Network Shapley-Shubik Power Index: Measuring Indirect Influence in Shareholding Networks

  • Takayuki Mizuno
  • Shohei DoiEmail author
  • Shuhei Kurizaki
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 882)


Extending the Shapley-Shubik power index to networks, we propose a new measure and numerical method to calculate the indirect influence of investors on companies: Network power index (NPI). While the original index, reflecting the characteristics of majority vote in a shareholders meeting, measures the direct voting power of a shareholder, NPI captures not only an investor’s direct influence over a company but also indirect influence over this company’s subsidiary. Since NPI is often incalculable in a large network, we present a new method to numerically compute NPI: label propagation. Applying this method to the global shareholding networks in 2016, we find NPIs and raw vote shares dramatically diverge for some investors and this discrepancy suggests the difference in investment strategies between governments and private financial institutions.


The Shapley-Shubik power index Complex networks analysis Indirect control Shareholding networks 



This work was partially supported by JSPS KAKENHI Grant Numbers 18H03627 and 16H05904.


  1. 1.
    Bureau van Dijk’s Orbis. Accessed 27 Apr 2019
  2. 2.
    Banzhaf III, J.F.: Weighted voting doesn’t work: a mathematical analysis. Rutgers L. Rev. 19, 317 (1964)Google Scholar
  3. 3.
    Bertini, C., Mercik, J., Stach, I.: Indirect control and power. Oper. Res. Decisions 2, 7–30 (2016)MathSciNetGoogle Scholar
  4. 4.
    Crama, Y., Leruth, L.: Control and voting power in corporate networks: concepts and computational aspects. Eur. J. Oper. Res. 178(3), 879–893 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Crama, Y., Leruth, L.: Power indices and the measurement of control in corporate structures. Int. Game Theory Rev. 15(03), 1340017 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dubey, P., Shapley, L.S.: Mathematical properties of the banzhaf power index. Math. Oper. Res. 4(2), 99–131 (1979)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gambarelli, G., Owen, G.: Indirect control of corporations. Int. J. Game Theory 23(4), 287–302 (1994)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Karos, D., Peters, H.: Indirect control and power in mutual control structures. Games Econ. Behav. 92, 150–165 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Laffont, J.J., Martimort, D.: The Theory of Incentives: The Principal-Agentmodel. Princeton University Press, Princeton (2009)CrossRefGoogle Scholar
  10. 10.
    Levy, M.: Control in pyramidal structures. Corp. Governance Int. Rev. 17(1), 77–89 (2009)CrossRefGoogle Scholar
  11. 11.
    Levy, M., Szafarz, A.: Cross-ownership: a device for management entrenchment? Rev. Finance 21(4), 1675–1699 (2016)zbMATHGoogle Scholar
  12. 12.
    Mercik, J., Łobos, K.: Index of implicit power as a measure of reciprocal ownership. In: Transactions on Computational Collective Intelligence XXIII, pp. 128–140. Springer, Heidelberg (2016)CrossRefGoogle Scholar
  13. 13.
    Penrose, L.S.: The elementary statistics of majority voting. J. Roy. Stat. Soc. 109(1), 53–57 (1946)CrossRefGoogle Scholar
  14. 14.
    Shapley, L.S., Shubik, M.: A method for evaluating the distribution of power in a committee system. Am. Polit. Sci. Rev. 48(3), 787–792 (1954)CrossRefGoogle Scholar
  15. 15.
    Shleifer, A., Vishny, R.W.: A survey of corporate governance. J. Finance 52(2), 737–783 (1997)CrossRefGoogle Scholar
  16. 16.
    Strøm, K.: Delegation and accountability in parliamentary democracies. Eur. J. Polit. Res. 37(3), 261–290 (2000)Google Scholar
  17. 17.
    Vitali, S., Glattfelder, J.B., Battiston, S.: The network of global corporate control. PloS One 6(10), e25995 (2011)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.National Institute of InformaticsTokyoJapan
  2. 2.Faculty of Political Science and EconomicsWaseda UniversityTokyoJapan

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