Top

Published in:

2018 | OriginalPaper | Chapter

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

Authors : Mustapha Ridaoui, Michel Grabisch, Christophe Labreuche

Published in: Modeling Decisions for Artificial Intelligence

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Extraction of Patterns to Support Dairy Culling Management

next chapter Fuzzy Positive Primitive Formulas

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

Choquet, G.: Theory of capacities. Ann. L’Institut Fourier 5, 131–295 (1953)MathSciNetCrossRef

Crama, Y., Hammer, P.: Boolean Functions. Number 142 in Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2011)

de Wolf, R.: A brief introduction to Fourier analysis on the Boolean cube. Theory Comput. Libr. Grad. Surv. 1, 1–20 (2008)

Dubey, P., Shapley, L.S.: Mathematical properties of the Banzhaf power index. Math. Oper. Res. 4(2), 99–131 (1979)MathSciNetCrossRef

Fujimoto, K., Kojadinovic, I., Marichal, J.-L.: Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices. Games Econ. Behav. 55(1), 72–99 (2006)MathSciNetCrossRef

Grabisch, M.: \(k\)-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst. 92(2), 167–189 (1997)MathSciNetCrossRef

Grabisch, M.: Set Functions, Games and Capacities in Decision Making. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-319-30690-2CrossRefMATH

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 2003

10.

Grabisch, M., Labreuche, C.: A note on the Sobol’ indices and interactive criteria. Fuzzy Sets Syst. 315, 99–108 (2017)MathSciNetCrossRef

11.

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

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)MathSciNetCrossRef

13.

Grabisch, M., Roubens, M.: Application of the Choquet integral in multicriteria decision making. Fuzzy Meas. Integr.-Theory Appl. 348–374 (2000)

14.

Hammer, P., Rudeanu, S.: Boolean Methods in Operations Research and Related Areas Econometrics and Operations Research, 7th edn. Springer, Heidelberg (1986). https://doi.org/10.1007/978-3-642-85823-9CrossRefMATH

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)

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)MathSciNetCrossRef

17.

Lange, F., Grabisch, M.: The interaction transform for functions on lattices. Discret. Math. 309(12), 4037–4048 (2009)MathSciNetCrossRef

18.

Lehrer, E.: An axiomatization of the Banzhaf value. Int. J. Game Theory 17(2), 89–99 (1988)MathSciNetCrossRef

19.

Moulin, H.: Fair Division and Collective Welfare. MIT Press, Cambridge (2003)

20.

Murofushi, T., Soneda, S.: Techniques for reading fuzzy measures (III): interaction index. In: 9th Fuzzy System Symposium, Sapporo, Japan, pp. 693–696 (1993)

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). https://doi.org/10.1007/978-3-319-67504-6_6CrossRef

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). https://doi.org/10.1007/978-3-319-61581-3_31CrossRef

23.

Ridaoui, M., Grabisch, M., Labreuche, C.: An interaction index for multichoice games. arXiv:1803.07541 (2018)

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)

25.

Sugeno, M.: Theory of fuzzy integrals and its applications. Ph.D thesis. Tokyo Institute of Technology (1974)

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)

Title: An Axiomatisation of the Banzhaf Value and Interaction Index for Multichoice Games
Authors: Mustapha Ridaoui
Michel Grabisch
Christophe Labreuche
Publisher: Springer International Publishing
Book: Modeling Decisions for Artificial Intelligence
Print ISBN: 978-3-030-00201-5

Electronic ISBN: 978-3-030-00202-2

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-030-00202-2_12

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner