, Volume 26, Issue 1, pp 164–186 | Cite as

A note on multinomial probabilistic values

  • Francesc Carreras
  • María Albina Puente
Original Paper


Multinomial values were previously introduced by one of the authors in reliability and extended later to all cooperative games. Here, we present for this subfamily of probabilistic values three new results, previously stated only for binomial semivalues in the literature. They concern the dimension of the subspace spanned by the multinomial values and two characterizations: one, individual, for each multinomial value; another, collective, for the whole subfamily they form. Finally, an application to simple games is provided.


Game theory Cooperative game Shapley value Probabilistic value Binomial semivalue 

Mathematics Subject Classification




The authors wish to thank two anonymous reviewers for their helpful comments.


  1. Alonso JM, Carreras F, Puente MA (2007) Axiomatic characterizations of the symmetric coalitional binomial semivalues. Discret Appl Math 155:2282–2293CrossRefGoogle Scholar
  2. Amer R, Giménez JM (2003) Modification of semivalues for games with coalition structures. Theory Decis 54:185–205CrossRefGoogle Scholar
  3. Carreras F (2004) \(\alpha \)-decisiveness in simple games. Theory Decis 56:77–91 [Also in: Gambarelli G (ed) Essays on cooperative games. Kluwer Academic Publishers, pp 77–91]Google Scholar
  4. Carreras F (2005) A decisiveness index for simple games. Eur J Oper Res 163:370–387CrossRefGoogle Scholar
  5. Carreras F, Puente MA (2012) Symmetric coalitional binomial semivalues. Group Decis Negot 21:637–662CrossRefGoogle Scholar
  6. Carreras F, Puente MA (2015a) Multinomial probabilistic values. Group Decis Negot 24:981–991CrossRefGoogle Scholar
  7. Carreras F, Puente MA (2015b) Coalitional multinomial probabilistic values. Eur J Oper Res 245:236–246CrossRefGoogle Scholar
  8. Domènech M, Giménez JM, Puente MA (2016) Some properties for probabilistic and multinomial (probabilistic) values on cooperative games. Optimization 65:1377–1395CrossRefGoogle Scholar
  9. Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Math Oper Res 4:99–131CrossRefGoogle Scholar
  10. Dubey P, Neyman A, Weber RJ (1981) Value theory without efficiency. Math Oper Res 6:122–128CrossRefGoogle Scholar
  11. Feltkamp V (1995) Alternative axiomatic characterizations of the Shapley and Banzhaf values. Int J Game Theory 24:179–186CrossRefGoogle Scholar
  12. Freixas J, Pons M (2015) An axiomatic characterization of the potential decisiveness index. J Oper Res Soc 66:353–359CrossRefGoogle Scholar
  13. 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
  14. Giménez JM, Llongueras MD, Puente MA (2014) Partnership formation and multinomial values. Discret Appl Math 170:7–20CrossRefGoogle Scholar
  15. Owen G (1972) Multilinear extensions of games. Manag Sci 18:64–79CrossRefGoogle Scholar
  16. Owen G (1975) Multilinear extensions and the Banzhaf value. Nav Res Logist Q 22:741–750CrossRefGoogle Scholar
  17. Owen G (1978) Characterization of the Banzhaf–Coleman index. SIAM J Appl Math 35:315–327CrossRefGoogle Scholar
  18. Puente MA (2000) Contributions to the representability of simple games and to the calculus of solutions for this class of games (in Spanish). Ph.D. Thesis. Universitat Politècnica de Catalunya, SpainGoogle Scholar
  19. Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games II. Annals of mathematical studies, vol 28. Princeton University Press, Princeton, pp 307–317Google Scholar
  20. Weber RJ (1988) Probabilistic values for games. In: Roth AE (ed) The Shapley value: essays in honor of Lloyd S. Shapley. Cambridge University Press, Cambridge, pp 101–119CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2017

Authors and Affiliations

  1. 1.Department of Mathematics, School of Industrial, Aerospace and Audiovisual Engineering of TerrassaUniversitat Politècnica de Catalunya (UPC)BarcelonaSpain
  2. 2.Department of Mathematics, Engineering School of ManresaUniversitat Politècnica de Catalunya (UPC)BarcelonaSpain

Personalised recommendations