Public Choice

, Volume 151, Issue 3–4, pp 757–787 | Cite as

Axiomatizations of a positional power score and measure for hierarchies

Open Access


The extant literature on positional power in hierarchies is mainly restricted to the analysis of power in terms of the authority structure. This is based upon the bare positions of actors and ignores the existence of decision-making mechanisms. The few studies, which also take into account decision-making mechanisms, are ‘membership-based’. In an earlier paper we demonstrated that such an approach is, in general, inappropriate. As a solution we suggested an ‘action-based’ approach and developed a power score and measure ascribing positional power to actors in hierarchies. In this paper we axiomatize this power score and measure and illustrate their applicability.


Hierarchies Decision-making mechanism Power Positional power Power score Power measure Binary voting Axiomatization 

JEL Classification

C79 D02 D71 


  1. Aumann, R. J., & Drèze, J. H. (1974). Cooperative games with coalition structures. International Journal of Game Theory, 3, 217–237. CrossRefGoogle Scholar
  2. Banzhaf, J. F. (1965). Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Review, 19, 317–343. Google Scholar
  3. Berg, S., & Paroush, J. (1998). Collective decision making in hierarchies. Mathematical Social Sciences, 35, 233–244. CrossRefGoogle Scholar
  4. Bolton, P., & Dewatripont, M. (1994). The firm as a communication network. Quarterly Journal of Economics, 109, 809–839. CrossRefGoogle Scholar
  5. Braham, M. (2008). Social power and social causation: towards a formal synthesis. In M. Braham & F. Steffen (Eds.), Power, freedom, and voting (pp. 1–21). Berlin: Springer. CrossRefGoogle Scholar
  6. Braham, M., & Holler, M. J. (2008). Distributing causal responsibility in collectivities. In T. Boylan & R. Gekker (Eds.), Economics, rational choice, and normative philosophy London: Routledge. Google Scholar
  7. Braham, M., & Steffen, F. (2002). Voting power and games with abstention. In M. J. Holler et al. (Ed.), Power and fairness (pp. 333–348). Tubingen: Mohr Siebeck. Google Scholar
  8. Brams, S. (1968). Measuring the concentration of power in political systems. American Political Science Review, 62, 461–475. CrossRefGoogle Scholar
  9. van den Brink, R. (1994). Relational power in hierarchical organizations. PhD dissertation, Tilburg University. Google Scholar
  10. van den Brink, R. (1997). An axiomatization of the disjunctive permission value for games with a permission structure. International Journal of Game Theory, 26, 27–43. CrossRefGoogle Scholar
  11. van den Brink, R. (1999). An axiomatization of the conjunctive permission value for games with a hierarchical permission structure. In H. de Swart (Ed.), Logic, game theory and social choice (pp. 125–139). Tilburg University Press. Google Scholar
  12. van den Brink, R. (2001). An axiomatization of the Shapley value using a fairness property. International Journal of Game Theory, 30, 309–319. CrossRefGoogle Scholar
  13. van den Brink, R. (2002). Apex games as relational power measures for directed graphs. Social Choice and Welfare, 19, 845–867. CrossRefGoogle Scholar
  14. van den Brink, R. (2008). Vertical wage differences in hierarchically structured firms. Social Choice and Welfare, 30, 225–243. CrossRefGoogle Scholar
  15. van den Brink, R., & Gilles, R. P. (1996). Axiomatizations of the conjunctive permission value for games with permission structures. Games and Economic Behavior, 12, 113–126. CrossRefGoogle Scholar
  16. van den Brink, R., & Gilles, R. P. (2000). Measuring domination in directed networks. Social Networks, 22, 1141–1157. Google Scholar
  17. van den Brink, R., & Steffen, F. (2008). Positional power in hierarchies. In M. Braham, F. Steffen (Eds.), Power, freedom, and voting (pp. 57–81). Berlin: Springer. CrossRefGoogle Scholar
  18. Chandler, A. D. (1966). In Strategy and structure New York: Doubleday. Google Scholar
  19. Chwe, M. S.-Y. (2000). Communication and coordination in social networks. Review of Economic Studies, 67, 1–16. CrossRefGoogle Scholar
  20. Copeland, A. H. (1951). A reasonable social welfare function. Mimeo, Seminar on Applications of Mathematics to Social. Sciences, University of Michigan. Google Scholar
  21. Daudi, P. (1986). Power in the organization. Basil: Blackwell. Google Scholar
  22. Felsenthal, D. S., & Machover, M. (1996). Alternative forms of the Shapley value and the Shapley-Shubik index. Public Choice, 87, 315–318. CrossRefGoogle Scholar
  23. Felsenthal, D. S., & Machover, M. (1997). Ternary voting games. International Journal of Game Theory, 26, 335–351. CrossRefGoogle Scholar
  24. Felsenthal, D. S., & Machover, M. (1998). The measurement of voting power. Cheltenham Glos: Edward Elgar. Google Scholar
  25. Felsenthal, D. S., & Machover, M. (2004). Analysis of QM rules in the draft constitution for Europe proposed by the European Convention, 2003. Social Choice and Welfare, 23, 1–20. CrossRefGoogle Scholar
  26. Felsenthal, D. S., Leech, D., List, C., & Machover, M. (2003). In defence of voting power analysis: responses to Albert. European Union Politics, 4, 473–497. CrossRefGoogle Scholar
  27. Fischer, A. J. (1999). The probability of being decisive. Public Choice, 101, 267–283. CrossRefGoogle Scholar
