Skip to main content
SpringerLink
Log in

The Shapley Value in Fuzzy Simple Cooperative Games

  • Conference paper
  • First Online:
Intelligent Information and Database Systems (ACIIDS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10751))

Included in the following conference series:

Abstract

This article presents an analysis of the Shapley value of simple cooperative games in situations where the weights of the players and the majority required to accept a decision (in the form of a winning coalition) are given by fuzzy numbers. We propose a modification of the concept of the additivity of the payoff function which is necessary in this framework.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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.

    The property of additivity (also called the “law of aggregation” by Shapley (1953)): for any two games \( \left( {N, v} \right),\left( {N, w} \right) \): \( \varphi \left( {v + w} \right) = \varphi \left( {v) + \varphi (w} \right) \) (i.e. \( \varphi_{i} \left( {v + w} \right) = \varphi_{i} \left( v \right) \) + \( \varphi_{i} \left( w \right) \) for all i in N, where the game \( \left( {v + w} \right) \) is defined by \( \left( {v + w} \right)\left( S \right) = v\left( S \right) + w\left( S \right) \) for any coalition S). This property states that when two independent games are combined, their values must be added player by player. This is a prime requisite for any scheme designed to be eventually applied to systems of interdependent games.

  2. 2.

    There exist several theorems related to additional properties of the Shapley value. For example, Young (1985) proved that the Shapley value is the only value satisfying the properties of effectiveness, symmetry and strong monotonicity, van den Brink (2001) showed that it is the only value preserving a fairness condition according to a modification of marginal contributions to a coalition and Myerson (1977) showed that it preserves fairness based on balanced contributions to a coalition.

  3. 3.

    q defines what proportion of the votes describes a winning majority.

References

  • Aumann, R.J.: Recent developments in the theory of the Shapley value. In: Proceedings of the International Congress of Mathematicians, Helsinki, pp. 995–1003 (1978)

    Google Scholar 

  • Bertini, C.: Shapley value. In: Dowding, K. (ed.) Encyclopedia of Power, pp. 600–603. SAGE Publications, Los Angeles (2011)

    Google Scholar 

  • Borkotokey, S.: Cooperative games with fuzzy coalition and fuzzy characteristic function. Fuzzy Sets Syst. 159, 138–151 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Borkotokey, S., Mesiar, R.: The Shapley value of cooperative games under fuzzy setting: a servey. Int. J. Gen. Syst. 43(1), 75–95 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • van den Brink, R.: An axiomatization of the Shapley value using a fairness property. Int. J. Game Theory 30, 309–319 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9(6), 613–626 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  • Gładysz, B.: Fuzzy-probabilistic PERT. Ann. Oper. Res. 245(1/2), 1–16 (2016)

    MathSciNet  MATH  Google Scholar 

  • Hart, S., Mas-Collel, A.: Potential, value, and consistency. Econometrica 57(3), 589–614 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Li, S., Zhang, Q.: A simplified expression of the Shapley function for fuzzy game. Eur. J. Oper. Res. 196, 234–245 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Meng, F., Zhang, Q.: The Shapley value on a kind of cooperative games. J. Comput. Inf. Syst. 7(6), 1846–1854 (2011)

    Google Scholar 

  • Mercik, J., Ramsey, D.M.: The effect of Brexit on the balance of power in the European Union Council: an approach based on pre-coalitions. In: Mercik, J. (ed.) Transactions on Computational Collective Intelligence XXVII. LNCS, vol. 10480, pp. 87–107. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70647-4_7

    Chapter  Google Scholar 

  • Myerson, R.: Graphs and cooperation in games. Math. Oper. Res. 2(3), 225–229 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  • Pang, J., Chen, X., Li, S.: The Shapley values on fuzzy coalition games with concave integral form. J. Appl. Math. 2014 (2014). http://dx.doi.org/10.1155/2014/231508

  • Shapley, L.S.: A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games II, Annals of Mathematics Studies, vol. 28, pp. 307–317. Princeton University Press, Princeton (1953)

    Google Scholar 

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

    Article  Google Scholar 

  • Tsurumi, M., Tanino, T., Inuiguchi, M.: A Shapley function on a class of cooperative fuzzy games. Eur. J. Oper. Res. 129, 596–618 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Yu, X., Zhang, Q.: An extension of fuzzy cooperative games. Fuzzy Sets Syst. 161, 1614–1634 (2010)

    Article  MATH  Google Scholar 

  • Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Wrocław University of Science and Technology, Wrocław, Poland

    Barbara Gładysz

  2. WSB University, Wrocław, Poland

    Jacek Mercik

Authors
  1. Barbara Gładysz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jacek Mercik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Barbara Gładysz .

Editor information

Editors and Affiliations

  1. Wrocław University of Science and Technology, Wrocław, Poland

    Ngoc Thanh Nguyen

  2. Quang Binh University, Dong Hoi City, Vietnam

    Duong Hung Hoang

  3. National University of Kaohsiung, Kaohsiung, Taiwan

    Tzung-Pei Hong

  4. Rutgers University, Piscataway, New Jersey, USA

    Hoang Pham

  5. Wrocław University of Science and Technology, Wrocław, Poland

    Bogdan Trawiński

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Gładysz, B., Mercik, J. (2018). The Shapley Value in Fuzzy Simple Cooperative Games. In: Nguyen, N., Hoang, D., Hong, TP., Pham, H., Trawiński, B. (eds) Intelligent Information and Database Systems. ACIIDS 2018. Lecture Notes in Computer Science(), vol 10751. Springer, Cham. https://doi.org/10.1007/978-3-319-75417-8_39

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-75417-8_39

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-75416-1

  • Online ISBN: 978-3-319-75417-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation