The Shapley Value in Knapsack Budgeted Games

  • Smriti Bhagat
  • Anthony Kim
  • S. Muthukrishnan
  • Udi Weinsberg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8877)


We propose the study of computing the Shapley value for a new class of cooperative games that we call budgeted games, and investigate in particular knapsack budgeted games, a version modeled after the classical knapsack problem. In these games, the “value” of a set S of agents is determined only by a critical subset T ⊆ S of the agents and not the entirety of S due to a budget constraint that limits how large T can be. We show that the Shapley value can be computed in time faster than by the naïve exponential time algorithm when there are sufficiently many agents, and also provide an algorithm that approximates the Shapley value within an additive error. For a related budgeted game associated with a greedy heuristic, we show that the Shapley value can be computed in pseudo-polynomial time. Furthermore, we generalize our proof techniques and propose what we term algorithmic representation framework that captures a broad class of cooperative games with the property of efficient computation of the Shapley value. The main idea is that the problem of determining the efficient computation can be reduced to that of finding an alternative representation of the games and an associated algorithm for computing the underlying value function with small time and space complexities in the representation size.


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  1. 1.
    Aadithya, K.V., Michalak, T.P., Jennings, N.R.: Representation of coalitional games with algebraic decision diagrams. In: AAMAS 2011 (2011)Google Scholar
  2. 2.
    Aziz, H., Sorensen, T.B.: Path coalitional games. In: CoopMAS 2011 (2011)Google Scholar
  3. 3.
    Bachrach, Y., Lev, O., Lovett, S., Rosenschein, J.S., Zadimoghaddam, M.: Cooperative weakest link games. In: AAMAS 2014 (to appear, 2014)Google Scholar
  4. 4.
    Bachrach, Y., Markakis, E., Resnick, E., Procaccia, A.D., Rosenschein, J.S., Saberi, A.: Approximating power indices: Theoretical and empirical analysis. In: Autonomous Agents and Multi-Agent Systems (March 2010)Google Scholar
  5. 5.
    Bachrach, Y., Porat, E.: Path disruption games. In: AAMAS 2010 (2010)Google Scholar
  6. 6.
    Bhagat, S., Kim, A., Muthukrishnan, S., Weinsberg, U.: The shapley value in knapsack budgeted games. arXiv:1409.5200 (2014)Google Scholar
  7. 7.
    Conitzer, V., Sandholm, T.: Computing shapley values, manipulating value division schemes, and checking core membership in multi-issue domains. In: AAAI 2004 (2004)Google Scholar
  8. 8.
    Deng, X., Papadimitriou, C.H.: On the complexity of cooperative solution concepts. Mathematics of Operations Research 19(2) (1994)Google Scholar
  9. 9.
    Faigle, U., Kern, W.: On some approximately balanced combinatorial cooperative games. Zeitschrift für Operations Research 38(2) (1993)Google Scholar
  10. 10.
    Fatima, S.S., Wooldridge, M., Jennings, N.R.: A linear approximation method for the shapley value. Artificial Intelligence 172(14) (2008)Google Scholar
  11. 11.
    Ieong, S., Shoham, Y.: Marginal contribution nets: A compact representation scheme for coalitional games. In: EC 2005 (2005)Google Scholar
  12. 12.
    Ieong, S., Shoham, Y.: Multi-attribute coalitional games. In: EC 2006 (2006)Google Scholar
  13. 13.
    Kuipers, J.: Bin packing games. Mathematical Methods of Operations Research 47(3) (1998)Google Scholar
  14. 14.
    Ma, R.T., Chiu, D., Lui, J.C., Misra, V., Rubenstein, D.: Internet economics: The use of shapley value for isp settlement. In: CoNEXT 2007 (2007)Google Scholar
  15. 15.
    Matsui, T., Matsui, Y.: A survey of algorithms for calculating power indices of weighted majority games. J. Oper. Res. Soc. Japan (2000)Google Scholar
  16. 16.
    Matsui, Y., Matsui, T.: Np-completeness for calculating power indices of weighted majority games. Theoretical Computer Science (2001)Google Scholar
  17. 17.
    Michalak, T.P., Aadithya, K.V., Szczepanski, P.L., Ravindran, B., Jennings, N.R.: Efficient computation of the shapley value for game-theoretic network centrality. J. Artif. Int. Res. (January 2013)Google Scholar
  18. 18.
    Misra, V., Ioannidis, S., Chaintreau, A., Massoulié, L.: Incentivizing peer-assisted services: A fluid shapley value approach. In: SIGMETRICS 2010 (2010)Google Scholar
  19. 19.
    Narayanam, R., Narahari, Y.: A shapley value-based approach to discover influential nodes in social networks. IEEE Transactions on Automation Science and Engineering 8(1), 130–147 (2011)CrossRefGoogle Scholar
  20. 20.
    Qiu, X.: Bin packing games. Master’s thesis, University of Twente (2010)Google Scholar
  21. 21.
    Shapley, L.S.: A value for n-person games. Contributions to the Theory of Games 2, 307–317 (1953)MathSciNetGoogle Scholar
  22. 22.
    Vazirani, V.V.: Approximation Algorithms. Springer-Verlag New York, Inc., New York (2001)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Smriti Bhagat
    • 1
  • Anthony Kim
    • 2
  • S. Muthukrishnan
    • 3
  • Udi Weinsberg
    • 1
  1. 1.Technicolor ResearchLos AltosUSA
  2. 2.Department of Computer ScienceStanford UniversityStanfordUSA
  3. 3.Department of Computer ScienceRutgers UniversityPiscatawayUSA

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