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The Multilinear Extension and the Symmetric Coalition Banzhaf Value

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Abstract

Alonso-Meijide and Fiestras-Janeiro (2002, Annals of Operations Research 109, 213–227) proposed a modification of the Banzhaf value for games where a coalition structure is given. In this paper we present a method to compute this value by means of the multilinear extension of the game. A real-world numerical example illustrates the application procedure.

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References

  • M. J. Albizuri (2001) ArticleTitleAn axiomatization of the modified Banzhaf–Coleman index International Journal of Game Theory 30 167–176 Occurrence Handle10.1007/s001820100071

    Article  Google Scholar 

  • M. J. Albizuri J. M. Zarzuelo (2004) ArticleTitleOn coalitional semivalues Games and Economic Behavior 49 221–243 Occurrence Handle10.1016/j.geb.2004.01.001

    Article  Google Scholar 

  • J. M. Alonso-Meijide M. G. Fiestras-Janeiro (2002) ArticleTitleModification of the Banzhaf value for games with a coalition structure Annals of Operations Research 109 213–227 Occurrence Handle10.1023/A:1016356303622

    Article  Google Scholar 

  • Alonso-Meijide, J. M. and Bowles, C. (2002), Generating functions for coalitional power indices: an application to the IMF, Forthcoming in Annals of Operations Research, special issue on Contributions to the Theory of Games.

  • R. Amer J. M. Giménez (2003) ArticleTitleModification of semivalues for games with coalition structure Theory and Decision 54 185–205 Occurrence Handle10.1023/A:1027353528853

    Article  Google Scholar 

  • R. Amer F. Carreras J. M. Giménez (2002) ArticleTitleThe modified Banzhaf value for games with a coalition structure: an axiomatic characterization Mathematical Social Sciences 43 45–54 Occurrence Handle10.1016/S0165-4896(01)00076-2

    Article  Google Scholar 

  • J. F. Banzhaf (1965) ArticleTitleWeighted voting doesn’t work: A mathematical analysis Rutgers Law Review 19 317–343

    Google Scholar 

  • F. Carreras J. Freixas (2002) ArticleTitleSemivalue versatility and applications Annals of Operations Research 109 343–358 Occurrence Handle10.1023/A:1016320723186

    Article  Google Scholar 

  • F. Carreras A. Magaña (1994) ArticleTitleThe multilinear extension and the modified Banzhaf–Coleman index Mathematical Social Sciences 28 215–222 Occurrence Handle10.1016/0165-4896(94)90004-3

    Article  Google Scholar 

  • F. Carreras A. Magaña (1997) ArticleTitleThe multilinear extension of the quotient game Games and Economic Behavior 18 22–31 Occurrence Handle10.1006/game.1997.0498

    Article  Google Scholar 

  • Coleman, J. S. (1971), Control of collectivities and the power of a collectivity to act, in Lieberman, B. (ed.), Social Choice, Gordon and Breach, pp. 269–300.

  • P. Dubey A. Neyman R. J. Weber (1981) ArticleTitleValue theory without efficiency Mathematics of Operations Research 6 122–128

    Google Scholar 

  • D. S. Felsenthal M. Machover (1995) ArticleTitlePostulates and paradoxes of relative voting power – A critical reappraisal Theory and Decision 38 195–229 Occurrence Handle10.1007/BF01079500

    Article  Google Scholar 

  • D. S. Felsenthal M. Machover (1998) The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes Edward Elgar Publishers London

    Google Scholar 

  • Laruelle, A. (1999), On the choice of a power index, Working Paper AD99-10, Instituto Valenciano de Investigaciones Económicas, University of Alicante.

  • A. Laruelle F. Valenciano (2001) ArticleTitleShapley–Shubik and Banzhaf indices revisited Mathematics of Operations Research 26 89–104 Occurrence Handle10.1287/moor.26.1.89.10589

    Article  Google Scholar 

  • G. Owen (1972) ArticleTitleMultilinear extensions of games Management Science 18 64–79 Occurrence Handle10.1287/mnsc.18.5.64

    Article  Google Scholar 

  • G. Owen (1975) ArticleTitleMultilinear extensions and the Banzhaf value Naval Research Logistics Quarterly 22 741–750

    Google Scholar 

  • G. Owen (1977) Values of games with a priori unions R. Henn O. Moeschlin (Eds) Mathematical Economics and Game Theory Springer Verlag Berlin 76–88

    Google Scholar 

  • G. Owen (1982) Modification of the Banzhaf–Coleman index for games with a priori unions M. J. Holler (Eds) Power, Voting and Voting Power Physica-Verlag Wurzburg 232–238

    Google Scholar 

  • G. Owen E. Winter (1992) ArticleTitleMultilinear extensions and the coalition value Games and Economic Behavior 4 582–587 Occurrence Handle10.1016/0899-8256(92)90038-T

    Article  Google Scholar 

  • L. S. Shapley (1953) A value for n-person games A. W. Tucker H. W. Kuhn (Eds) Contributions to the Theory of Games Princeton University Press Princeton 307–317

    Google Scholar 

  • P. D. Straffin (1988) The Shapley-Shubik and Banzhaf power indices as probabilities A. E. Roth (Eds) The Shapley Value Cambridge University Press Cambridge 71–81

    Google Scholar 

  • R.J. Weber (1979) Subjectivity in the valuation of games O. Moeschlin D. Pallaschke (Eds) Game Theory and Related Topics North-Holland Amsterdam 129–136

    Google Scholar 

  • R. J. Weber (1988) Probabilistic values for games A. E. Roth (Eds) The Shapley Value Cambridge University Press Cambridge 101–119

    Google Scholar 

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Authors and Affiliations

  1. Departamento de Estatística e I. O., Facultade de Ciencias, Universidade de Santiago de Compostela, Alfonso X s/n, Campus de Lugo, E-27002, Lugo, España

    J. M. Alonso-Meijide

  2. Departamento de Matemática Aplicada II, Universidad Politécnica de Cataluña, ETSEIT, Colom 11, E-08222, Terrassa, España

    F. Carreras

  3. Departamento de Estatística e I. O, Facultade de Ciencias Económicas, Universidade de Vigo, E-36200, Vigo, España

    M. G. Fiestras-Janeiro

Authors
  1. J. M. Alonso-Meijide
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  2. F. Carreras
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  3. M. G. Fiestras-Janeiro
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Correspondence to J. M. Alonso-Meijide.

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MSC (2000) Classification: 91A12

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Alonso-Meijide, J.M., Carreras, F. & Fiestras-Janeiro, M.G. The Multilinear Extension and the Symmetric Coalition Banzhaf Value. Theor Decis 59, 111–126 (2005). https://doi.org/10.1007/s11238-005-0944-x

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