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Theory of Computing Systems

, Volume 63, Issue 1, pp 150–174 | Cite as

Analyzing Power in Weighted Voting Games with Super-Increasing Weights

  • Yuval FilmusEmail author
  • Joel Oren
  • Yair Zick
  • Yoram Bachrach
Article
  • 79 Downloads
Part of the following topical collections:
  1. Special Issue on Algorithmic Game Theory (SAGT 2016)

Abstract

Weighted voting games (WVGs) are a class of cooperative games that capture settings of group decision making in various domains, such as parliaments or committees. Earlier work has revealed that the effective decision making power, or influence of agents in WVGs is not necessarily proportional to their weight. This gave rise to measures of influence for WVGs. However, recent work in the algorithmic game theory community have shown that computing agent voting power is computationally intractable. In an effort to characterize WVG instances for which polynomial-time computation of voting power is possible, several classes of WVGs have been proposed and analyzed in the literature. One of the most prominent of these are super increasing weight sequences. Recent papers show that when agent weights are super-increasing, it is possible to compute the agents’ voting power (as measured by the Shapley value) in polynomial-time. We provide the first set of explicit closed-form formulas for the Shapley value for super-increasing sequences. We bound the effects of changes to the quota, and relate the behavior of voting power to a novel function. This set of results constitutes a complete characterization of the Shapley value in weighted voting games, and answers a number of open questions presented in previous work.

Keywords

Weighted voting games Shapley values 

Notes

Acknowledgements

Significant portions of the research presented in this work were done while Filmus and Oren were affiliated with the University of Toronto; Zick was affiliated with Nanyang Technological University and then with Carnegie Mellon University; Bachrach was affiliated with Microsoft Research, Cambridge. A preliminary version of this work was presented in SAGT 2016; the authors express their gratitude to the anonymous SAGT reviewers for their useful suggestions.

References

  1. 1.
    Aziz, H., Bachrach, Y., Elkind, E., Paterson, M.: False-name manipulations in weighted voting games. J. Artif. Intell. Res. 40, 57–93 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aziz, H., Paterson, M.: Computing voting power in easy weighted voting games. CoRR arXiv:0811.2497 (2008)
  3. 3.
    Bachrach, Y.: Honor among thieves: collusion in multi-unit auctions. In: Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems (AAMAS-10), pp. 617–624 (2010)Google Scholar
  4. 4.
    Bachrach, Y., Filmus, Y., Oren, J., Zick, Y.: A characterization of voting power for discrete weight distributions. In: Proceedings of the 25th International Joint Conference on Artificial Intelligence (IJCAI-16), pp. 74–80 (2016)Google Scholar
  5. 5.
    Bachrach, Y., Graepel, T., Kasneci, G., Kosinski, M., Gael, J.V.: Crowd IQ: aggregating opinions to boost performance. In: Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems (AAMAS-12), pp. 535–542 (2012)Google Scholar
  6. 6.
    Bachrach, Y., Lev, O., Lovett, S., Rosenschein, J.S., Zadimoghaddam, M.: Cooperative weakest link games. In: Proceedings of the 13th International Conference on Autonomous Agents and Multiagent Systems (AAMAS-14), pp. 589–596 (2014)Google Scholar
  7. 7.
    Bachrach, Y., Markakis, E., Resnick, E., Procaccia, A., Rosenschein, J., Saberi, A.: Approximating power indices: theoretical and empirical analysis. Auton. Agent. Multi-Agent Syst. 20(2), 105–122 (2010)Google Scholar
  8. 8.
    Bachrach, Y., Meir, R., Feldman, M., Tennenholtz, M.: Solving cooperative reliability games. CoRR arXiv:1202.3700 (2012)
  9. 9.
    Bachrach, Y., Parkes, D.C., Rosenschein, J.S.: Computing cooperative solution concepts in coalitional skill games. Artif. Intell. 204, 1–21 (2013)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bachrach, Y., Savani, R., Shah, N.: Cooperative max games and agent failures. In: Proceedings of the 13th International Conference on Autonomous Agents and Multiagent Systems (AAMAS-14), pp. 29–36 (2014)Google Scholar
  11. 11.
    Bachrach, Y., Shah, N.: Reliability weighted voting games. In: The 6th International Symposium on Algorithmic Game Theory (SAGT-13), pp. 38–49 (2013)Google Scholar
  12. 12.
    Bachrach, Y., Zuckerman, M., Wooldridge, M., Rosenschein, J.S.: Proof systems and transformation games. Ann. Math. Artif. Intell. 67(1), 1–30 (2013)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Banzhaf, J.: Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Rev. 19, 317 (1964)Google Scholar
  14. 14.
    Blocq, G., Bachrach, Y., Key, P.: The shared assignment game and applications to pricing in cloud computing. In: Proceedings of the 13th International Conference on Autonomous Agents and Multiagent Systems (AAMAS-14), pp. 605–612 (2014)Google Scholar
  15. 15.
    Bork, P., Jensen, L., von Mering, C., Ramani, A., Lee, I., Marcott, E.: Protein interaction networks from yeast to human. Curr. Opin. Struct. Biol. 14 (3), 292–299 (2004)Google Scholar
  16. 16.
    Chakravarty, N., Goel, A., Sastry, T.: Easy weighted majority games. Math. Soc. Sci. 40(2), 227–235 (2000)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Chalkiadakis, G., Elkind, E., Wooldridge, M.: Computational Aspects of Cooperative Game Theory. Morgan and Claypool (2011)Google Scholar
  18. 18.
    Chalkiadakis, G., Wooldridge, M.: Weighted voting games. In: Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A. (eds.) Handbook of Computational Social Choice, chap. 16. Cambridge University Press (2016)Google Scholar
  19. 19.
    Datta, A., Sen, S., Zick, Y.: Algorithmic transparency via quantitative input influence: theory and experiments with learning systems. In: Proceedings of The 37th IEEE Symposium on Security and Privacy (Oakland-16), pp. 598–617 (2016)Google Scholar
  20. 20.
    Deng, X., Papadimitriou, C.: On the complexity of cooperative solution concepts. Math. Oper. Res. 19(2), 257–266 (1994)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Dubey, P., Shapley, L.: Mathematical properties of the banzhaf power index. Math. Oper. Res. 4(2), 99–131 (1979)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Elkind, E., Chalkiadakis, G., Jennings, N.: Coalition structures in weighted voting games. In: Proceedings of the 18th European Conference on AI (ECAI-08), pp. 393–397 (2008)Google Scholar
  23. 23.
    Elkind, E., Goldberg, L., Goldberg, P., Wooldridge, M.: Computational complexity of weighted threshold games. In: Proceedings of the 22nd AAAI Conference on Artificial Intelligence (AAAI-07), pp. 718–723 (2007)Google Scholar
  24. 24.
    Faliszewski, P., Hemaspaandra, L.: The complexity of power-index comparison. Theor. Comput. Sci. 410(1), 101–107 (2009)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Fatima, S., Wooldridge, M., Jennings, N.: An approximation method for power indices for voting games. In: Proceedings of the 2nd International Workshop on Agent-Based Complex Automated Negotiations (ACAN-09), pp. 72–86 (2009)Google Scholar
  26. 26.
    Felsenthal, D., Machover, M.: The Measurement of Voting Power. Edward Elgar Publishing (1998)Google Scholar
  27. 27.
    Felsenthal, D., Machover, M.: Voting power measurement: a story of misreinvention. Soc. Choice Welfare 25(2), 485–506 (2005)zbMATHGoogle Scholar
  28. 28.
    Girvan, M., Newman, M.: Community structure in social and biological networks. Proc. Natl. Acad. Sci. USA 99(12), 7821–7826 (2002)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Jelnov, A., Tauman, Y.: Voting power and proportional representation of voters. Int. J. Game Theory 43(4), 747–766 (2014)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Keinan, A., Sandbank, B., Hilgetag, C., Meilijson, I., Ruppin, E.: Fair attribution of functional contribution in artificial and biological networks. Neural Comput. 16(9), 1887–1915 (2004)zbMATHGoogle Scholar
  31. 31.
    Lasisi, R., Allan, V.: New bounds on false-name manipulation in weighted voting games. In: Proceedings of the 27th International Florida Artificial Intelligence Research Society Conference (FLAIRS-14), pp. 57–62 (2014)Google Scholar
  32. 32.
    Lasisi, R.O., Allan, V.H.: Manipulation of weighted voting games via annexation and merging. In: Proceedings of the 4th International Conference on Agents and Artificial Intelligence (ICAART-12), pp. 364–378 (2012)Google Scholar
  33. 33.
    Leech, D.: Designing the voting system for the council of the european union. Publ. Choice 113(3-4), 437–464 (2002)Google Scholar
  34. 34.
    Leech, D.: Voting power in the governance of the international monetary fund. Ann. Oper. Res. 109(1–4), 375–397 (2002)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Leech, D.: Computing power indices for large voting games. Manag. Sci. 49 (6), 831–837 (2003)zbMATHGoogle Scholar
  36. 36.
    Lehrer, E.: An axiomatization of the Banzhaf value. Int. J. Game Theory 17 (2), 89–99 (1988)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Lindelauf, R., Hamers, H., Husslage, B.: Cooperative game theoretic centrality analysis of terrorist networks: The cases of Jemaah Islamiyah and Al Qaeda. Eur. J. Oper. Res. 229(1), 230–238 (2013)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Littlechild, S.C., Owen, G.: A simple expression for the shapely value in a special case. Manag. Sci. 20(3), 370–372 (1973)zbMATHGoogle Scholar
  39. 39.
    Machover, M.: Penrose’s square-root rule and the EU council of ministers: significance of the quota. Distribution of power and voting procedures in the EU (2007)Google Scholar
  40. 40.
    Maleki, S., Tran-Thanh, L., Hines, G., Rahwan, T., Rogers, A.: Bounding the estimation error of sampling-based shapley value approximation with/without stratifying. CoRR arXiv:1306.4265 (2013)
  41. 41.
    Mann, I., Shapley, L.S.: Values of large games, IV: evaluating the electoral college by Monte-Carlo techniques. Tech. rep. RAND Corporation (1960)Google Scholar
  42. 42.
    Mann, I., Shapley, L.S.: Values of large games, VI: evaluating the electoral college exactly. Tech. rep. RAND Corporation (1962)Google Scholar
  43. 43.
    Matsui, Y., Matsui, T.: N P-completeness for calculating power indices of weighted majority games. Theor. Comput. Sci. 263 (1–2), 305–310 (2001)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Merrill, S.: Approximations to the Banzhaf index. Amer. Math. Monthly 89, 108–110 (1982)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Michalak, T., Aaditha, K., Szczepanski, P., Ravindran, B., Jennings, N.: Efficient computation of the shapley value for game-theoretic network centrality. J. Artif. Intell. Res. 46, 607–650 (2013)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Michalak, T., Rahwan, T., Szczepanski, P., Skibski, O., Narayanam, R., Wooldridge, M., Jennings, N.: Computational analysis of connectivity games with applications to the investigation of terrorist networks. In: Proceedings of the 23rd International Joint Conference on Artificial Intelligence (IJCAI-13), pp. 293–301 (2013)Google Scholar
  47. 47.
    Peleg, B.: An axiomatization of the core of cooperative games without side payments. J. Math. Econ. 14(2), 203–214 (1985)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Peleg, B., Sudhölter, P.: Introduction to the Theory of Cooperative Games, Theory and Decision Library. Series C: Game Theory, Mathematical Programming and Operations Research, 2nd edn, vol. 34. Springer (2007)Google Scholar
  49. 49.
    Penrose, L.S.: The elementary statistics of majority voting. J. R. Stat. Soc. 109(1), 53–57 (1946)Google Scholar
  50. 50.
    del Pozo, M., Manuel, C., González-Arangüena, E., Owen, G.: Centrality in directed social networks. A game theoretic approach. Soc. Netw. 33(3), 191–200 (2011)zbMATHGoogle Scholar
  51. 51.
    Prasad, K., Kelly, J.S.: NP-completeness of some problems concerning voting games. Int. J. Game Theory 19, 1–9 (1990)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Rey, A., Rothe, J.: False-name manipulation in weighted voting games is hard for probabilistic polynomial time. J. Artif. Intell. Res. 50, 573–601 (2014)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Shapley, L.: A value for n-person games. In: Contributions to the Theory of Games, vol. 2, Annals of Mathematics Studies, no. 28, pp 307–317. Princeton University Press, Princeton (1953)Google Scholar
  54. 54.
    Shapley, L., Shubik, M.: A method for evaluating the distribution of power in a committee system. Am. Polit. Sci. Rev. 48(3), 787–792 (1954)Google Scholar
  55. 55.
    Słomczyński, W., Życzkowski, K.: Penrose voting system and optimal quota. Acta Phys. Pol. B 37(11), 3133–3143 (2006)Google Scholar
  56. 56.
    Snijders, C.: Axiomatization of the nucleolus. Math. Oper. Res. 20(1), 189–196 (1995)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Szczepański, P., Michalak, T., Rahwan, T.: Efficient algorithms for game-theoretic betweenness centrality. Artif. Intell. 231, 39–63 (2016)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Szczepański, P., Tarkowski, M., Michalak, T., Harrenstein, P., Wooldridge, M.: Efficient computation of semivalues for game-theoretic network centrality. In: Proceedings of the 29th AAAI Conference on Artificial Intelligence (AAAI-15), pp. 461–469 (2015)Google Scholar
  59. 59.
    Winter, E.: The shapley value. Handbook Game Theory Econ. Appl. 3, 2025–2054 (2002)Google Scholar
  60. 60.
    Young, H.: Monotonic solutions of cooperative games. Int. J. Game Theory 14(2), 65–72 (1985)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Zick, Y.: On random quotas and proportional representation in weighted voting games. In: Proceedings of the 23rd International Joint Conference on Artificial Intelligence (IJCAI-13), pp. 432–438 (2013)Google Scholar
  62. 62.
    Zick, Y., Skopalik, A., Elkind, E.: The shapley value as a function of the quota in weighted voting games. In: Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI-11), pp. 490–495 (2011)Google Scholar
  63. 63.
    Zuckerman, M., Faliszewski, P., Bachrach, Y., Elkind, E.: Manipulating the quota in weighted voting games. Artif. Intell. 180–181, 1–19 (2012)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Technion — Israel Institute of TechnologyHaifaIsrael
  2. 2.Yahoo! ResearchHaifaIsrael
  3. 3.National University of SingaporeSingaporeSingapore
  4. 4.DeepMindLondonUK

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