of weights in the voting system. Once the three rules were obtained and the rationale underlying them justified, a compromise framework integrating all the rules was formulated. The compromise framework is computationally based on goal programming. In this way, it is possible to establish directly trade-offs and balance compromise solutions between the three social rules. Thus, not only is the weighted voting system formulated within a theoretically sound framework, but the applicability of the approach to real political situations is also more flexible and pragmatic.

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  1. Banzhaf JF (1965) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19:317–343

    Google Scholar 

  2. Felsenthal DS, Machover M (1998) The measurement of voting power. Edward Elgar, Cheltenham

    Google Scholar 

  3. González-Pachón J, Romero C (2011) The design of socially optimal decisions in a consensus scenario. OMEGA Int J Manag Sci 39:179–185. https://doi.org/10.1016/j.omega.2010.06.004

    Article  Google Scholar 

  4. González-Pachón J, Romero C (2016) Bentham, Marx and Rawls ethical principles: in search for a compromise. OMEGA Int J Manag Sci 62:47–51. https://doi.org/10.1016/j.omega.2015.08.008

    Article  Google Scholar 

  5. Holler MJ, Packel EW (1983) Power, luck and the right index. Zeitschrift fur National ökonomie (J Econ) 43:21–29

    Google Scholar 

  6. Johnston RJ (1978) On the measurement of power: some reactions to Laver. Environ Plan 10A:907–914

    Article  Google Scholar 

  7. Jones DF, Tamiz M (2010) Practical goal programming. Springer, New York

    Google Scholar 

  8. Penrose LS (1946) The elementary statistics of majority voting. J R Stat Soc 109:53–57

    Article  Google Scholar 

  9. Romero C (2001) Extended lexicographic goal programming: a unifying approach. OMEGA Int J Manag Sci 29:63–71. https://doi.org/10.1016/S0305-0483(00)00026-8

    Article  Google Scholar 

  10. Shapley LS, Shubik M (1954) A method for evaluating the distribution power in a committee system. Am Polit Sci Rev 48:787–792

    Article  Google Scholar 

  11. Steuer RE (1989) Multicriteria optimization. Wiley, NewYork

    Google Scholar 

  12. Turnovec F (1996) Weights and votes in European Union: extension and institutional reform. Prague Econ Pap 4:161–174

    Google Scholar 

  13. Yu PL (1973) A class of solutions for group decision problems. Manag Sci 19:936–946. https://doi.org/10.1287/mnsc.19.8.936

    MathSciNet  Article  MATH  Google Scholar 

  14. Yu PL (1985) Multiple-criteria decision making. Concepts, techniques, and extensions. Plenum Press, New York

    Google Scholar 

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This study forms part of the Project “Gestión forestal en un contexto de uso múltiple y toma de decisiones colectivas” (AGL2015-68657-R) funded by the Ministry of Economy and Competitiveness of Spain. Besides, we have received funding from the European Union’s H2020 research and innovation program under the Marie Skłdowska-Curie Grant Agreement No. 691149 (SuFoRun). We appreciate the comments raised by the referees that have helped us in order to increase the clarity and accuracy of our work. Finally, thanks are due to Diana Badder for her editing of the English.

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Correspondence to Luis Diaz-Balteiro.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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Communicated by V. Loia.

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González-Pachón, J., Diaz-Balteiro, L. & Romero, C. A multi-criteria approach for assigning weights in voting systems. Soft Comput 23, 8181–8186 (2019). https://doi.org/10.1007/s00500-018-3453-x

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  • Compromise
  • Goal programming
  • Weighted voting system
  • p-norms