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Core Stability of Minimum Coloring Games

  • Thomas Bietenhader
  • Yoshio Okamoto
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3353)

Abstract

In cooperative game theory, a characterization of games with stable cores is known as one of the most notorious open problems. We study this problem for a special case of the minimum coloring games, introduced by Deng, Ibaraki & Nagamochi, which arises from a cost allocation problem when the players are involved in conflict. In this paper, we show that the minimum coloring game on a perfect graph has a stable core if and only if every vertex of the graph belongs to a maximum clique. We also consider the problem on the core largeness, the extendability, and the exactness of minimum coloring games.

Keywords

Cooperative Game Maximum Clique Cost Allocation Chordal Graph Large Core 
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.
    Biswas, A.K., Parthasarathy, T., Potters, J.A.M., Voorneveld, M.: Large cores and exactness. Games and Economic Behavior 28, 1–12 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chudnovsky, M., Seymour, P.: Recognizing Berge graphs (Submitted)Google Scholar
  3. 3.
    Corneil, D.G., Perl, Y.: Clustering and domination in perfect graphs. Discrete Applied Mathematics 9, 27–39 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cornuéjols, G., Liu, X., Vus̆ković, K.: A polynomial algorithm for recognizing perfect graphs. In: Proc. 44th FOCS, pp. 20–27 (2003)Google Scholar
  5. 5.
    Curiel, I.J.: Cooperative Game Theory and Applications: Cooperative Games Arising from Combinatorial Optimization Problems. Kluwer Academic Publishers, Dordrecht (1997)Google Scholar
  6. 6.
    Deng, X., Ibaraki, T., Nagamochi, H.: Algorithmic aspects of the core of combinatorial optimization games. Math. Oper. Res. 24, 751–766 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Deng, X., Ibaraki, T., Nagamochi, H., Zang, W.: Totally balanced combinatorial optimization games. Math. Program. 87, 441–452 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Deng, X., Papadimitriou, C.H.: On the complexity of cooperative solution concepts. Math. Oper. Res. 19, 257–266 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Farber, M.: Independent domination in chordal graphs. Oper. Res. Lett. 1, 134–138 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pacific J. Math. 15, 835–855 (1965)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Gillies, D.B.: Some theorems on n-person games. Ph.D. Thesis. Princeton University, Princeton (1953)Google Scholar
  12. 12.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization, 2nd edn. Springer, Berlin (1993)zbMATHGoogle Scholar
  13. 13.
    Kikuta, K., Shapley, L.S.: Core stability in n-person games. (1986) (Manuscript)Google Scholar
  14. 14.
    Kratsch, D.: Algorithms. In: Haynes, T.W., Hedetniemi, S.T., Slater, P.J. (eds.) Domination in Graphs (Advanced Topics), pp. 191–231. Marcel Dekker Inc., New York (1998)Google Scholar
  15. 15.
    Kratsch, D., Stewart, L.: Domination on cocomparability graphs. SIAM J. Discrete Math. 6, 400–417 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lovász, L.: Normal hypergraphs and the perfect graph conjecture. Discrete Math. 2, 253–267 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Okamoto, Y.: Submodularity of some classes of the combinatorial optimization games. Math. Methods Oper. Res. 58, 131–139 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Okamoto, Y.: Fair cost allocations under conflicts — A game-theoretic point of view —. In: Ibaraki, T., Katoh, N., Ono, H. (eds.) ISAAC 2003. LNCS, vol. 2906, pp. 686–695. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  19. 19.
    Schmeidler, D.: Cores of exact games I. J. Math. Anal. Appl. 40, 214–225 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Shapley, L.S.: Cores of convex games. Internat. J. Game Theory 1, 11–26 (1971); Errata is in the same volume, pp. 199 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sharkey, W.W.: Cooperative games with large cores. Internat. J. Game Theory 11, 175–182 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Solymosi, T., Raghavan, T.E.S.: Assignment games with stable cores. Internat. J. Game Theory 30, 177–185 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    van Gellekom, J.R.G., Potters, J.A.M., Reijnierse, J.H.: Prosperity properties of TU-games. Internat. J. Game Theory 28, 211–227 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behaviour. Princeton University Press, Princeton (1944)Google Scholar
  25. 25.
    Zverovich, I.E.: Independent domination on 2P 3-free perfect graphs. DIMACS Technical Report 2003-22 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Thomas Bietenhader
    • 1
  • Yoshio Okamoto
    • 1
  1. 1.Department of Computer ScienceETH ZurichZurichSwitzerland

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