Skip to main content
SpringerLink
Log in

A New Axiomatization of the Banzhaf Index for Games with Abstention

  • Published:
Group Decision and Negotiation Aims and scope Submit manuscript

Abstract

(3, 2)-Simple games are a model for voting situation in which players can vote not only in favour or against a proposal but they can also abstain. Also in this model, power indices are used to evaluate the power of players. In particular, the Banzhaf index and the Shapley–Shubik index have been generalized to define analogous power indices in the context of games with abstention. In this work we provide a new axiomatization of the Banzhaf index for games with abstention, to underline its properties and increase the justification of the use of this index as a solution concept also in the family of games with abstention.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. For simple games, super-additivity is equivalent to the condition of being proper: it is not possible that a coalition and its complement are both winning.

  2. To simplify the notation we omit the braces to denote the sets in a tripartition, for instance the informal notation (a, b, c) stands for \((\{a\},\{b\},\{c\}) \).

  3. We use \(S \subset T\) if \(S\subseteq T\) and \(S\ne T\).

  4. This property is analogous to the Symmetric gain-loss axiom defined in Laruelle and Valenciano (2001) for simple games.

References

  • Albizuri MJ (2001) An axiomatization of the modified Banzhaf Coleman index. Int J Game Theory 30(2):167–176

    Article  Google Scholar 

  • Banzhaf JF (1964) Weighted voting doesn’t work: a mathematical analysis. Rutgers L Rev 19:317–343

    Google Scholar 

  • Barua R, Chakravarty SR, Roy S (2005) A new characterization of the Banzhaf index of power. Int Game Theory Rev 7(4):545–553

    Article  Google Scholar 

  • Dubey P (1975) On the uniqueness of the Shapley value. Int J Game Theory 4(3):131–139

    Article  Google Scholar 

  • Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Math Oper Res 4(2):99–131

    Article  Google Scholar 

  • Felsenthal DS, Machover M (1996) Alternative forms of the Shapley value and the Shapley–Shubik index. Public Choice 87(3):315–318

    Article  Google Scholar 

  • Felsenthal DS, Machover M (1997) Ternary voting games. Int J Game Theory 26(3):335–351

    Article  Google Scholar 

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

    Book  Google Scholar 

  • Feltkamp V (1995) Alternative axiomatic characterizations of the Shapley and Banzhaf values. Int J Game Theory 24(2):179–186

    Article  Google Scholar 

  • Freixas J (2005a) Banzhaf measures for games with several levels of approval in the input and output. Ann Oper Res 137(1):45–66

    Article  Google Scholar 

  • Freixas J (2005b) The Shapley-Shubik power index for games with several levels of approval in the input and output. Decis Support Syst 39(2):185–195

    Article  Google Scholar 

  • Freixas J, Lucchetti R (2016) Power in voting rules with abstention: an axiomatization of a two components power index. Ann Oper Res 244(2):455–474

    Article  Google Scholar 

  • Freixas J, Pons M (2017) Using the multilinear extension to study some probabilistic power indices. Group Decis Negot 26(3):437–452

    Article  Google Scholar 

  • Freixas J, Zwicker WS (2003) Weighted voting, abstention, and multiple levels of approval. Soc Choice Welf 21(3):399–431

    Article  Google Scholar 

  • Laruelle A, Valenciano F (2001) Shapley–Shubik and Banzhaf indices revisited. Math Oper Res 26(1):89–104

    Article  Google Scholar 

  • Lehrer E (1988) An axiomatization of the Banzhaf value. Int J Game Theory 17(2):89–99

    Article  Google Scholar 

  • Monderer D, Samet D (2002) Variations on the Shapley value. Handb Game Theory Econ Appl 3:2055–2076

    Google Scholar 

  • Nowak AS (1997) On an axiomatization of the Banzhaf value without the additivity axiom. Int J Game Theory 26(1):137–141

    Article  Google Scholar 

  • Penrose LS (1946) The elementary statistics of majority voting. J R Stat Soc 109(1):53–57

    Article  Google Scholar 

  • Shapley LS (1953) A value for n-person games. Contrib Theory Games 2(28):307–317

    Google Scholar 

  • Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48(3):787–792

    Article  Google Scholar 

Download references

Acknowledgements

The author is grateful to Roberto Lucchetti and Josep Freixas which provided some ideas to develop this work and she would also like to thank two anonymous referees for their helpful comments, which allowed to improve the final version of the paper.

Author information

Authors and Affiliations

  1. Politecnico di Milano, Piazza Leonardo da Vinci, 32, Milan, Italy

    Giulia Bernardi

Authors
  1. Giulia Bernardi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giulia Bernardi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bernardi, G. A New Axiomatization of the Banzhaf Index for Games with Abstention. Group Decis Negot 27, 165–177 (2018). https://doi.org/10.1007/s10726-017-9546-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10726-017-9546-6

Keywords

Search

Navigation