Social Choice and Welfare

, Volume 47, Issue 1, pp 65–88 | Cite as

The inverse problem for power distributions in committees

  • Sascha Kurz
Original Paper


Several power indices have been introduced in the literature in order to measure the influence of individual committee members on an aggregated decision. Here we ask the inverse question and aim to design voting rules for a committee such that a given desired power distribution is met as closely as possible. We generalize the approach of Alon and Edelman who studied power distributions for the Banzhaf index, where most of the power is concentrated on few coordinates. It turned out that each Banzhaf vector of an n-member committee that is near to such a desired power distribution, also has to be near to the Banzhaf vector of a k-member committee. We show that such Alon-Edelman type results exist for other power indices like e.g. the Public Good index or the Coleman index to prevent actions, while such results are principally impossible to derive for e.g. the Johnston index.

Mathematics Subject Classification

91B12 94C10 



We would like to thank two anonymous referees for their very valuable and extensive comments on an earlier draft of this paper.


  1. Alon N, Edelman P (2010) The inverse Banzhaf problem. Soc Choice Welf 34(3):371–377CrossRefGoogle Scholar
  2. Alonso-Meijide J, Freixas J (2010) A new power index based on minimal winning coalitions without any surplus. Decis Support Syst 49(1):70–76CrossRefGoogle Scholar
  3. Alonso-Meijide J, Freixas J, Molinero X (2012) Computation of several power indices by generating functions. Appl Math Comput 219(8):3395–3402Google Scholar
  4. Banzhaf J (1965) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19:317–343Google Scholar
  5. Bertini C, Freixas J, Gambarelli G, Stach I (2013) Comparing power indices. Int. Game Theory Rev 15(2)Google Scholar
  6. Bertini C, Gambarelli G, Stach I (2008) A public help index. In: Braham M, Steffen F (eds) Power, freedom, and voting. Springer, Berlin, pp 83–98Google Scholar
  7. Bolger E (1986) Power indices for multicandidate voting games. Internat J Game Theory 15(3):175–186CrossRefGoogle Scholar
  8. Braham Me, Steffen Fe (2008) Power, freedom, and voting. In: Essays in honour of Manfred J. Holler. Papers presented at the Festschrift conference, Hamburg, August 17–20, 2006. Springer, Berlin, p 438 (xiv)Google Scholar
  9. Brams S, Kilgour D, Affuso P (1989) Presidential power: a game-theoretic analysis. In: Brace P, Harrington C, King G (eds) The presidency in american politics. New York University Press, pp 55–74Google Scholar
  10. Chang P-L, Chua V, Machover M (2006) LS Penrose’s limit theorem: tests by simulation. Math Soc Sci 51(1):90–106CrossRefGoogle Scholar
  11. Coleman J (1971) Control of collectivities and the power of a collectivity to act. In: Lieberman B (ed) Social Choice. Gordon and Breach, pp 269–300Google Scholar
  12. De A, Diakonikolas I, Feldman V, Servedio R (2012a) Nearly optimal solutions for the chow parameters problem and low-weight approximation of halfspaces. In: Proceedings of the 44th symposium on theory of computing, STOC ’12, pp 729–746. ACM, New YorkGoogle Scholar
  13. De A, Diakonikolas I, Servedio R (2012b) The inverse Shapley value problem. In: Automata, Languages, and Programming, pp 266–277. SpringerGoogle Scholar
  14. de Keijzer B, Klos T, Zhang Y (2010) Enumeration and exact design of weighted voting games. In: Proceedings of the 9th international conference on autonomous agents and multiagent systems, vol 1, pp 391–398. International Foundation for Autonomous Agents and Multiagent SystemsGoogle Scholar
  15. de Keijzer B, Klos T, Zhang Y (2014) Finding optimal solutions for voting game design problems. J Artif Intell Res 50:105–140Google Scholar
  16. Deegan J Jr, Packel E (1978) A new index of power for simple \(n\)-person games. Internat J Game Theory 7(2):113–123CrossRefGoogle Scholar
  17. Dragan I (2005) On the inverse problem for semivalues of cooperative TU games. Int J Pure Appl Math 22(4):545–561Google Scholar
  18. Dragan I (2012) On the inverse problem for multiweighted Shapley values of cooperative TU games. Int J Pure Appl Math 75(3):279–287Google Scholar
  19. Dragan I (2013) The inverse problem for binomial semivalues of cooperative TU games. In: Petrosyan LA, Zenkevich NA (eds) Game theory and management. Proceedings of the seventh international conference game theory and management. SPb.: Graduate School of Management SPbU, 2013, p 274, vol 26, p 72Google Scholar
  20. Dubey P, Neyman A, Weber R (1981) Value theory without efficiency. Math Oper Res 6(1):122–128CrossRefGoogle Scholar
  21. Dubey P, Shapley L (1979) Mathematical properties of the Banzhaf power index. Math Oper Res 4(2):99–131CrossRefGoogle Scholar
  22. Felsenthal D, Machover M (1998) The measurement of voting power: Theory and practice, problems and paradoxes, vol xviii, p 322. Edward Elgar, CheltenhamGoogle Scholar
  23. Felsenthal D, Machover M (2005) Voting power measurement: a story of misreinvention. Soc Choice Welf 25(2–3):485–506CrossRefGoogle Scholar
  24. Freixas J, Kaniovski S (2014) The minimum sum representation as an index of voting power. Eur J Oper Res 233(3):739–748CrossRefGoogle Scholar
  25. Freixas J, Kurz S (2014) On \(\alpha \)-roughly weighted games. Internat J Game Theory 43(3):659–692CrossRefGoogle Scholar
  26. Freixas J, Zwicker W (2003) Weighted voting, abstention, and multiple levels of approval. Soc Choice Welf 21(3):399–431CrossRefGoogle Scholar
  27. Freixas J, Zwicker W (2009) Anonymous yes-no voting with abstention and multiple levels of approval. Games Econ Behav 67(2):428–444CrossRefGoogle Scholar
  28. Gvozdeva T, Hemaspaandra L, Slinko A (2013) Three hierarchies of simple games parameterized by “resource” parameters. Internat J Game Theory 42(1):1–17CrossRefGoogle Scholar
  29. Holler M (1982) Forming coalitions and measuring voting power. Polit Stud 30(2):262–271CrossRefGoogle Scholar
  30. Imrie R (1973) The impact of the weighted vote on representation in municipal governing bodies of New York State. Ann New York Acad Sci 219(1):192–199CrossRefGoogle Scholar
  31. Isbell J (1958) A class of simple games. Duke Math J 25:423–439CrossRefGoogle Scholar
  32. Johnston R (1978) On the measurement of power: some reactions to Laver. Environ Plann A 10(8):907–914CrossRefGoogle Scholar
  33. König T, Bräuninger T (2001) Decisiveness and inclusiveness: Intergovernmental choice of European decision rules. In: Holler MJ, Owen G (eds) Power indices and coalition formation. Kluwer, pp 273–290Google Scholar
  34. Kurz S (2012a) On minimum sum representations for weighted voting games. Ann Oper Res 196(1):361–369CrossRefGoogle Scholar
  35. Kurz S (2012b) On the inverse power index problem. Optimization 61(8):989–1011CrossRefGoogle Scholar
  36. Kurz S (2014) Measuring voting power in convex policy spaces. Economies 2(1):45–77CrossRefGoogle Scholar
  37. Kurz S, Napel S (2014) Heuristic and exact solutions to the inverse power index problem for small voting bodies. Ann Oper Res 215(1):137–163CrossRefGoogle Scholar
  38. Kurz S, Napel S, Nohn A (2014) The nucleolus of large majority games. Econ Lett 123(3):139–143CrossRefGoogle Scholar
  39. Laruelle A, Valenciano F (2011) Voting and collective decision-making. Bargaining and power. Reprint of the 2008 hardback ed., vol xvii, p 184. Cambridge University Press, CambridgeGoogle Scholar
  40. Laruelle A, Valenciano F (2013) Voting and power. In: Power, Voting, and voting power: 30 years after, pp 137–149. SpringerGoogle Scholar
  41. Laruelle A, Widgrén M (1998) Is the allocation of voting power among EU states fair? Public Choice 94(3–4):317–339CrossRefGoogle Scholar
  42. Le Breton M, Montero M, Zaporozhets V (2012) Voting power in the EU Council of Ministers and fair decision making in distributive politics. Math Soc Sci 63(2):159–173CrossRefGoogle Scholar
  43. Lindner I, Machover M (2004) LS Penrose’s limit theorem: proof of some special cases. Math Soc Sci 47(1):37–49CrossRefGoogle Scholar
  44. Lindner I, Owen G (2007) Cases where the Penrose limit theorem does not hold. Math Soc Sci 53(3):232–238CrossRefGoogle Scholar
  45. Malawski M (2004) “Counting” power indices for games with a priori unions. Theory Decis 56(1–2):125–140CrossRefGoogle Scholar
  46. Milnor J, Shapley L (1978) Values of large games II: oceanic games. Math Oper Res 3(4):290–307CrossRefGoogle Scholar
  47. Neyman A (1982) Renewal theory for sampling without replacement. Ann Prob, pp 464–481Google Scholar
  48. Nurmi H (1980) Game theory and power indices. Zeitschrift für Nationalökonomie 40(1–2):35–58CrossRefGoogle Scholar
  49. Nurmi H (1982) The problem of the right distribution of voting power. In: Power, voting, and voting power, pp 203–212. SpringerGoogle Scholar
  50. O’Donnell R, Servedio R (2011) The chow parameters problem. SIAM J Comput 40(1):165–199CrossRefGoogle Scholar
  51. Papayanopoulos L (1983) On the partial construction of the semi-infinite Banzhaf polyhedron. In: Fiacco A, Kortanek K (eds) Semi-infinite programming and applications, lecture notes in economics and mathematical systems, vol 215, pp 208–218. Springer, Berlin, HeidelbergGoogle Scholar
  52. Penrose L (1946) The elementary statistics of majority voting. J R Stat Soc 109:53–57CrossRefGoogle Scholar
  53. Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM J Appl Math 17:1163–1170CrossRefGoogle Scholar
  54. Shapiro N, Shapley L (1978) Values of large games, I: a limit theorem. Math Oper Res 3(1):1–9CrossRefGoogle Scholar
  55. Shapley L (1953) A value for \(n\)-person games. Contrib. Theory of Games. Ann Math StudGoogle Scholar
  56. Shapley L, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48(03):787–792CrossRefGoogle Scholar
  57. Straffin P (1977) Homogeneity, independence, and power indices. Public Choice 30(1):107–118CrossRefGoogle Scholar
  58. Taylor A, Zwicker W (1999) Simple games, desirability relations, trading, pseudoweightings. Princeton University Press, PrincetonGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Physics, and Computer ScienceUniversity of BayreuthBayreuthGermany

Personalised recommendations