Skip to main content
SpringerLink
Log in

Axiomatic of the Shapley value of a game with a priori unions

  • Original Paper
  • Published:
TOP Aims and scope Submit manuscript

Abstract

In this paper, we define a modification of the Shapley value for the model of TU games with a priori unions. We provide two characterizations of this value and a new characterization of the Banzhaf–Owen coalitional value.

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. A simple game (N,v) is a TU game such that v(S)=0 or v(S)=1 for all S⊆N; v(S)≤v(T) for all S⊆T⊆N; and v(N)=1.

  2. A minimal winning coalition is a coalition S⊆N, which satisfies that v(S)=1 and v(T)=0 for all T⊆̷S. Note that a simple game can be interpreted as a pair (N,MW), where MW is the set of minimal winning coalitions.

  3. The notion of indivisible player has been introduced by Alonso-Meijide et al. (2012). A player a is indivisible when there is no pair of players i and j such that {i,j}=a.

References

  • Alonso-Meijide JM, Fiestras-Janeiro MG (2002) Modification of the Banzhaf value for games with a coalition structure. Ann Oper Res 109:213–227

    Article  Google Scholar 

  • Alonso-Meijide JM, Fiestras-Janeiro MG, Carreras F (2005) The multilinear extension and the symmetric coalition Banzhaf value. Theory Decis 59:111–126

    Article  Google Scholar 

  • Alonso-Meijide JM, Carreras F, Fiestras-Janeiro MG, Owen G (2007a) A comparative axiomatic characterization of the Banzhaf–Owen coalitional value. Decis Support Syst 43:701–712

    Article  Google Scholar 

  • Alonso-Meijide JM, Carreras F, Puente MA (2007b) Axiomatic characterizations of the symmetric coalitional binomial semivalues. Discrete Appl Math 155:2282–2293

    Article  Google Scholar 

  • Alonso-Meijide JM, Álvarez-Mozos M, Fiestras-Janeiro MG (2012) Notes on a comment on 2-efficiency and the Banzhaf value. Appl Math Lett 25:1098–1100

    Article  Google Scholar 

  • Álvarez-Mozos M, Tejada O (2011) Parallel characterizations of a generalized Shapley value and a generalized Banzhaf value for cooperative games with levels structure of cooperation. Decis Support Syst 52:21–27

    Article  Google Scholar 

  • Álvarez-Mozos M, van den Brink R, van der Laan G, Tejada O (2013) Share functions for cooperative games with levels structure of cooperation. Eur J Oper Res 224:167–179

    Article  Google Scholar 

  • Amer R, Carreras F (1995) Cooperation indices and coalitional value. Top 3:117–135

    Article  Google Scholar 

  • Amer R, Carreras F, Giménez JM (2002) The modified Banzhaf value for games with coalition structure: an axiomatic characterization. Math Soc Sci 43:45–54

    Article  Google Scholar 

  • Aumann RJ, Drèze J (1974) Cooperative games with coalition structures. Int J Game Theory 3:217–237

    Article  Google Scholar 

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

    Google Scholar 

  • Carreras F, Magaña A (1994) The multilinear extension and the modified Banzhaf–Coleman index. Math Soc Sci 28:215–222

    Article  Google Scholar 

  • Carreras F, Llongueras MD, Magaña A (2005) On the convenience to form coalitions or partnerships in simple games. Ann Oper Res 137:67–89

    Article  Google Scholar 

  • Gómez-Rúa M, Vidal-Puga J (2010) The axiomatic approach to three values in games with coalition structure. Eur J Oper Res 207:795–806

    Article  Google Scholar 

  • Hamiache G (1999) A new axiomatization of the Owen value for games with coalition structures. Math Soc Sci 37:281–305

    Article  Google Scholar 

  • Kamijo Y (2013) The collective value: a new solution for games with coalition structures. Top. Article in press. doi:10.1007/s11750-011-0191-y

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

    Article  Google Scholar 

  • Meng FY (2011) The Banzhaf–Owen value for fuzzy games with a coalition structure. Eng Technol 80:855–859

    Google Scholar 

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

    Article  Google Scholar 

  • Owen G (1972) Multilinear extensions of games. Manag Sci 18:64–79

    Article  Google Scholar 

  • Owen G (1975) Multilinear extensions and the Banzhaf value. Nav Res Logist Q 22:741–750

    Article  Google Scholar 

  • Owen G (1977) Values of games with a priori unions. In: Henn R, Moeschlin O (eds) Mathematical economics and game theory. Springer, Berlin, pp 76–88

    Chapter  Google Scholar 

  • Owen G (1982) Modification of the Banzhaf–Coleman index for games with a priori unions. In: Holler MJ (ed) Power, voting and voting power. Physica-Verlag, Würzburg, pp 232–238

    Google Scholar 

  • Owen G, Winter E (1992) The multilinear extension and the coalition structure value. Games Econ Behav 4:582–587

    Article  Google Scholar 

  • Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games II. Princeton University Press, Princeton, pp 307–317

    Google Scholar 

  • Vázquez-Brage M, van den Nouweland A, García-Jurado I (1997) Owen’s coalitional value and aircraft landing fees. Math Soc Sci 34:273–286

    Article  Google Scholar 

  • Winter E (1992) The consistency and potential for values of games with coalition structure. Games Econ Behav 4:132–144

    Article  Google Scholar 

  • Young HP (1985) Monotonic solutions of cooperative games. Int J Game Theory 14:65–72

    Article  Google Scholar 

Download references

Acknowledgements

Financial support from Ministerio de Ciencia y Tecnología and FEDER through grants MTM2011-27731-C03-02 and ECO2011-23460 is gratefully acknowledged. The authors are also very grateful for the interesting suggestions given by three anonymous referees.

Author information

Authors and Affiliations

  1. MODESTYA Research Group, Department of Statistics and Operations Research and Faculty of Sciences, University of Santiago de Compostela, Santiago de Compostela, Spain

    J. M. Alonso-Meijide

  2. MODESTYA Research Group, Department of Statistics and Operations Research and Faculty of Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain

    B. Casas-Méndez

  3. Department of Statistics and Operations Research and Faculty of Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain

    A. M. González-Rueda

  4. MODES Research Group, Department of Mathematics and Faculty of Computer Science, University of A Coruña, 15071, A Coruña, Spain

    S. Lorenzo-Freire

Authors
  1. J. M. Alonso-Meijide
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Casas-Méndez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. M. González-Rueda
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. Lorenzo-Freire
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Lorenzo-Freire.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alonso-Meijide, J.M., Casas-Méndez, B., González-Rueda, A.M. et al. Axiomatic of the Shapley value of a game with a priori unions. TOP 22, 749–770 (2014). https://doi.org/10.1007/s11750-013-0298-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11750-013-0298-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation