Skip to main content
SpringerLink
Log in

Least Square Approximations and Linear Values of Cooperative Games

  • Chapter
  • First Online:
Algebraic Techniques and Their Use in Describing and Processing Uncertainty

Part of the book series: Studies in Computational Intelligence ((SCI,volume 878))

  • 255 Accesses

Abstract

Many important values for cooperative games are known to arise from least square optimization problems. The present investigation develops an optimization framework to explain and clarify this phenomenon in a general setting. The main result shows that every linear value results from some least square approximation problem and that, conversely, every least square approximation problem with linear constraints yields a linear value. This approach includes and extends previous results on so-called least square values and semivalues in the literature. In particular, it is demonstrated how known explicit formulas for solutions under additional assumptions easily follow from the general results presented here.

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
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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 general treatment of set functions, games, capacities and their application in decision making, see [7].

  2. 2.

    see Faigle et al. [5] or any other textbook on mathematical optimization.

  3. 3.

    Later published in [11].

  4. 4.

    See also Ding [4], and Marichal and Mathonet [14].

  5. 5.

    See also Sun et al. [20] for similar problems.

References

  1. Banzhaf, J.F.: Weighted voting does not work: a mathematical analysis. Rutgers Law Rev. 19, 317–343 (1965)

    Google Scholar 

  2. Charnes, A., Golany, B., Keane, M., Rousseau, J.: Extremal principle solutions of games in characteristic function form: core, Chebychev and Shapley value generalizations. In: Sengupta, J.K., Kadekodi, G.K. (eds.) Econometrics of Planning and Efficiency, pp. 123–133. Kluwer Academic Publisher (1988)

    Google Scholar 

  3. Derks, J., Haller, H., Peters, H.: The selectope for cooperative games. Int. J. Game Theory 29, 23–38 (2000)

    Article  MathSciNet  Google Scholar 

  4. Ding, G., Lax, R., Chen, J., Chen, P., Marx, B.: Transforms of pseudo-Boolean random variables. Discrete Appl. Math. 158, 13–24 (2010)

    Article  MathSciNet  Google Scholar 

  5. Faigle, U., Kern, W., Still, G.: Algorithmic Principles of Mathematical Programming. Springer, Dordrecht (2002)

    Book  Google Scholar 

  6. Faigle, U., Voss, J.: A system-theoretic model for cooperation, interaction and allocation. Discrete Appl. Math. 159, 1736–1750 (2011)

    Article  MathSciNet  Google Scholar 

  7. Grabisch, M.: Set Functions, Games and Capacities in Decision Making. Theory and Decision Library C, vol. 46, Springer (2016)

    Google Scholar 

  8. Grabisch, M., Labreuche, C.: Fuzzy measures and integrals in MCDA. In: Figueira, J., Greco, S., Ehrgott, M. (eds.) Multiple Criteria Decision Analysis, pp. 563–608. Kluwer Academic Publishers (2005)

    Google Scholar 

  9. Grabisch, M., Marichal, J.-L., Roubens, M.: Equivalent representations of set functions. Math. Oper. Res. 25(2), 157–178 (2000)

    Article  MathSciNet  Google Scholar 

  10. Hammer, P.L., Holzman, R.: On approximations of pseudo-Boolean functions. RUTCOR Research Report RRR 29–87. State University of New Jersey (1987)

    Google Scholar 

  11. Hammer, P.L., Holzman, R.: On approximations of pseudo-Boolean functions. ZOR—Methods Models Oper. Res. 36, 3–21 (1992)

    Article  Google Scholar 

  12. Hammer, P.L., Rudeanu, S.: Boolean Methods in Operations Research and Related Areas. Springer, Berlin (1968)

    Book  Google Scholar 

  13. Kultti, K., Salonen, H.: Minimum norm solutions for cooperative games. Int. J. Game Theory 35, 591–602 (2007)

    Article  MathSciNet  Google Scholar 

  14. Marichal, J.-L., Mathonet, P.: Weighted Banzhaf power and interaction indexes through weighted approximations of games. Eur. J. Oper. Res. 211, 352–358 (2011)

    Article  MathSciNet  Google Scholar 

  15. Peleg, B., Sudhölter, P.: Introduction to the Theory of Cooperative Games. Kluwer Academic Publisher (2003)

    Google Scholar 

  16. Ruiz, L.M., Valenciano, F., Zarzuelo, J.M.: The family of least-square values for transferable utility games. Games Econ. Behav. 24, 109–130 (1998)

    Article  MathSciNet  Google Scholar 

  17. Ruiz, L.M., Valenciano, F., Zarzuelo, J.M.: Some new results on least square values for TU games. TOP 6, 139–158 (1998)

    Article  MathSciNet  Google Scholar 

  18. Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57(3), 571–587 (1989)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  20. Sun, H., Hao, Z., Xu, G.: Optimal Solutions for TU-Games with Decision Approach. Preprint, Northwestern Polytechnical University, Xi’an, Shaanxi, China (2013)

    Google Scholar 

  21. Tanino, T.: One-point solutions obtained from best approximation problems for cooperative games. Kybernetika 49, 395–403 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Weber, R.J.: Probabilistic values for games. In: Roth, A.E. (ed.) The Shapley Value, pp. 101–120. Cambridge University Press, Cambridge (1988)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität zu Köln, Weyertal 80, 50931, Köln, Germany

    Ulrich Faigle

  2. Paris School of Economics, University of Paris I, 106-112, Bd. de l’Hôpital, Paris, 75013, France

    Michel Grabisch

Authors
  1. Ulrich Faigle
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michel Grabisch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michel Grabisch .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, USA

    Hung T. Nguyen

  2. Department of Computer Science, The University of Texas at El Paso, El Paso, TX, USA

    Vladik Kreinovich

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Faigle, U., Grabisch, M. (2020). Least Square Approximations and Linear Values of Cooperative Games. In: Nguyen, H., Kreinovich, V. (eds) Algebraic Techniques and Their Use in Describing and Processing Uncertainty. Studies in Computational Intelligence, vol 878. Springer, Cham. https://doi.org/10.1007/978-3-030-38565-1_2

Download citation

Publish with us

Policies and ethics

Search

Navigation