Abstract
In recent years, there has been increasing awareness of the importance of formal measures of voting power and of the relevance of such measures to real life political issues. Nevertheless, existing measures have been criticized, especially because of their dependence on the unrealistic assumption that different coalitions have equal probabilities. In this paper we show that the classical problem of measuring voting power can be naturally embedded in information theory. This perspective on voting power allows us to extend measures of voting power to cases in which there are dependencies among voters. In doing so, we distinguish between two different notions of a given voter’s power—‘control’ and ‘informativeness’—corresponding, respectively, to the average uncertainty regarding the outcome of a vote that remains when all others have voted and the average uncertainty that is eliminated when only the given voter has voted. This distinction settles a number of well-known paradoxes and enables the study of voting power on the basis of actual political behavior at all levels.
Similar content being viewed by others
References
Albert M (2003) The voting power approach: measurement without theory. Eur Union Polit 4(3): 351–366. doi:10.1177/14651165030043005
Albert M (2004) The voting power approach: unresolved ambiguities. Eur Union Polit 5(1): 139–146. doi:10.1177/1465116504040449
Aleskerov F, Blagoveschenski N, Satarov G, Sokolova A, Yakuba V (2004) Power distribution among groups and fractions in Russian parliament. LSE-CPNSS 3rd annual conference
Axelrod R (1970) Conflict of interest: a theory of divergent goals with applications to politics. Markham, Chicago
Bachrach P, Baratz M (1962) Two faces of power. Am Polit Sci Rev 56: 947–952. doi:10.2307/1952796
Banzhaf JF III (1966) Multi-member electoral districts—do they violate the one man, one vote principle?. Yale Law J 75: 1309–1338. doi:10.2307/795047
Banzhaf JF III (1968) One man 3.312 votes: a mathematical analysis of the electoral college. Villanova Law Rev 13: 304–332
Beisbart C (2007) Measuring influence for dependent voters: a generalisation of the Banzhaf measure. Fifth annual LSE workshop on voting power and procedures
Berg S (1999) On voting power indices and a class of probability distributions: with applications to EU. Group Decis Negot 8(1): 17–31. doi:10.1023/A:1008673712816
Bovens L, Beisbart C (2007) Measuring influence for dependent voters: a generalisation of the Banzhaf Measure. Fifth annual LSE workshop on voting power and procedures
Braham M, Holler M (2005) The impossibility of a preference-based power index. J Theor Polit 17: 137–157. doi:10.1177/0951629805047801
Brams SJ (1975) Game theory and politics. Free Press, New York
Brams SJ, Sanver R (2006) Critical strategies under approval voting: who gets ruled in and ruled out. Elect Stud 25(2): 287–305. doi:10.1016/j.electstud.2005.05.007
Coleman JS (1971) Control of collectivities and the power of a collectivity to act. In: Lieberman B, Social choice. Gordon and Breach, New York, pp 269–300
De Swaan A (1973) Coalition theories and cabinet formations: a study of formal theories of coalition formation applied to nine European parliaments after 1948. Jossey-Bass Publishers, San Francisco
Downs A (1957) An economic theory of democracy. Harper and Row, New York
Feix MR, Lepelley D, Merlin VR, Rouet JL (2004) The probability of conflicts in a U.S. presidential type election. Econ Theory 23: 227–258. doi:10.1007/s00199-003-0375-2
Felsenthal DS, Machover M (1997) The weighted voting rule in the EU’s council of ministers, 1958–1995: intentions and outcomes. Elect Stud 16: 33–47. doi:10.1016/S0261-3794(96)00055-8
Felsenthal DS, Machover M (1998) The measurement of voting power: theory and practice problems and paradoxes. Edward Elgar, Cheltenham
Felsenthal DS, Machover M (2005) Voting power measurement: a story of misreinvention. Soc Choice Welf 25: 485–506. doi:10.1007/s00355-005-0015-9
Felsenthal DS, Machover M (2003) The voting power approach: response to a philosophical reproach. Eur Union Polit 4(4):493–499, 513–517 doi:10.1177/146511650344005
Garrett G, Tsebelis G (1999) Why resist the temptation to apply power indices to the European union?. J Theor Polit 11(3): 291–308. doi:10.1177/0951692899011003001
Garrett G, Tsebelis G (1999) More reasons to resist the temptation of power indices in the European union. J Theor Polit 11(3): 331–338. doi:10.1177/0951692899011003004
