Skip to main content
SpringerLink
Log in

Preorders in Simple Games

  • Chapter
  • First Online:
Transactions on Computational Collective Intelligence XXVII

Part of the book series: Lecture Notes in Computer Science ((TCCI,volume 10480))

  • 560 Accesses

Abstract

Any power index defines a total preorder in a simple game and, thus, induces a hierarchy among its players. The desirability relation, which is also a preorder, induces the same hierarchy as the Banzhaf and the Shapley indices on linear games, i.e., games in which the desirability relation is total. The desirability relation is a sub–preorder of another preorder, the weak desirability relation, and the class of weakly linear games, i.e., games for which the weak desirability relation is total, is larger than the class of linear games. The weak desirability relation induces the same hierarchy as the Banzhaf and the Shapley indices on weakly linear games. In this paper, we define a chain of preorders between the desirability and the weak desirability preorders. From them we obtain new classes of totally preordered games between linear and weakly linear games.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For a detailed discussion of some issues raised by allowing abstentions, see Felsenthal and Machover [6] and for several levels of approval in input and output, see Freixas and Zwicker [9].

  2. 2.

    Linear games are also called complete, ordered or directed games in the literature, see Taylor and Zwicker [16] for references on these names.

References

  1. Banzhaf, J.F.: Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev. 19(2), 317–343 (1965)

    Google Scholar 

  2. Carreras, F., Freixas, J.: Some theoretical reasons for using regular semivalues. In: De Swart, H. (ed.) Proceedings of the International Conference on Logic, Game Theory and Social Choice, LGS, Tilburg, The Netherlands, pp. 140–154 (1999)

    Google Scholar 

  3. Carreras, F., Freixas, J.: On ordinal equivalence of power measures given by regular semivalues. Math. Soc. Sci. 55(2), 221–234 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Coleman, J.S.: Control of collectivities and the power of a collectivity to act. In: Lieberman, B. (ed.) Social Choice, pp. 269–300. Gordon and Breach, New York (1971)

    Google Scholar 

  5. Diffo Lambo, L., Moulen, J.: Ordinal equivalence of power notions in voting games. Theor. Decis. 53(4), 313–325 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Felsenthal, D.S., Machover, M.: The Measurament of Voting Power: Theory and Practice, Problems and Paradoxes. Edward Elgar, Cheltenham (1998)

    Book  MATH  Google Scholar 

  7. Freixas, J., Marciniak, D., Pons, M.: On the ordinal equivalence of the Johnston, Banzhaf and Shapley power indices. Eur. J. Oper. Res. 216(2), 367–375 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Freixas, J., Pons, M.: Hierarchies achievable in simple games. Theor. Decis. 68(4), 393–404 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Freixas, J., Zwicker, W.S.: Weighted voting, abstention, and multiple levels of approval. Soc. Choice Welfare 21(3), 399–431 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Isbell, J.R.: A class of simple games. Duke Math. J. 25(3), 423–439 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  11. Johnston, R.J.: On the measurement of power: some reactions to Laver. Environ. Plann. A 10(8), 907–914 (1978)

    Article  Google Scholar 

  12. Penrose, L.S.: The elementary statistics of majority voting. J. Roy. Stat. Soc. 109(1), 53–57 (1946)

    Article  Google Scholar 

  13. Roy, S.: The ordinal equivalence of the jhonston index and the established notions of power. In: Econophisics and Economics of Games, Social Choices and Quantitative Techniques, pp. 372–380. Nex Economics Windows, Part II (2010)

    Google Scholar 

  14. Shapley, L.S.: A value for n-person games. In: Tucker, A.W., Kuhn, H.W. (eds.) Contributions to the Theory of Games II, pp. 307–317. Princeton University Press, Princeton (1953)

    Google Scholar 

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

    Article  Google Scholar 

  16. Taylor, A.D., Zwicker, W.S.: Simple games: desirability relations, trading, and pseudoweightings. Princeton University Press, New Jersey (1999)

    MATH  Google Scholar 

Download references

Acknowledgements

This research was partially supported by grant MTM2015–66818-P(MINECO/FEDER) from the Spanish Ministry of Economy and Competitiveness (MINECO) and from the European Union (FEDER funds).

Author information

Authors and Affiliations

  1. Department of Mathematics, Universitat Politècnica de Catalunya, Manresa, Spain

    Josep Freixas & Montserrat Pons

Authors
  1. Josep Freixas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Montserrat Pons
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Montserrat Pons .

Editor information

Editors and Affiliations

  1. Department of Finance and Management, The WSB University, Wroclaw, Poland

    Jacek Mercik

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Freixas, J., Pons, M. (2017). Preorders in Simple Games. In: Mercik, J. (eds) Transactions on Computational Collective Intelligence XXVII. Lecture Notes in Computer Science(), vol 10480. Springer, Cham. https://doi.org/10.1007/978-3-319-70647-4_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-70647-4_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-70646-7

  • Online ISBN: 978-3-319-70647-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation