Computing a Payoff Division in the Least Core for MC-nets Coalitional Games

  • Katsutoshi Hirayama
  • Kenta Hanada
  • Suguru Ueda
  • Makoto Yokoo
  • Atsushi Iwasaki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8861)


MC-nets is a concise representation of the characteristic functions that exploits a set of rules to compute payoffs. Given a MC-nets instance, the problem of computing a payoff division in the least core, which is a generalization of the core-non-emptiness problem that is known to be coNP-complete, is definitely a hard computational problem. In fact, to the best of our knowledge, no algorithm can actually compute such a payoff division for MC-nets instances with dozens of agents. We propose a new algorithm for this problem, that exploits the constraint generation technique to solve the linear programming problem that potentially has a huge number of constraints. Our experimental results are striking since, using 8 GB memory, our proposed algorithm can successfully compute a payoff division in the least core for the instances with up to 100 agents, but the naive algorithm fails due to a lack of memory for instances with 30 or more agents.


coalitional games least core MC-nets constraint generation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chalkiadakis, G., Elkind, E., Wooldridge, M.: Computational Aspects of Cooperative Game Theory. Morgan & Claypool Publishers (2011)Google Scholar
  2. 2.
    Conitzer, V., Sandholm, T.: Complexity of constructing solutions in the core based on synergies among coalitions. Artificial Intelligence 170, 607–619 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gillies, D.: Some Theorems on n-Person Games. Ph.D. thesis, Princeton University (1953)Google Scholar
  4. 4.
    Greco, G., Malizia, E., Palopoli, L., Scarcello, F.: On the complexity of core, kernel, and bargaining set. Artificial Intelligence 175(12–13), 1877–1910 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hillier, F.S., Lieberman, G.J.: Introduction to Operations Research, 9th edn. McGraw-Hill (2010)Google Scholar
  6. 6.
    Ieong, S., Shoham, Y.: Marginal contribution nets: A compact representation scheme for coalitional games. In: Proceedings of the 6th ACM Conference on Electronic Commerce (EC 2005), pp. 193–202 (2005)Google Scholar
  7. 7.
    Liao, X., Koshimura, M., Fujita, H., Hasegawa, R.: Solving the coalition structure generation problem with MaxSAT. In: Proceedings of the 2012 IEEE 24th International Conference on Tools with Artificial Intelligence (ICTAI 2012), pp. 910–915 (2012)Google Scholar
  8. 8.
    Malizia, E., Palopoli, L., Scarcello, F.: Infeasibility certificates and the complexity of the core in coalitional games. In: Proceedings of the 20th International Joint Conference on Artificial Intelligence (IJCAI 2007), pp. 1402–1407 (2007)Google Scholar
  9. 9.
    Sandholm, T.: Algorithm for optimal winner determination in combinatorial auctions. Artificial Intelligence 135(1–2), 1–54 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Shapley, L.S.: A value for n-person games. In: Contributions to the Theory of Games, vol. 2. Princeton University Press (1953)Google Scholar
  11. 11.
    Tombuş, Ö., Bilgiç, T.: A column generation approach to the coalition formation problem in multi-agent systems. Computers & Operations Research 31, 1635–1653 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Tran-Thanh, L., Nguyen, T.D., Rahwan, T., Rogers, A., Jennings, N.R.: An efficient vector-based representation for coalitional games. In: Proceedings of the 23rd International Joint Conference on Artificial Intelligence (IJCAI 2013), pp. 383–389 (2013)Google Scholar
  13. 13.
    Ueda, S., Hasegawa, T., Hashimoto, N., Ohta, N., Iwasaki, A., Yokoo, M.: Handling negative value rules in MC-net-based coalition structure generation. In: Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2012), pp. 795–802 (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Katsutoshi Hirayama
    • 1
  • Kenta Hanada
    • 1
  • Suguru Ueda
    • 2
  • Makoto Yokoo
    • 3
  • Atsushi Iwasaki
    • 4
  1. 1.Kobe UniversityJapan
  2. 2.National Institute of InformaticsJapan
  3. 3.Kyushu UniversityJapan
  4. 4.University of Electro-CommnunicationsJapan

Personalised recommendations