Advertisement

A New Look at the Study of Solutions for Games in Partition Function Form

  • Joss Sánchez-PérezEmail author
Chapter
Part of the Static & Dynamic Game Theory: Foundations & Applications book series (SDGTFA)

Abstract

This chapter studies the structure of games in partition function and according to an axiomatic point of view, we provide a global description of linear symmetric solutions by means of a decomposition of the set of such games (as well as of a decomposition of the space of payoff vectors). The exhibition of relevant subspaces in such decomposition and based on the idea that every permutation of the set of players may be thought of as a linear map, allow for a new look at linear symmetric solutions.

Keywords

Cooperative games in partition function form Axiomatic solutions Symmetric group 

Notes

Acknowledgements

I thank the participants of the European Meeting on Game Theory (St. Petersburg, Russia, 2015) for comments, interesting discussions, and encouragement. I am also grateful to the Editors and two anonymous referees for their useful comments and suggestions. J. Sánchez-Pérez acknowledges support from CONACYT research grant 130515.

References

  1. 1.
    Albizuri, M.J., Arin, J., Rubio, J.: An axiom system for a value for games in partition function form. Int. Game Theory Rev. 7 (1), 63–72 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bolger, E.M.: A class of efficient values for games in partition function form. J. Algebraic Discrete Methods 8 (3), 460–466 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bolger E.M.: A set of axioms for a value for partition function games. Int. J. Game Theory 18 (1), 37–44 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Fulton, W., Harris, J.: Representation Theory: A First Course. Graduate Texts in Mathematics, vol. 129. Springer, New York (1991)Google Scholar
  5. 5.
    Hernández-Lamoneda, L., Juárez, R., Sánchez-Sánchez, F.: Dissection of solutions in cooperative game theory using representation techniques. Int. J. Game Theory 35 (3), 395–426 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Hernández-Lamoneda, L., Sánchez-Pérez, J. Sánchez-Sánchez, F.: The class of efficient linear symmetric values for games in partition function form. Int. Game Theory Rev. 11 (3), 369–382 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Hu, C.C., Yang, Y.Y.: An axiomatic characterization of a value for games in partition function form. SERIEs: J. Span. Econ. Assoc. 1 (4), 475–487 (2010)CrossRefGoogle Scholar
  8. 8.
    Ju, Y.: The consensus value for games in partition function form. Int. Game Theory Rev. 9 (3), 437–452 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kleinberg, N.L., Weiss, J.H.: Equivalent n-person games and the null space of the Shapley value. Math. Oper. Res. 10 (2), 233–243 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kleinberg, N.L., Weiss, J.H.: Weak values, the core and new axioms for the Shapley value. Math. Soc. Sci. 12, 21–30 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Lucas, W.F., Thrall, R.M.: n-person games in partition function form. Nav. Res. Logist. Q. 10, 281–298 (1963)Google Scholar
  12. 12.
    Macho-Stadler, I., Pérez-Castrillo, D., Wettstein, D.: Sharing the surplus: an extension of the Shapley value for environments with externalities. J. Econ. Theory 135, 339–356 (2007)Google Scholar
  13. 13.
    Myerson R.B.: Values of games in partition function form. Int. J. Game Theory 6 (1), 23–31 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Pham, D.K., Norde, H.: The Shapley value for partition function games. Int. Game Theory Rev. 9 (2), 353–360 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Sánchez-Pérez, J.: Application of the representations of symmetric groups to characterize solutions of games in partition function form. Oper. Res. Decis. 24 (2), 97–122 (2014)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Faculty of EconomicsUASLP, Av. Pintores S/N, Burócratas del EstadoSan Luis PotosíMexico

Personalised recommendations