Gelman A, Katz JN, Tuerlinckx F (2002) The mathematics and statistics of voting power. Stat Sci 17(4): 420–435. doi:10.1214/ss/1049993201
Heard AD, Swartz TB (1999) Extended voting measures. Can J Stat 27: 177–186
Kaniovski S (2008) The exact bias of the Banzhaf measure of power when votes are neither equiprobable nor independent. Soc Choice Welf (in press)
Holler M, Widgrén M (1999) Why power indices for assissing European union decision-making?. J Theor Polit 11(3): 321–330. doi:10.1177/0951692899011003003
Kilgour MD (1974) A Shapley value for cooperative games with quarreling. In: Rapoport A, Game theory as a theory of conflict resolution. Dordrecht, Holland, pp 193–206
Lane J-E (2005) International organization analyzed with the power index method. LSE-CPNSS 4th annual workshop
Lane JE, Berg S (1999) Relevance of voting power. J Theor Polit 11(3): 309–320
Laruelle A, Widgren M (1998) Is the allocation of voting power among EU states fair?. Public Choice 94(3–4): 317–339
Laruelle A, Valenciano F (2005) Assessing success and decisiveness in voting situations. Soc Choice Welf 24: 171–197. doi:10.1007/s00355-003-0298-7
Laruelle A, Valenciano F (2008) Voting and collective decision-making: bargaining and power. Cambridge University Press, London
Leech D (2002) Power indices as an aid to institutional design: the generalized apportionment problem. LSE-CPNSS 1st annual conference
Leech D (2002) Voting power in the governance of the international monetary fund. Ann Oper Res 109: 375–397. doi:10.1023/A:1016324824094
Leech D, Leech R (2006) Voting power and voting blocs. Public Choice 127(3): 285–303. doi:10.1007/s11127-006-1914-8
List C (2003) The voting power approach: a theory of measurement: a response to Max Albert. Eur Union Polit 4(4): 473–497. doi:10.1177/146511650344005
Luce D, Rogow A (1956) A game-theoretic analysis of congressional power distributions for a stable two-party system. Behav Sci 1: 83–95
Machover M (2007) Discussion topic: voting power when voters’ independence is not assumed. LSE research online (http://eprints.lse.ac.uk/2966), pp 1–4
Maaser M, Napel S (2005) Equal representation in two-tier voting systems. LSE-CPNSS 4th annual conference
Napel S, Widgrén M (2005) The possibility of a preference-based power index. J Theor Polit 17: 377–387. doi:10.1177/0951629805052886
Nurmi H (1997) The representation of voter groups in the European parliament: a Penrose–Banzhaf index analysis. Elect Stud 16: 317–327. doi:10.1016/S0261-3794(97)00027-9
Nurmi M, Meskanen T (1999) A priori power measures and the institutions of the European union. Eur J Polit Res 35: 161–179
Penrose LS (1952) On the objective study of crowd behaviour. H.K. Lewis and Co., London
Rablen M (2005) United Nations security council reform: a proposal for weighted voting. LSE-CPNSS 4th annual conference
Riker WH (1986) The first power index. Soc Choice Welf 3: 293–295. doi:10.1007/BF00292733
Russell B (1938) Power: a new social analysis. George Allen and Unwin, London
Shannon CE (1948) A mathematical theory of communication. Bell Syst Technol J 27:379–423, 623–656.
Shapley LS (1953) A value for N-person games. In: Kuhn HW, Tucker AW, Contributions to the theory of games II (Annals of Mathematic Studies 28). Princeton University Press, Princeton
Shapley LS, Shubik M (1954) A method of evaluating the distribution of power in a committee system. Am Polit Sci Rev 48: 787–792. doi:10.2307/1951053
Steuneberg B, Smidtchen D, Koboldt C (1999) Strategic power in the European union: evaluating the distribution of power in policy games. J Theor Polit 11(3): 339–366. doi:10.1177/0951692899011003005
Straffin PD (1977) Homogeneity, independence and power indices. Public Choice 30: 107–118. doi:10.1007/BF01718820
Straffin PD (1978) Probability models for power indices. In: Peter O, Game theory and political science. New York University Press, New York
Straffin PD (1988) The Shapley–Shubik and Banzhaf power indices as probabilities. In: Alvin R, The Shapley value: essays in honor of Lloyd Shapley. Cambridge University Press, Cambridge
Weber M (1978, 1921–1922) In: Guenther R, Wittich C (eds) Economy and society: an outline of interpretive sociology. University of California Press, Berkeley
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Diskin, A., Koppel, M. Voting power: an information theory approach. Soc Choice Welf 34, 105–119 (2010). https://doi.org/10.1007/s00355-009-0390-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00355-009-0390-8