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A Hierarchical Model for Cooperative Games

  • Ulrich Faigle
  • Britta Peis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4997)

Abstract

Classically, a cooperative game is given by a normalized real-valued function v on the collection of all subsets of the set N of players. Shapley has observed that the core of the game is non-empty if v is a non-negative convex (a.k.a. supermodular) set function. In particular, the Shapley value of a convex game is a member of the core. We generalize the classical model of games such that not all subsets of N need to form feasible coalitions. We introduce a model for ranking individual players which yields natural notions of Weber sets and Shapley values in a very general context. We establish Shapley’s theorem on the nonemptyness of the core of monotone convex games in this framework. The proof follows from a greedy algorithm that, in particular, generalizes Edmonds’ polymatroid greedy algorithm.

Keywords

Greedy Algorithm Cooperative Game Choice Function Submodular Function Convex Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Algaba, E., Bilbao, J.M., van den Brink, R., Jiménez-Losada, A.: Cooperative games on antimatroids. Discr. Mathematics 282, 1–15 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bilbao, J.M., Jiménez, N., Lebrón, E., López, J.J.: The marginal operators for games on convex geometries. Intern. Game Theory Review 8, 141–151 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bilbao, J.M., Lebrón, E., Jiménez, N.: The core of games on convex geometries. Europ. J. Operational Research 119, 365–372 (1999)CrossRefzbMATHGoogle Scholar
  4. 4.
    Derks, J., Gilles, R.P.: Hierarchical organization structures and constraints in coalition formation. Intern. J. Game Theory 24, 147–163 (1995)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dietrich, B.L., Hoffman, A.J.: On greedy algorithms, partially ordered sets, and submodular functions. IBM J. Res. & Dev. 47, 25–30 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Danilov, V., Koshevoy, G.: Choice functions and extending operators (preprint, 2007)Google Scholar
  7. 7.
    Edmonds, J.: Submodular functions, matroids and certain polyhedra. In: Proc. Int. Conf. on Combinatorics (Calgary), pp. 69–87 (1970)Google Scholar
  8. 8.
    Edelman, P.H., Jamison, R.E.: The theory of convex geometries. Geometriae Dedicata 19, 247–270 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fujishige, S.: Submodular Functions and Optimization. 2nd edn.; Ann. Discrete Mathematics 58 (2005)Google Scholar
  10. 10.
    Faigle, U.: Cores of games with restricted cooperation. Methods and Models of Operations Research 33, 405–422 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Faigle, U., Kern, W.: The Shapley value for cooperative games under precedence constraints. Intern. J. Game Theory 21, 249–266 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Faigle, U., Kern, W.: Submodular linear programs on forests. Math. Programming 72, 195–206 (1996)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Faigle, U., Kern, W.: On the core of ordered submodular cost games. Math. Programming 87, 483–489 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Faigle, U., Kern, W.: An order-theoretic framework for the greedy algorithm with applications to the core and Weber set of cooperative games. Order 17, 353–375 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Faigle, U., Peis, B.: Two-phase greedy algorithms for some classes of combinatorial linear programs. In: SODA 2008 (accepted, 2008)Google Scholar
  16. 16.
    Frank, A.: Increasing the rooted-connectivity of a digraph by one. Math. Programming 84, 565–576 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fujishige, S.: Dual greedy polyhedra, choice functions, and abstract convex geometries. Discrete Optimization 1, 41–49 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grabisch, M., Xie, L.J.: The core of games on distributive lattices (working paper)Google Scholar
  19. 19.
    Grabisch, M., Xie, L.J.: A new investigation about the core and Weber set of multichoice gamse. Mathematical Methods of Operations Research (to appear)Google Scholar
  20. 20.
    Gilles, R.P., Owen, G., van den Brink, R.: Games with permission structures: the conjunctive approach. Intern. J. Game Theory 20, 277–293 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hoffman, A.J., Schwartz, D.E.: On lattice polyhedra. In: Hajnal, A., Sós, V.T. (eds.) Proc. 5th Hungarian Conference in Combinatorics, pp. 593–598. North-Holland, Amsterdam (1978)Google Scholar
  22. 22.
    Hsiao, C.-R., Raghavan, T.E.S.: Shapley value for multi-choice cooperative games. Games and Economic Behavior 5, 240–256 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Koshevoy, G.: Choice functions and abstract convex geometries. Math. Soc. Sci. 38, 35–44 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Moulin, H.: Choice functions over a finite set: a summary. Soc. Choice Welfare 2, 147–160 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Shapley, L.S.: A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games, Ann. Math. Studies, vol. 28, pp. 307–317. Princeton University Press, Princeton (1953)Google Scholar
  26. 26.
    Shapley, L.S.: Cores of convex games. Intern. J. Game Theory 1, 12–26 (1971)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Weber, R.J.: Probabilistic values for games. In: Roth, A.E. (ed.) The Shapley Value, pp. 101–120. Cambridge University Press, Cambridge (1988)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ulrich Faigle
    • 1
  • Britta Peis
    • 2
  1. 1.Zentrum für Angewandte Informatik Köln (ZAIK)KölnGermany
  2. 2.Technische Universität BerlinBerlinGermany

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