Group Decision and Negotiation

, Volume 21, Issue 5, pp 637–662 | Cite as

Symmetric Coalitional Binomial Semivalues

Article

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.

Keywords

Cooperative game Binomial semivalue Coalition structure Multilinear extension 

Mathematics Subject Classification (2000)

91A12 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Albizuri MJ (2001) An axiomatization of the modified Banzhaf–Coleman index. Int J Game Theory 30: 167–176CrossRefGoogle Scholar
  2. Albizuri MJ (2002) Axiomatizations of Owen value without efficiency. Discussion Paper 25, Department of Applied Economics IV. Basque Country University, SpainGoogle Scholar
  3. Albizuri MJ, Zarzuelo JM (2004) On coalitional semivalues. Games Econ Behav 49: 221–243CrossRefGoogle Scholar
  4. Alonso JM, Fiestras MG (2002) Modification of the Banzhaf value for games with a coalition structure. Ann Oper Res 109: 213–227CrossRefGoogle Scholar
  5. Alonso JM, Carreras F, Fiestras MG (2005) The multilinear extension and the symmetric coalition Banzhaf value. Theory Decis 59: 111–126CrossRefGoogle Scholar
  6. Amer R, Carreras F (1995) Cooperation indices and coalition value. TOP 3: 117–135CrossRefGoogle Scholar
  7. 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 153–173Google Scholar
  8. Amer R, Giménez JM (2003) Modification of semivalues for games with coalition structures. Theory Decis 54: 185–205CrossRefGoogle Scholar
  9. Amer R, Carreras F, Giménez JM (2002) The modified Banzhaf value for games with a coalition structure: an axiomatic characterization. Math Soc Sci 43: 45–54CrossRefGoogle Scholar
  10. Aumann RJ, Drèze J (1974) Cooperative games with coalition structures. Int J Game Theory 3: 217–237CrossRefGoogle Scholar
  11. Banzhaf JF (1965) Weigthed voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19: 317–343Google Scholar
  12. Carreras F (2001) Elementary theory of simple games. Working Paper MA2-IT-01-00001, Department of Applied Mathematics II. Technical University of Catalonia, SpainGoogle Scholar
  13. Carreras F (2005) A decisiveness index for simple games. Eur J Oper Res 163: 370–387CrossRefGoogle Scholar
  14. Carreras F, Freixas J (1996) Complete simple games. Math Soc Sci 32: 139–155CrossRefGoogle Scholar
  15. 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 140–154Google Scholar
  16. Carreras F, Freixas J (2000) A note on regular semivalues. Int Game Theory Rev 2: 345–352CrossRefGoogle Scholar
  17. Carreras F, Freixas J (2002) Semivalue versatility and applications. Ann Oper Res 109: 343–358CrossRefGoogle Scholar
  18. Carreras F, Magaña A (1994) The multilinear extension and the modified Banzhaf–Coleman index. Math Soc Sci 28: 215–222CrossRefGoogle Scholar
  19. Carreras F, Magaña A (1997) The multilinear extension of the quotient game. Games Econ Behav 18: 22–31CrossRefGoogle Scholar
  20. Carreras F, Freixas J, Puente MA (2003) Semivalues as power indices. Eur J Oper Res 149: 676–687CrossRefGoogle Scholar
  21. 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 269–300Google Scholar
  22. Dragan I (1996) New mathematical properties of the Banzhaf value. Eur J Oper Res 95: 451–463CrossRefGoogle Scholar
  23. 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, USAGoogle Scholar
  24. Dubey P (1975) On the uniqueness of the Shapley value. Int J Game Theory 4: 131–139CrossRefGoogle Scholar
  25. Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Math Oper Res 4: 99–131CrossRefGoogle Scholar
  26. Dubey P, Neyman A, Weber RJ (1981) Value theory without efficiency. Math Oper Res 6: 122–128CrossRefGoogle Scholar
  27. Einy E (1987) Semivalues of simple games. Math Oper Res 12: 185–192CrossRefGoogle Scholar
  28. Feltkamp V (1995) Alternative axiomatic characterizations of the Shapley and Banzhaf values. Int J Game Theory 24: 179–186CrossRefGoogle Scholar
  29. Freixas J, Puente MA (2002) Reliability importance measures of the components in a system based on semivalues and probabilistic values. Ann Oper Res 109: 331–342CrossRefGoogle Scholar