  28. Gilles, R. P., & Owen, G. (1994). Cooperative games and disjunctive permission structures. Department of. Economics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia. Google Scholar
  29. Gilles, R. P., Owen, G., & van den Brink, R. (1992). Games with permission structures: the conjunctive approach. International Journal of Game Theory, 20, 277–293. CrossRefGoogle Scholar
  30. Grofman, B., & Owen, G. (1982). A game theoretic approach to measuring degree of centrality in social networks. Social Networks, 4, 213–224. CrossRefGoogle Scholar
  31. Harré, R. (1970). Powers. British Journal of the Philosophy of Science, 21, 81–101. CrossRefGoogle Scholar
  32. Herings, P. J. J., van der Laan, G., & Talman, A. J. J. (2005). The positional power of nodes in digraphs. Social Choice and Welfare, 24, 439–454. CrossRefGoogle Scholar
  33. Hu, X., & Shapley, L. S. (2003). On authority distributions in organisations: controls. Games and Economic Behavior, 45, 153–170. CrossRefGoogle Scholar
  34. Johnston, P., & Gill, J. (1993). Management control and organizational behaviour. London: Paul Chapman. Google Scholar
  35. Kaniovski, S. (2008). Straffin meets condorcet: what can a voting power theorist learn from a Jury theorist? Homo Oeconomicus, 25, 1–22. Google Scholar
  36. Laruelle, A., & Valenciano, F. (2008). Voting and collective decision-making. Cambridge: Cambridge University Press. CrossRefGoogle Scholar
  37. Leech, D. (2002). Designing the voting system for the council of the European Union. Public Choice, 113, 437–464. CrossRefGoogle Scholar
  38. Mackenzie, K. D. (1976). A theory of group structures, vol. I: Basic theory. New York: Gordon & Breach. Google Scholar
  39. Martin, J. (1998). Organizational behaviour. International Thompson Business Press. Google Scholar
  40. Matthews, K., & Thompson, J. (2008). The economics of banking (2nd edn.). New York: Wiley. Google Scholar
  41. Mizruchi, M. S., & Potts, B. B. (1998). Centrality and power revisited: actor success in group decision making. Social Networks, 20, 353–387. CrossRefGoogle Scholar
  42. Morgenstern, O. (1951). Prolegomena to a theory of organization. Rand Research Memorandum RM-5438-PR. Google Scholar
  43. Morriss, P. (1987/2002). Power: a philosophical analysis. Manchester: Manchester University Press. Google Scholar
  44. Napel, S., & Widgrén, M. (2006). The inter-institutional distribution of power in EU codecision. Social Choice and Welfare, 27, 129–154. CrossRefGoogle Scholar
  45. Packel, E. W., & Deegan, J. (1980). An axiomated family of power indices for simple n-person games. Public Choice, 35, 229–239. CrossRefGoogle Scholar
  46. Radner, R. (1992). Hierarchy: the economics of managing. Journal of Economic Literature, 30, 1382–1415. Google Scholar
  47. Russett, B. M. (1968). Probalism and the number of units: measuring influence concentration. American Political Science Review, 62, 476–480. CrossRefGoogle Scholar
  48. Sah, R. K., & Stiglitz, J. E. (1985). The theory of economic organizations. AEA Papers and Proceedings, 75, 292–297. Google Scholar
  49. Sah, R. K., & Stiglitz, J. E. (1986). The architecture of economic systems: hierarchies and polyarchies. American Economic Review, 76, 716–727. Google Scholar
  50. Shapley, L. S. (1962). Simple games: an outline of the descriptive theory. Behavioral Science, 7, 59–66. CrossRefGoogle Scholar
  51. Shapley, L. S., & Palamara, J. R. (2000a). Control games and organizations. UCLA Working Paper 795. Google Scholar
  52. Shapley, L. S., & Palamara, J. R. (2000b). Simple games and authority structure. UCLA Working Paper 796. Google Scholar
  53. Shapley, L. S., & Shubik, M. (1954). A method for evaluating the distribution of power in a committee system. American Political Science Review, 48, 787–792. CrossRefGoogle Scholar
  54. Shubik, M. (1962). Incentives, decentralized control, the assignment of joint costs and internal pricing. Management Science, 8, 325–343. CrossRefGoogle Scholar
  55. Simon, H. A. (1962). The architecture of complexity. Proceedings of the American Philosophical Society, 106, 467–482. Google Scholar
  56. Smith, C. G., & Tannenbaum, A. S. (1963). Organizational control structure: a comparative analysis. Human Relations 16, 299–316. CrossRefGoogle Scholar
  57. Steffen, F. (2002). Essays in the theory of voting power. PhD Dissertation, University of Hamburg. Google Scholar
  58. Straffin, P.D. Jr. (1977). Homogeneity, independence, and power indices. Public Choice 30, 107–118. CrossRefGoogle Scholar
  59. Straffin, P. D. Jr. (1978). Probability models for power indices. In P. C. Ordeshook (Ed.), Game theory and political science (pp. 477–510). New York: New York University Press. Google Scholar
  60. Sutter, M. (2000). Flexible integration, EMU and relative voting power in the EU. Public Choice, 104, 41–62. CrossRefGoogle Scholar
  61. Tannenbaum, A. S. (1961). Control and effectiveness in a voluntary organization. American Journal of Sociology, 67, 33–46. CrossRefGoogle Scholar
  62. Tannenbaum, A. S., & Kahn, R. L. (1957). Organizational control structure Human Relations, 10, 127–140. CrossRefGoogle Scholar
  63. Williamson, O. E. (1967). Hierarchical control and optimum firm size. Journal of Political Economy, 75, 123–138. CrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of Econometrics and Tinbergen InstituteVU UniversityAmsterdamThe Netherlands
  2. 2.Management SchoolThe University of LiverpoolLiverpoolUK

Personalised recommendations