Symmetries of Quasi-Values

  • Ales A. Kubena
  • Peter Franek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8146)


According to Shapley’s game-theoretical result, there exists a unique game value of finite cooperative games that satisfies axioms on additivity, efficiency, null-player property and symmetry. The original setting requires symmetry with respect to arbitrary permutations of players. We analyze the consequences of weakening the symmetry axioms and study quasi-values that are symmetric with respect to permutations from a  group G ≤ S n . We classify all the permutation groups G that are large enough to assure a unique G-symmetric quasi-value, as well as the structure and dimension of the space of all such quasi-values for a general permutation group G.

We show how to construct G-symmetric quasi-values algorithmically by averaging certain basic quasi-values (marginal operators).


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  1. 1.
    Beaumont, R., Peterson, R.: Set-transitive permutation groups. Canadian Journal of Mathematics 7(1), 35–42 (1955)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brânzei, R., Dimitrov, D., Tijs, S.: Models in Cooperative Game Theory: Crisp, Fuzzy, and Multi-Choice Games. Lecture Notes in Economics and Mathematical Systems. Springer (2005)Google Scholar
  3. 3.
    Brink, R.: An axiomatization of the shapley value using a fairness property. International Journal of Game Theory 30, 309–319 (2002)CrossRefGoogle Scholar
  4. 4.
    Carnahan, S.: Small finite sets (2007)Google Scholar
  5. 5.
    David, S.: The nucleolus of a characteristic function game. Siam Journal on Applied Mathematics 17(6), 1163–1166 (1969)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Derks, J., Haller, H., Peters, H.: The selectope for cooperative games. Open access publications from maastricht university, Maastricht University (2000)Google Scholar
  7. 7.
    Dixon, J., Mortimer, B.: Permutation groups. Springer (1996)Google Scholar
  8. 8.
    Flajolet, P., Sedgewick, R.: Analytic combinatorics. Cambridge University Press (2009)Google Scholar
  9. 9.
    Fulman, J.: Cycle indices for the finite classical groups (1997)Google Scholar
  10. 10.
    Gilboa, I., Monderer, D.: Quasi-value on subspaces. International Journal of Game Theory 19(4), 353–363 (1991)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gilles, R.: The Cooperative Game Theory of Networks and Hierarchies. Theory and decision library: Game theory, mathematical programming, and operations research. Springer (2010)Google Scholar
  12. 12.
    Janusz, G., Rotman, J.: Outer automorphisms of S 6. The American Mathematical Monthly 89(6), 407–410 (1982)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jordan, M.: Super-transitive group action (mathoverflow contribution),
  14. 14.
    Kalai, E., Samet, D.: On Weighted Shapley Values. International Journal of Game Theory 16(3), 205–222 (1987)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kar, A.: Axiomatization of the shapley value on minimum cost spanning tree games. Games and Economic Behavior 38(2), 265–277 (2002)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Monderer, D., Ruckle, W.H.: On the Symmetry Axiom for Values of Nonatomic Games. Int. Journal of Math. and Math. Sci. 13(1), 165–170 (1990)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Neumann, J., Morgenstern, O., Rubinstein, A., Kuhn, H.: Theory of Games and Economic Behavior. Princeton Classic Editions. Princeton University Press (2007)Google Scholar
  18. 18.
    Neyman, A.: Uniqueness of the shapley value. Games and Economic Behavior 1(1), 116–118 (1989)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Neyman, A.: Values of Games with Infinitely Many Players. Handbook of Game Theory with Economic Applications, vol. 3, ch. 56, pp. 2121–2167. Elsevier (2002)Google Scholar
  20. 20.
    Owen, G.: Values of Games with A Priori Unions. In: Henn, R., Moeschlin, O. (eds.) Mathematical Economics and Game Theory. Lecture Notes in Economics and Mathematical Systems, vol. 141, pp. 76–88. Springer, Heidelberg (1977)CrossRefGoogle Scholar
  21. 21.
    Rotman, J.: An introduction to the theory of groups. Springer (1995)Google Scholar
  22. 22.
    Shapley, L.S.: A value for n-person games. Annals of Mathematics Studies 2(28), 307–317 (1953)MathSciNetGoogle Scholar
  23. 23.
    Weber, R.: Probabilistic Values of Games. In: Roth, A. (ed.) The Shapley Value: Essays in Honor of Lloyd S. Shapley, pp. 101–120. Cambridge Univ. Press (1988)Google Scholar
  24. 24.
    Young, H.P.: Monotonic solutions of cooperative games. International Journal of Game Theory 14, 65–72 (1985)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ales A. Kubena
    • 1
  • Peter Franek
    • 2
  1. 1.Institute of Information Theory and Automation of the ASCRPragueCzech Republic
  2. 2.Institute of Information TechnologiesCzech Technical UniversityPragueCzech Republic

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