  30. Giménez JM (2001) Contribuciones al estudio de soluciones para juegos cooperativos (in Spanish). Ph.D. Thesis. Technical University of Catalonia, SpainGoogle Scholar
  31. Hamiache G (1999) A new axiomatization of the Owen value for games with coalition structures. Math Soc Sci 37: 281–305CrossRefGoogle Scholar
  32. Hamiache G (2001) The Owen value values friendship. Int J Game Theory 29: 517–532CrossRefGoogle Scholar
  33. Hart S, Kurz M (1983) Endogeneous formation of coalitions. Econometrica 51: 1047–1064CrossRefGoogle Scholar
  34. Laruelle A (1999) On the choice of a power index. IVIE Discussion Paper WP–AD99–10, Instituto Valenciano de Investigaciones Económicas. Valencia, SpainGoogle Scholar
  35. Laruelle A, Valenciano F (2001) Shapley–Shubik and Banzhaf indices revisited. Math Oper Res 26: 89–104CrossRefGoogle Scholar
  36. Laruelle A, Valenciano F (2001b) Semivalues and voting power. Discussion Paper 13, Department of Applied Economics IV, Basque Country University, SpainGoogle Scholar
  37. 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, SpainGoogle Scholar
  38. Lehrer E (1988) An axiomatization of the Banzhaf value. Int J Game Theory 17: 89–99CrossRefGoogle Scholar
  39. Owen G (1972) Multilinear extensions of games. Manage Sci 18: 64–79CrossRefGoogle Scholar
  40. Owen G (1975) Multilinear extensions and the Banzhaf value. Naval Res Logist Q 22: 741–750CrossRefGoogle Scholar
  41. Owen G (1977) Values of games with a priori unions. In: Henn R, Moeschlin O (eds) Mathematical economics and game theory. Springer, Berlin, pp 76–88CrossRefGoogle Scholar
  42. Owen G (1978) Characterization of the Banzhaf–Coleman index. SIAM J Appl Math 35: 315–327CrossRefGoogle Scholar
  43. 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 232–238Google Scholar
  44. Owen G (1995) Game theory, 3rd edn. Academic Press Inc, LondonGoogle Scholar
  45. Owen G, Winter E (1992) Multilinear extensions and the coalitional value. Games Econ Behav 4: 582–587CrossRefGoogle Scholar
  46. 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, IsraelGoogle Scholar
  47. Penrose LS (1946) The elementary statistics of majority voting. J R Stat Soc 109: 53–57CrossRefGoogle Scholar
  48. 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, SpainGoogle Scholar
  49. Roth, AE (eds) (1988) The Shapley value: essays in Honor of Lloyd S. Shapley. Cambridge University Press, CambridgeGoogle Scholar
  50. 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 307–317Google Scholar
  51. Shapley LS (1962) Simple games: an outline of the descriptive theory. Behav Sci 7: 59–66CrossRefGoogle Scholar
  52. Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48: 787–792CrossRefGoogle Scholar
  53. 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 71–81CrossRefGoogle Scholar
  54. Taylor AD, Zwicker WS (1999) Simple games: desirability relations, trading, and pseudoweightings. Princeton University Press, PrincetonGoogle Scholar
  55. Vázquez M (1998) Contribuciones a la teoría del valor en juegos con utilidad transferible (in Spanish). Ph.D. Thesis. University of Santiago de Compostela, SpainGoogle Scholar
  56. Vázquez M, van den Nouweland A, García–Jurado I (1997) Owen’s coalitional value and aircraft landing fees. Math Soc Sci 34: 273–286CrossRefGoogle Scholar
  57. Weber RJ (1979) Subjectivity in the valuation of games. In: Moeschlin O, Pallaschke D (eds) Game theory and related topics. North–Holland, Amsterdam, pp 129–136Google Scholar
  58. 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 101–119CrossRefGoogle Scholar
  59. Winter E (1992) The consistency and potential for values with coalition structure. Games Econ Behav 4: 132–144CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Applied Mathematics II and Industrial and Aeronautical Engineering School of TerrassaTechnical University of CataloniaCataloniaSpain
  2. 2.Department of Applied Mathematics III and Engineering School of ManresaTechnical University of CataloniaCataloniaSpain

Personalised recommendations