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
Albizuri MJ, Aurrekoetxea J (2006) Coalition configurations and the Banzhaf index. Soc Choice Welfare 26: 571–596
Alonso-Meijide JM, Fiestras-Janeiro M (2002) Modification of the Banzhaf value for games with a coalition structure. Ann Oper Res 109: 213–227
Alonso-Meijide JM, Carreras F, Fiestras-Janeiro M (2005) The multilinear extension and the symmetric coalition Banzhaf value. Theory Decis 59: 111–126
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
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
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
Banzhaf JF (1965) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19: 317–343
Branzei R, Dimitrov D, Tijs S (2005) Models in cooperative game theory. Springer, Berlin
Carreras F, Magana A (1997) The multilinear extension of the quotient game. Games Econ Behav 18: 22–31
Carreras F, Magana A (1994) The multilinear extension and the modified Banzhaf–Colemen index. Math Soc Sci 28: 215–222
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
Felsenthal D, Machover M (1997) Ternary voting games. Int J Game Theory 26: 335–351
Fiestras-Janeiro MG, Garcia-Jurado I, Mosquera MA (2011) Cooperative games and cost allocation problems. Top 19: 1–22
Fragnelli V, Iandolino A (2004) A cost allocation problem in urban solid wastes collection and disposal. Math Methods Oper Res 59: 447–463
Grabisch M, Lange F (2007) Games on lattices, multichoice games and the Shapley value: a new approach. Math Methods Oper Res 65: 153–167
Hart S, Kurz M (1983) Endogeneous formation of coalitions. Econometrica 51: 1047–1064
Hsiao C, Raghavan T (1993) Shapley value for multichoice cooperative games I. Games Econ Behav 5: 240–256
Jones MA, Wilson JM (2010) Multilinear extensions and values for multichoice games. Math Methods Oper Res 72: 145–169
Kamijo Y (2009) A two-step Shapley value for cooperative games with coalition structures. Int Game Theory Rev 11: 207–214
Khmelnitskaya A, Yanovskaya E (2007) Owen coalitional value without additivity axiom. Math Methods Oper Res 66: 225–261
Owen G (1972) Multilinear extensions of games. Manag Sci 18: 64–79
Owen G (1975) Multilinear extensions and the Banzhaf value. Naval Res Logist Q 22: 741–750
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
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
Owen G, Winter E (1992) The multilinear extension and the coalition structure value. Games Econ Behav 4: 582–587
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
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
Vázquez-Brage M, van den Nouweland A, García-Jurado I (1997) Owens coalitional value and aircraft landing fees. Math Soc Sci 34: 273–286
Winter E (1992) The consistency and potential for values of games with coalition structures. Games Econ Behav 4: 132–144
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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
Keywords
- Multichoice games
- Coalition values
- Owen coalition value
- Shapley value
- Banzhaf value
- Axiomatic characterizations
- Multilinear extension