ICALP 2012: Automata, Languages, and Programming pp 266-277

The Inverse Shapley Value Problem

• Anindya De
• Ilias Diakonikolas
• Rocco Servedio
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

For f a weighted voting scheme used by n voters to choose between two candidates, the n Shapley-Shubik Indices (or Shapley values) of f provide a measure of how much control each voter can exert over the overall outcome of the vote. Shapley-Shubik indices were introduced by Lloyd Shapley and Martin Shubik in 1954 [SS54] and are widely studied in social choice theory as a measure of the “influence” of voters. The Inverse Shapley Value Problem is the problem of designing a weighted voting scheme which (approximately) achieves a desired input vector of values for the Shapley-Shubik indices. Despite much interest in this problem no provably correct and efficient algorithm was known prior to our work.

We give the first efficient algorithm with provable performance guarantees for the Inverse Shapley Value Problem. For any constant ε > 0 our algorithm runs in fixed poly(n) time (the degree of the polynomial is independent of ε) and has the following performance guarantee: given as input a vector of desired Shapley values, if any “reasonable” weighted voting scheme (roughly, one in which the threshold is not too skewed) approximately matches the desired vector of values to within some small error, then our algorithm explicitly outputs a weighted voting scheme that achieves this vector of Shapley values to within error ε. If there is a “reasonable” voting scheme in which all voting weights are integers at most poly(n) that approximately achieves the desired Shapley values, then our algorithm runs in time poly(n) and outputs a weighted voting scheme that achieves the target vector of Shapley values to within error ε = n − 1/8.

Keywords

Boolean Function Power Index Vote Scheme Full Version Performance Guarantee
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. [APL07]
Aziz, H., Paterson, M., Leech, D.: Efficient algorithm for designing weighted voting games. In: IEEE Intl. Multitopic Conf., pp. 1–6 (2007)Google Scholar
2. [Ban65]
Banzhaf, J.: Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review 19, 317–343 (1965)Google Scholar
3. [BKS99]
Benjamini, I., Kalai, G., Schramm, O.: Noise sensitivity of Boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math. 90, 5–43 (1999)
4. [BMR+10]
Bachrach, Y., Markakis, E., Resnick, E., Procaccia, A., Rosenschein, J., Saberi, A.: Approximating power indices: theoretical and empirical analysis. Autonomous Agents and Multi-Agent Systems 20(2), 105–122 (2010)
5. [Cho61]
Chow, C.K.: On the characterization of threshold functions. In: Proc. 2nd FOCS 1961, pp. 34–38 (1961)Google Scholar
6. [DDFS12]
De, A., Diakonikolas, I., Feldman, V., Servedio, R.: Near-optimal solutions for the Chow Parameters Problem and low-weight approximation of halfspaces. To appear in STOC (2012)Google Scholar
7. [dK08]
de Keijzer, B.: A survey on the computation of power indices (2008), http://www.st.ewi.tudelft.nl/~tomas/theses/DeKeijzerSurvey.pdf
8. [dKKZ10]
de Keijzer, B., Klos, T., Zhang, Y.: Enumeration and exact design of weighted voting games. In: AAMAS 2010, pp. 391–398 (2010)Google Scholar
9. [DP78]
Deegan, J., Packel, E.: A new index of power for simple n-person games. International Journal of Game Theory 7, 113–123 (1978)
10. [EGGW07]
Elkind, E., Goldberg, L.A., Goldberg, P.W., Wooldridge, M.: Computational complexity of weighted voting games. In: AAAI 2007, pp. 718–723 (2007)Google Scholar
11. [FWJ08]
Fatima, S., Wooldridge, M., Jennings, N.: An Anytime Approximation Method for the Inverse Shapley Value Problem. In: AAMAS 2008, pp. 935–942 (2008)Google Scholar
12. [Gol06]
Goldberg, P.: A Bound on the Precision Required to Estimate a Boolean Perceptron from its Average Satisfying Assignment. SIDMA 20, 328–343 (2006)
13. [Hol82]
Holler, M.J.: Forming coalitions and measuring voting power. Political Studies 30, 262–271 (1982)
14. [Imp95]
Impagliazzo, R.: Hard-core distributions for somewhat hard problems. In: Proc. 36th FOCS 1995, pp. 538–545 (1995)Google Scholar
15. [KS06]
Kalai, G., Safra, S.: Threshold phenomena and influence. In: Computational Complexity and Statistical Physics, pp. 25–60. Oxford University Press (2006)Google Scholar
16. [Kur11]
Kurz, S.: On the inverse power index problem. Optimization (2011), doi:10.1080/02331934.2011.587008Google Scholar
17. [Lee03]
Leech, D.: Computing power indices for large voting games. Management Science 49(6) (2003)Google Scholar
18. [MTT61]
Muroga, S., Toda, I., Takasu, S.: Theory of majority switching elements. J. Franklin Institute 271, 376–418 (1961)
19. [OS08]
O’Donnell, R., Servedio, R.: The Chow Parameters Problem. In: Proc. 40th STOC 2008, pp. 517–526 (2008)Google Scholar
20. [Owe72]
Owen, G.: Multilinear extensions of games. Management Science 18(5), 64–79 (1972); Part 2, Game theory and GamingGoogle Scholar
21. [Rot88]
Roth, A.E. (ed.): The Shapley value. University of Cambridge Press (1988)Google Scholar
22. [SS54]
Shapley, L., Shubik, M.: A Method for Evaluating the Distribution of Power in a Committee System. American Political Science Review 48, 787–792 (1954)
23. [TTV08]
Trevisan, L., Tulsiani, M., Vadhan, S.: Regularity, Boosting and Efficiently Simulating every High Entropy Distribution. Technical Report 103, ECCC, 2008. Conference version in Proc. CCC (2009)Google Scholar
24. [ZFBE08]
Zuckerman, M., Faliszewski, P., Bachrach, Y., Elkind, E.: Manipulating the quota in weighted voting games. In: AAAI, pp. 215–220 (2008)Google Scholar

Authors and Affiliations

• Anindya De
• 1
• Ilias Diakonikolas
• 1
• Rocco Servedio
• 2
1. 1.UC BerkeleyUSA
2. 2.Columbia UniversityUSA