Abstract
We introduce here a family of mixed coalitional values. They extend the binomial semivalues to games endowed with a coalition structure, satisfy the property of symmetry in the quotient game and the quotient game property, generalize the symmetric coalitional Banzhaf value introduced by Alonso and Fiestras and link and merge the Shapley value and the binomial semivalues. A computational procedure in terms of the multilinear extension of the original game is also provided and an application to political science is sketched.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Albizuri MJ (2001) An axiomatization of the modified Banzhaf–Coleman index. Int J Game Theory 30: 167–176
Albizuri MJ (2002) Axiomatizations of Owen value without efficiency. Discussion Paper 25, Department of Applied Economics IV. Basque Country University, Spain
Albizuri MJ, Zarzuelo JM (2004) On coalitional semivalues. Games Econ Behav 49: 221–243
Alonso JM, Fiestras MG (2002) Modification of the Banzhaf value for games with a coalition structure. Ann Oper Res 109: 213–227
Alonso JM, Carreras F, Fiestras MG (2005) The multilinear extension and the symmetric coalition Banzhaf value. Theory Decis 59: 111–126
Amer R, Carreras F (1995) Cooperation indices and coalition value. TOP 3: 117–135
Amer R, Carreras F (2001) Power, cooperation indices and coalition structures. In: Holler MJ, Owen G (eds) Power indices and coalition formation. Kluwer, Dordrecht, pp 153–173
Amer R, Giménez JM (2003) Modification of semivalues for games with coalition structures. Theory Decis 54: 185–205
Amer R, Carreras F, Giménez JM (2002) The modified Banzhaf value for games with a coalition structure: an axiomatic characterization. Math Soc Sci 43: 45–54
Aumann RJ, Drèze J (1974) Cooperative games with coalition structures. Int J Game Theory 3: 217–237
Banzhaf JF (1965) Weigthed voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19: 317–343
Carreras F (2001) Elementary theory of simple games. Working Paper MA2-IT-01-00001, Department of Applied Mathematics II. Technical University of Catalonia, Spain
Carreras F (2005) A decisiveness index for simple games. Eur J Oper Res 163: 370–387
Carreras F, Freixas J (1996) Complete simple games. Math Soc Sci 32: 139–155
Carreras F, Freixas J. (1999) Some theoretical reasons for using (regular) semivalues. In: Swart H (eds) Logic, game theory and social choice. Tilburg University Press, Tilburg, pp 140–154
Carreras F, Freixas J (2000) A note on regular semivalues. Int Game Theory Rev 2: 345–352
Carreras F, Freixas J (2002) Semivalue versatility and applications. Ann Oper Res 109: 343–358
Carreras F, Magaña A (1994) The multilinear extension and the modified Banzhaf–Coleman index. Math Soc Sci 28: 215–222
Carreras F, Magaña A (1997) The multilinear extension of the quotient game. Games Econ Behav 18: 22–31
Carreras F, Freixas J, Puente MA (2003) Semivalues as power indices. Eur J Oper Res 149: 676–687
Coleman JS (1971) Control of collectivities and the power of a collectivity to act. In: Lieberman B (eds) Social choice. Gordon and Breach, New York, pp 269–300
Dragan I (1996) New mathematical properties of the Banzhaf value. Eur J Oper Res 95: 451–463
Dragan I (1997) Some recursive definitions of the Shapley value and other linear values of cooperative TU games. Working paper 328. University of Texas at Arlington, USA
Dubey P (1975) On the uniqueness of the Shapley value. Int J Game Theory 4: 131–139
Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Math Oper Res 4: 99–131
Dubey P, Neyman A, Weber RJ (1981) Value theory without efficiency. Math Oper Res 6: 122–128
Einy E (1987) Semivalues of simple games. Math Oper Res 12: 185–192
Feltkamp V (1995) Alternative axiomatic characterizations of the Shapley and Banzhaf values. Int J Game Theory 24: 179–186
Freixas J, Puente MA (2002) Reliability importance measures of the components in a system based on semivalues and probabilistic values. Ann Oper Res 109: 331–342
Giménez JM (2001) Contribuciones al estudio de soluciones para juegos cooperativos (in Spanish). Ph.D. Thesis. Technical University of Catalonia, Spain
Hamiache G (1999) A new axiomatization of the Owen value for games with coalition structures. Math Soc Sci 37: 281–305
Hamiache G (2001) The Owen value values friendship. Int J Game Theory 29: 517–532
Hart S, Kurz M (1983) Endogeneous formation of coalitions. Econometrica 51: 1047–1064
Laruelle A (1999) On the choice of a power index. IVIE Discussion Paper WP–AD99–10, Instituto Valenciano de Investigaciones Económicas. Valencia, Spain
Laruelle A, Valenciano F (2001) Shapley–Shubik and Banzhaf indices revisited. Math Oper Res 26: 89–104
Laruelle A, Valenciano F (2001b) Semivalues and voting power. Discussion Paper 13, Department of Applied Economics IV, Basque Country University, Spain
Laruelle A, Valenciano F (2003) On the meaning of the Owen–Banzhaf coalitional value in voting situations. Discussion Paper 35, Department of Applied Economics IV, Basque Country University, Spain
Lehrer E (1988) An axiomatization of the Banzhaf value. Int J Game Theory 17: 89–99
Owen G (1972) Multilinear extensions of games. Manage 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: Henn R, Moeschlin O (eds) Mathematical economics and game theory. Springer, Berlin, pp 76–88
Owen G (1978) Characterization of the Banzhaf–Coleman index. SIAM J Appl Math 35: 315–327
Owen G (1982) Modification of the Banzhaf-Coleman index for games with a priori unions. In: Holler MJ (ed) Power, voting and voting power, pp 232–238
Owen G (1995) Game theory, 3rd edn. Academic Press Inc, London
Owen G, Winter E (1992) Multilinear extensions and the coalitional value. Games Econ Behav 4: 582–587
Peleg B (1989) Introduction to the theory of cooperative games. Chapter 8: The Shapley value RM 88, Center for Research in Mathematical Economics and Game Theory. the Hebrew University, Israel
Penrose LS (1946) The elementary statistics of majority voting. J R Stat Soc 109: 53–57
Puente MA (2000) Aportaciones a la representabilidad de juegos simples y al cálculo de soluciones de esta clase de juegos (in Spanish). Ph.D. Thesis. Technical University of Catalonia, Spain
Roth, AE (eds) (1988) The Shapley value: essays in Honor of Lloyd S. Shapley. Cambridge University Press, Cambridge
Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games II. Princeton University Press, Princeton, pp 307–317
Shapley LS (1962) Simple games: an outline of the descriptive theory. Behav Sci 7: 59–66
Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48: 787–792
Straffin PD (1988) The Shapley–Shubik and Banzhaf power indices. In: Kuhn HW, Roth AE (eds) The Shapley value: essays in Honor of Lloyd S. Shapley. Cambridge University Press, Cambridge, pp 71–81
Taylor AD, Zwicker WS (1999) Simple games: desirability relations, trading, and pseudoweightings. Princeton University Press, Princeton
Vázquez M (1998) Contribuciones a la teoría del valor en juegos con utilidad transferible (in Spanish). Ph.D. Thesis. University of Santiago de Compostela, Spain
Vázquez M, van den Nouweland A, García–Jurado I (1997) Owen’s coalitional value and aircraft landing fees. Math Soc Sci 34: 273–286
Weber RJ (1979) Subjectivity in the valuation of games. In: Moeschlin O, Pallaschke D (eds) Game theory and related topics. North–Holland, Amsterdam, pp 129–136
Weber RJ (1988) Probabilistic values for games. In: Kuhn HW, Roth AE (eds) The Shapley value: essays in Honor of Lloyd S. Shapley. Cambridge University Press, Cambridge, pp 101–119
Winter E (1992) The consistency and potential for values with coalition structure. Games Econ Behav 4: 132–144
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by Grant SGR 2009-01029 of the Catalonia Government (Generalitat de Catalunya) and Grants MTM 2006-06064 and MTM 2009-08037 of the Science and Innovation Spanish Ministry and the European Regional Development Fund.
Rights and permissions
About this article
Cite this article
Carreras, F., Puente, M.A. Symmetric Coalitional Binomial Semivalues. Group Decis Negot 21, 637–662 (2012). https://doi.org/10.1007/s10726-011-9239-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10726-011-9239-5