A randomized method for the shapley value for the voting game

ABSTRACT

The Shapley value is one of the key solution concepts for coalition games. Its main advantage is that it provides a unique and fair solution, but its main problem is that, for many coalition games, the Shapley value cannot be determined in polynomial time. In particular, the problem of finding this value for the voting game is known to be #P-complete in the general case. However, in this paper, we show that there are some specific voting games for which the problem is computationally tractable. For other general voting games, we overcome the problem of computational complexity by presenting a new randomized method for determining the approximate Shapley value. The time complexity of this method is linear in the number of players. We also show, through empirical studies, that the percentage error for the proposed method is always less than 20% and, in most cases, less than 5%.

REFERENCES

• 1: R. Aumann. Acceptable points in general cooperative n-person games. In Contributions to the Theory of Games volume IV. Princeton University Press, 1959.
• 2: Giorgio Ausiello , M. Protasi , A. Marchetti-Spaccamela , G. Gambosi , P. Crescenzi , V. Kann, Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties, Springer-Verlag New York, Inc., Secaucus, NJ, 1999
• 3: J. M. Bilbao, J. R. Fernandez, A. J. Losada, and J. J. Lopez. Generating functions for computing power indices efficiently. Top 8, 2:191--213, 2000.
• 4: P. Bork, H. Grote, D. Notz, and M. Regler. Data Analysis Techniques in High Energy Physics Experiments. Cambridge University Press, 1993.
• 5: V. Conitzer and T. Sandholm. Computing Shapley values, manipulating value division schemes, and checking core membership in multi-issue domains. In Proceedings of the National Conference on Artificial Intelligence, pages 219--225, San Jose, California, 2004.
• 6: Xiaotie Deng , Christos H. Papadimitriou, On the complexity of cooperative solution concepts, Mathematics of Operations Research, v.19 n.2, p.257-266, May 1994 [doi>10.1287/moor.19.2.257]
• 7: S. S. Fatima, M. Wooldridge, and N. R. Jennings. An analysis of the shapley value and its uncertainty for the voting game. In Proc. 7th Int. Workshop on Agent Mediated Electronic Commerce, pages 39--52, 2005.
• 8: A. Francis. Advanced Level Statistics. Stanley Thornes Publishers, 1979.
• 9: Samuel Ieong , Yoav Shoham, Marginal contribution nets: a compact representation scheme for coalitional games, Proceedings of the 6th ACM conference on Electronic commerce, p.193-202, June 05-08, 2005, Vancouver, BC, Canada [doi>10.1145/1064009.1064030]
• 10: Samuel Ieong , Yoav Shoham, Multi-attribute coalitional games, Proceedings of the 7th ACM conference on Electronic commerce, p.170-179, June 11-15, 2006, Ann Arbor, Michigan, USA [doi>10.1145/1134707.1134726]
• 11: J. P. Kahan and A. Rapoport. Theories of Coalition Formation. Lawrence Erlbaum Associates Publishers, 1984.
• 12: I. Mann and L. S. Shapley. Values for large games iv: Evaluating the electoral college exactly. Technical report, The RAND Corporation, Santa Monica, 1962.
• 13: A. MasColell, M. Whinston, and J. R. Green. Microeconomic Theory. Oxford University Press, 1995.
• 14: M. J. Osborne and A. Rubinstein. A Course in Game Theory. The MIT Press, 1994.
• 15: C. H. Papadimitriou. Computational Complexity. Addison Wesley Longman, 1994.
• 16: A. Rapoport. N-person Game Theory: Concepts and Applications. Dover Publications, Mineola, NY, 2001.
• 17: A. E. Roth. Introduction to the shapley value. In A. E. Roth, editor, The Shapley value, pages 1--27. University of Cambridge Press, Cambridge, 1988.
• 18: Tuomas W. Sandholm , Victor R. Lesser, Coalitions among computationally bounded agents, Artificial Intelligence, v.94 n.1-2, p.99-137, July 1997 [doi>10.1016/S0004-3702(97)00030-1]
• 19: L. S. Shapley. A value for n person games. In A. E. Roth, editor, The Shapley value, pages 31--40. University of Cambridge Press, Cambridge, 1988.
• 20: O. Shehory and S. Kraus. A kernel-oriented model for coalition-formation in general environments: Implemetation and results. In In Proceedings of the National Conference on Artificial Intelligence (AAAI-96), pages 131--140, 1996.
• 21: Onn Shehory , Sarit Kraus, Methods for task allocation via agent coalition formation, Artificial Intelligence, v.101 n.1-2, p.165-200, May 1998 [doi>10.1016/S0004-3702(98)00045-9]
• 22: J. R. Taylor. An introduction to error analysis: The study of uncertainties in physical measurements. University Science Books, 1982.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.2 ARTIFICIAL INTELLIGENCE
        I.2.11 Distributed Artificial Intelligence

General Terms:
Algorithms, Design, Theory

Keywords:
Shapley value, approximation, coalition formation, game-theory