Agent Failures in Totally Balanced Games and Convex Games

  • Yoram Bachrach
  • Ian Kash
  • Nisarg Shah
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7695)

Abstract

We examine the impact of independent agents failures on the solutions of cooperative games, focusing on totally balanced games and the more specific subclass of convex games. We follow the reliability extension model, recently proposed in [1] and show that a (approximately) totally balanced (or convex) game remains (approximately) totally balanced (or convex) when independent agent failures are introduced or when the failure probabilities increase. One implication of these results is that any reliability extension of a totally balanced game has a non-empty core. We propose an algorithm to compute such a core imputation with high probability. We conclude by outlining the effect of failures on non-emptiness of the core in cooperative games, especially in totally balanced games and simple games, thereby extending observations in [1].

Keywords

Totally Balanced Games Convex Games Agent Failures Cooperative Game Theory 

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References

  1. 1.
    Bachrach, Y., Meir, R., Feldman, M., Tennenholtz, M.: Solving cooperative reliability games. In: UAI, pp. 27–34 (2011)Google Scholar
  2. 2.
    Bachrach, Y., Meir, R., Jung, K., Kohli, P.: Coalitional structure generation in skill games (2010)Google Scholar
  3. 3.
    Bachrach, Y., Meir, R., Zuckerman, M., Rothe, J., Rosenschein, J.: The cost of stability in weighted voting games. In: AAMAS (2009)Google Scholar
  4. 4.
    Bala, V., Goyal, S.: A strategic analysis of network reliability. Review of Economic Design 5, 205–228 (2000)CrossRefGoogle Scholar
  5. 5.
    Billand, P., Bravard, C., Sarangi, S.: Nash networks with imperfect reliability and heterogeous players. IGTR 13(2), 181–194 (2011)MathSciNetGoogle Scholar
  6. 6.
    Caillou, P., Aknine, S., Pinson, S.: Multi-agent models for searching pareto optimal solutions to the problem of forming and dynamic restructuring of coalitions. In: ECAI, pp. 13–17 (2002)Google Scholar
  7. 7.
    Chalkiadakis, G., Elkind, E., Wooldridge, M.: Computational aspects of cooperative game theory. Synth. Lect. on Artif. Intell. and Machine Learning (2011)Google Scholar
  8. 8.
    Chalkiadakis, G., Markakis, E., Boutilier, C.: Coalition formation under uncertainty: bargaining equilibria and the Bayesian core. In: AAMAS (2007)Google Scholar
  9. 9.
    Deng, X., Fang, Q., Sun, X.: Finding nucleolus of flow game. J. Comb. Opt. (2009)Google Scholar
  10. 10.
    Deng, X., Ibaraki, T., Nagamochi, H., Zang, W.: Totally balanced combinatorial optimization games. Math. Prog. 87, 441–452 (2000)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Gillies, D.: Some Theorems on n-Person Games. PhD thesis, Princeton U. (1953)Google Scholar
  12. 12.
    Granot, D., Huberman, G.: Minimum cost spanning tree games. Math. Prog. 21, 1–18 (1981)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Kalai, E., Zemel, E.: Totally balanced games and games of flow. MOR (1982)Google Scholar
  14. 14.
    Kannan, R., Sarangi, S., Iyengar, S.: A simple model for reliable query reporting in sensor networks. In: Information Fusion, vol. 2, pp. 1070–1075 (2002)Google Scholar
  15. 15.
    Kern, W., Paulusma, D.: Matching games: The least core and the nucleolus. MOR (2003)Google Scholar
  16. 16.
    Kuipers, J.: A Polynomial Time Algorithm for Computing the Nucleolus of Convex Games. Reports in Operations Research and Systems Theory (1996)Google Scholar
  17. 17.
    Myerson, R.B.: Game Theory: Analysis of Conflict. Harvard U. Press (1997)Google Scholar
  18. 18.
    Osborne, M., Rubinstein, A.: A course in game theory. The MIT press (1994)Google Scholar
  19. 19.
    Owen, G.: On the core of linear production games. Math. Prog. 9, 358–370 (1975)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Penn, M., Polukarov, M., Tennenholtz, M.: Congestion games with failures. Discrete Applied Mathematics 159(15), 1508–1525 (2011)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Resnick, E., Bachrach, Y., Meir, R., Rosenschein, J.S.: The Cost of Stability in Network Flow Games. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 636–650. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  22. 22.
    Sandholm, T., Lesser, V.: Coalitions among computationally bounded agents. Artif. Intell. 94(1-2), 99–137 (1997)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Shapley, L.: Cores of convex games. IJGT 1, 11–26 (1971)MathSciNetMATHGoogle Scholar
  24. 24.
    Shapley, L., Shubik, M.: On market games. JET 1(1), 9–25 (1969)MathSciNetGoogle Scholar
  25. 25.
    Shapley, L., Shubik, M.: The assignment game I: The core. IJGT 1, 111–130 (1971)MathSciNetMATHGoogle Scholar
  26. 26.
    Solymosi, T., Raghavan, T.: An algorithm for finding the nucleolus of assignment games. IJGT 23, 119–143 (1994)MathSciNetMATHGoogle Scholar
  27. 27.
    Tijs, S.H., Parthasarathy, T., Potters, J., Prasad, V.: Permutation games: Another class of totally balanced games. OR Spectrum 6, 119–123 (1984)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yoram Bachrach
    • 1
  • Ian Kash
    • 1
  • Nisarg Shah
    • 2
  1. 1.Microsoft Research LtdCambridgeUK
  2. 2.Computer Science DepartmentCarnegie Mellon UniversityUSA

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