Summary
The Shapley value and Banzhaf index are two well known indices for measuring the power a player has in a voting game. However, the problem of computing these indices is computationally hard. To overcome this problem, we analyze approximation methods for computing these indices. Although these methods have polynomial time complexity, finding an approximate Shapley value using them is easier than finding an approximate Banzhaf index. We also find the absolute error for the methods and show that this error for the Shapley value is lower than that for the Banzhaf index.
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References
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and approximation: Combinatorial optimization problems and their approximability properties. Springer, Heidelberg (2003)
Banzhaf, J.F.: Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review 19, 317–343 (1965)
Bork, P.V., Grote, H., Notz, D., Regler, M.: Data Analysis Techniques in High Energy Physics Experiments. Cambridge University Press, Cambridge (1993)
Conitzer, V., Sandholm, T.: Complexity of constructing solutions in the core based on synergies among coalitions. Artificial Intelligence Journal 170, 607–619 (2006)
Deng, X., Papadimitriou, C.H.: On the complexity of cooperative solution concepts. Mathematics of Operations Research 19(2), 257–266 (1994)
Elkind, E., Goldberg, L.A., Goldberg, P., Wooldridge, M.: Computational complexity of weighted threshold games. In: Proceedings of the National Conference on Artificial Intelligence (AAAI 2007) (2007)
Fatima, S.S., Wooldridge, M., Jennings, N.R.: A randomized method for the Shapley value for the voting game. In: Proceedings of the Sixth International Conference on Autonomous Agents and Multi-Agent Systems, pp. 955–962 (2007)
Francis, A.: Advanced Level Statistics. Stanley Thornes Publishers (1979)
Mann, I., Shapley, L.S.: Values for large games iv: Evaluating the electoral college by monte carlo techniques. Technical report, The RAND Corporation, Santa Monica (1960)
Osborne, M.J., Rubinstein, A.: A Course in Game Theory. The MIT Press, Cambridge (1994)
Owen, G.: Multilinear extensions of games. Management Science 18(5), 64–79 (1972)
Prasad, K., Kelly, J.S.: NP-completeness of some problems concerning voting games. International Journal of Game Theory 19, 1–9 (1990)
Rahwan, T., Jennings, N.R.: An algorithm for distributing coalitional value calculations among cooperating agents. Artificial Intelligence Journal 171, 535–567 (2007)
Sandholm, T., Lesser, V.: Coalitions among computationally bounded agents. Artificial Intelligence Journal 94(1), 99–137 (1997)
Shapley, L.S.: A value for n person games. In: Roth, A.E. (ed.) The Shapley value, pp. 31–40. University of Cambridge Press, Cambridge (1988)
Shehory, O., Kraus, S.: A kernel-oriented model for coalition-formation in general environments: Implemetation and results. In: Proceedings of the National Conference on Artificial Intelligence (AAAI 1996), pp. 131–140 (1996)
Shehory, O., Kraus, S.: Methods for task allocation via agent coalition formation. Artificial Intelligence Journal 101(2), 165–200 (1998)
Taylor, J.R.: An introduction to error analysis: The study of uncertainties in physical measurements. University Science Books (1982)
Zlotkin, G., Rosenschein, J.: Coalition, cryptography, and stability: mechanisms foe coalition formation in task oriented domains. In: Proceedings of the National Conference on Artificial Intelligence (AAAI 1994), pp. 432–437 (1994)
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Fatima, S.S., Wooldridge, M., Jennings, N.R. (2010). An Approximation Method for Power Indices for Voting Games. In: Ito, T., Zhang, M., Robu, V., Fatima, S., Matsuo, T., Yamaki, H. (eds) Innovations in Agent-Based Complex Automated Negotiations. Studies in Computational Intelligence, vol 319. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15612-0_10
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DOI: https://doi.org/10.1007/978-3-642-15612-0_10
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