An Axiomatisation of the Banzhaf Value and Interaction Index for Multichoice Games

  • Mustapha RidaouiEmail author
  • Michel Grabisch
  • Christophe Labreuche
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11144)


We provide an axiomatisation of the Banzhaf value (or power index) and the Banzhaf interaction index for multichoice games, which are a generalisation of cooperative games with several levels of participation. Multichoice games can model any aggregation model in multicriteria decision making, provided the attributes take a finite number of values. Our axiomatisation uses standard axioms of the Banzhaf value for classical games (linearity, null axiom, symmetry), an invariance axiom specific to the multichoice context, and a generalisation of the 2-efficiency axiom, characteristic of the Banzhaf value.


Banzhaf value Multicriteria decision aid Multichoice games Interaction 


  1. 1.
    Banzhaf, J.: Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev. 19, 317–343 (1965)Google Scholar
  2. 2.
    Choquet, G.: Theory of capacities. Ann. L’Institut Fourier 5, 131–295 (1953)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Crama, Y., Hammer, P.: Boolean Functions. Number 142 in Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2011)Google Scholar
  4. 4.
    de Wolf, R.: A brief introduction to Fourier analysis on the Boolean cube. Theory Comput. Libr. Grad. Surv. 1, 1–20 (2008)Google Scholar
  5. 5.
    Dubey, P., Shapley, L.S.: Mathematical properties of the Banzhaf power index. Math. Oper. Res. 4(2), 99–131 (1979)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fujimoto, K., Kojadinovic, I., Marichal, J.-L.: Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices. Games Econ. Behav. 55(1), 72–99 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Grabisch, M.: \(k\)-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst. 92(2), 167–189 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Grabisch, M.: Set Functions, Games and Capacities in Decision Making. Springer, Heidelberg (2016). Scholar
  9. 9.
    Grabisch, M., Labreuche, C.: Capacities on lattices and \(k\)-ary capacities. In: International Conference Of the Euro Society for Fuzzy Logic and Technology (EUSFLAT), Zittau, Germany, 10–12 September 2003Google Scholar
  10. 10.
    Grabisch, M., Labreuche, C.: A note on the Sobol’ indices and interactive criteria. Fuzzy Sets Syst. 315, 99–108 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Grabisch, M., Marichal, J.-L., Roubens, M.: Equivalent representations of set functions. Math. Oper. Res. 25(2), 157–178 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Grabisch, M., Roubens, M.: An axiomatic approach to the concept of interaction among players in cooperative games. Int. J. Game Theory 28(4), 547–565 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Grabisch, M., Roubens, M.: Application of the Choquet integral in multicriteria decision making. Fuzzy Meas. Integr.-Theory Appl. 348–374 (2000)Google Scholar
  14. 14.
    Hammer, P., Rudeanu, S.: Boolean Methods in Operations Research and Related Areas Econometrics and Operations Research, 7th edn. Springer, Heidelberg (1986). Scholar
  15. 15.
    Hsiao, C.R., Raghavan, T.E.S.: Shapley value for multi-choice cooperative games I. Discussion Paper of the University of Illinois at Chicago, Chicago (1990)Google Scholar
  16. 16.
    Kojadinovic, I.: A weight-based approach to the measurement of the interaction among criteria in the framework of aggregation by the bipolar Choquet integral. Eur. J. Oper. Res. 179, 498–517 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lange, F., Grabisch, M.: The interaction transform for functions on lattices. Discret. Math. 309(12), 4037–4048 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lehrer, E.: An axiomatization of the Banzhaf value. Int. J. Game Theory 17(2), 89–99 (1988)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Moulin, H.: Fair Division and Collective Welfare. MIT Press, Cambridge (2003)Google Scholar
  20. 20.
    Murofushi, T., Soneda, S.: Techniques for reading fuzzy measures (III): interaction index. In: 9th Fuzzy System Symposium, Sapporo, Japan, pp. 693–696 (1993)Google Scholar
  21. 21.
    Ridaoui, M., Grabisch, M., Labreuche, C.: An alternative view of importance indices for multichoice games. In: Rothe, J. (ed.) ADT 2017. LNCS (LNAI), vol. 10576, pp. 81–92. Springer, Cham (2017). Scholar
  22. 22.
    Ridaoui, M., Grabisch, M., Labreuche, C.: Axiomatization of an importance index for generalized additive independence models. In: Antonucci, A., Cholvy, L., Papini, O. (eds.) ECSQARU 2017. LNCS (LNAI), vol. 10369, pp. 340–350. Springer, Cham (2017). Scholar
  23. 23.
    Ridaoui, M., Grabisch, M., Labreuche, C.: An interaction index for multichoice games. arXiv:1803.07541 (2018)
  24. 24.
    Shapley, L.S.: A value for \(n\)-person games. In: Kuhn, H.W., Tucker, A.W. (eds.), Contributions to the Theory of Games. Number 28 in Annals of Mathematics Studies, vol. II, pp. 307–317. Princeton University Press (1953)Google Scholar
  25. 25.
    Sugeno, M.: Theory of fuzzy integrals and its applications. Ph.D thesis. Tokyo Institute of Technology (1974)Google Scholar
  26. 26.
    Weber, R.J.: Probabilistic values for games. In: Roth, A.E. (ed.), The Shapley Value: Essays in Honor of Lloyd S. Shapley, pp. 101–120. Cambridge University Press (1988)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Mustapha Ridaoui
    • 1
    Email author
  • Michel Grabisch
    • 1
  • Christophe Labreuche
    • 2
  1. 1.Paris School of EconomicsUniversité Paris I - Panthéon-SorbonneParisFrance
  2. 2.Thales Research and TechnologyPalaiseauFrance

Personalised recommendations