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Two-step coalition values for multichoice games

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Abstract

We introduce and compare several coalition values for multichoice games. Albizuri defined coalition structures and an extension of the Owen coalition value for multichoice games using the average marginal contribution of a player over a set of orderings of the player’s representatives. Following an approach used for cooperative games, we introduce a set of nested or two-step coalition values on multichoice games which measure the value of each coalition and then divide this among the players in the coalition using either a Shapley or Banzhaf value at each step. We show that when a Shapley value is used in both steps, the resulting coalition value coincides with that of Albizuri. We axiomatize the three new coalition values and show that each set of axioms, including that of Albizuri, is independent. Further we show how the multilinear extension can be used to compute the coalition values. We conclude with a brief discussion about the applicability of the different values.

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References

  • Albizuri MJ (2009) The multichoice coalition value. Ann Oper Res 172: 363–374

    Article  MathSciNet  MATH  Google Scholar 

  • Albizuri MJ, Aurrekoetxea J (2006) Coalition configurations and the Banzhaf index. Soc Choice Welfare 26: 571–596

    Article  MathSciNet  MATH  Google Scholar 

  • Alonso-Meijide JM, Fiestras-Janeiro M (2002) Modification of the Banzhaf value for games with a coalition structure. Ann Oper Res 109: 213–227

    Article  MathSciNet  MATH  Google Scholar 

  • Alonso-Meijide JM, Carreras F, Fiestras-Janeiro M (2005) The multilinear extension and the symmetric coalition Banzhaf value. Theory Decis 59: 111–126

    Article  MathSciNet  MATH  Google Scholar 

  • Alonso-Meijide JM, Carreras F, Fiestras-Janeiro M, Owen G (2007) A comparative axiomatic characterization of the Banzhaf-Owen coalitional value. Decis Support Syst 43: 701–712

    Article  Google Scholar 

  • Amer R, Carreras F, Giménez JM (2002) The modified Banzhaf value for games with coalition structure: an axiomatic characterization. Math Soc Sci 43: 45–54

    Article  MATH  Google Scholar 

  • Amer R, Giménez JM (2008) A general procedure to compute mixed modified semivalues for cooperative games with structure of coalition blocks. Math Soc Sci 56: 269–282

    Article  MATH  Google Scholar 

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

    Google Scholar 

  • Branzei R, Dimitrov D, Tijs S (2005) Models in cooperative game theory. Springer, Berlin

    MATH  Google Scholar 

  • Carreras F, Magana A (1997) The multilinear extension of the quotient game. Games Econ Behav 18: 22–31

    Article  MathSciNet  MATH  Google Scholar 

  • Carreras F, Magana A (1994) The multilinear extension and the modified Banzhaf–Colemen index. Math Soc Sci 28: 215–222

    Article  MathSciNet  MATH  Google Scholar 

  • Coleman J (1968) Control of collectivities and the power of a collectivity to act. RAND Paper P-3902: 39 p

  • Derks J, Peters H (1993) A Shapley value for games with restricted coalitions. Int J Game Theory 21: 351–360

    Article  MathSciNet  MATH  Google Scholar 

  • Felsenthal D, Machover M (1997) Ternary voting games. Int J Game Theory 26: 335–351

    Article  MathSciNet  MATH  Google Scholar 

  • Fiestras-Janeiro MG, Garcia-Jurado I, Mosquera MA (2011) Cooperative games and cost allocation problems. Top 19: 1–22

    Article  MathSciNet  MATH  Google Scholar 

  • Fragnelli V, Iandolino A (2004) A cost allocation problem in urban solid wastes collection and disposal. Math Methods Oper Res 59: 447–463

    MathSciNet  MATH  Google Scholar 

  • Grabisch M, Lange F (2007) Games on lattices, multichoice games and the Shapley value: a new approach. Math Methods Oper Res 65: 153–167

    Article  MathSciNet  MATH  Google Scholar 

  • Hart S, Kurz M (1983) Endogeneous formation of coalitions. Econometrica 51: 1047–1064

    Article  MathSciNet  MATH  Google Scholar 

  • Hsiao C, Raghavan T (1993) Shapley value for multichoice cooperative games I. Games Econ Behav 5: 240–256

    Article  MathSciNet  MATH  Google Scholar 

  • Jones MA, Wilson JM (2010) Multilinear extensions and values for multichoice games. Math Methods Oper Res 72: 145–169

    Article  MathSciNet  MATH  Google Scholar 

  • Kamijo Y (2009) A two-step Shapley value for cooperative games with coalition structures. Int Game Theory Rev 11: 207–214

    Article  MathSciNet  MATH  Google Scholar 

  • Khmelnitskaya A, Yanovskaya E (2007) Owen coalitional value without additivity axiom. Math Methods Oper Res 66: 225–261

    Article  MathSciNet  Google Scholar 

  • Owen G (1972) Multilinear extensions of games. Manag Sci 18: 64–79

    Article  MathSciNet  MATH  Google Scholar 

  • Owen G (1975) Multilinear extensions and the Banzhaf value. Naval Res Logist Q 22: 741–750

    Article  MathSciNet  MATH  Google Scholar 

  • Owen G (1977) Values of games with a priori unions. In: Hernn R, Moschlin O (eds) Lecture notes in economics and mathematical systems: essays in honor of Oskar Morgenstern. Springer, New York, pp 76–88

    Google Scholar 

  • Owen G (1981) Modification of the Banzhaf–Coleman index for games with a priori unions. In: Holler MJ (ed) Voting and voting power. Physica-Verlag, Wurzburg, pp 232–238

    Chapter  Google Scholar 

  • Owen G, Winter E (1992) The multilinear extension and the coalition structure value. Games Econ Behav 4: 582–587

    Article  MathSciNet  MATH  Google Scholar 

  • Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games, vol II. (Annals of Mathematical Studies), vol 28. pp 307–317

  • Straffin P (1977) Homogeneity, independence, and power indices. Public Choice 30: 107–119

    Article  Google Scholar 

  • Van den Nouweland A, Potters J, Tijs S, Zarzuelo J (1995) Cores and related solution concepts for multi-choice games. Math Methods Oper Res 41: 289–311

    Article  MathSciNet  MATH  Google Scholar 

  • Vázquez-Brage M, van den Nouweland A, García-Jurado I (1997) Owens coalitional value and aircraft landing fees. Math Soc Sci 34: 273–286

    Article  MATH  Google Scholar 

  • Winter E (1992) The consistency and potential for values of games with coalition structures. Games Econ Behav 4: 132–144

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematical Reviews, 416 Fourth St., Ann Arbor, MI, 48103, USA

    Michael A. Jones

  2. Department of Natural Science and Mathematics, Eugene Lang College The New School for Liberal Arts, New York, NY, 10011, USA

    Jennifer M. Wilson

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  1. Michael A. Jones
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  2. Jennifer M. Wilson
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Correspondence to Michael A. Jones.

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Jones, M.A., Wilson, J.M. Two-step coalition values for multichoice games. Math Meth Oper Res 77, 65–99 (2013). https://doi.org/10.1007/s00186-012-0415-4

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  • DOI: https://doi.org/10.1007/s00186-012-0415-4

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