WINE 2014: Web and Internet Economics pp 106-119 | Cite as
The Shapley Value in Knapsack Budgeted Games
Abstract
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.
Keywords
Cooperative Game Marginal Contribution Greedy Heuristic Coalitional Game Algorithmic RepresentationPreview
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References
- 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.Aziz, H., Sorensen, T.B.: Path coalitional games. In: CoopMAS 2011 (2011)Google Scholar
- 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.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.Bachrach, Y., Porat, E.: Path disruption games. In: AAMAS 2010 (2010)Google Scholar
- 6.Bhagat, S., Kim, A., Muthukrishnan, S., Weinsberg, U.: The shapley value in knapsack budgeted games. arXiv:1409.5200 (2014)Google Scholar
- 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.Deng, X., Papadimitriou, C.H.: On the complexity of cooperative solution concepts. Mathematics of Operations Research 19(2) (1994)Google Scholar
- 9.Faigle, U., Kern, W.: On some approximately balanced combinatorial cooperative games. Zeitschrift für Operations Research 38(2) (1993)Google Scholar
- 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.Ieong, S., Shoham, Y.: Marginal contribution nets: A compact representation scheme for coalitional games. In: EC 2005 (2005)Google Scholar
- 12.Ieong, S., Shoham, Y.: Multi-attribute coalitional games. In: EC 2006 (2006)Google Scholar
- 13.Kuipers, J.: Bin packing games. Mathematical Methods of Operations Research 47(3) (1998)Google Scholar
- 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.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.Matsui, Y., Matsui, T.: Np-completeness for calculating power indices of weighted majority games. Theoretical Computer Science (2001)Google Scholar
- 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.Misra, V., Ioannidis, S., Chaintreau, A., Massoulié, L.: Incentivizing peer-assisted services: A fluid shapley value approach. In: SIGMETRICS 2010 (2010)Google Scholar
- 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.Qiu, X.: Bin packing games. Master’s thesis, University of Twente (2010)Google Scholar
- 21.Shapley, L.S.: A value for n-person games. Contributions to the Theory of Games 2, 307–317 (1953)MathSciNetGoogle Scholar
- 22.Vazirani, V.V.: Approximation Algorithms. Springer-Verlag New York, Inc., New York (2001)Google Scholar