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Feature Column

Voting Games: Part I


6. References

Algaba, E. and J. Bilbao, J. Garcia, J. Lopez, Computing power indices in weighted multiple majority games, Math. Social Sciences 46 (2003) 63-80.

Amer, R. and F. Carreras, and A. Magana, Extension of values to games with multiple alternatives, Annals of OR, 84 (1998) 63-78.

Banzhaf, J., Weighted voting does not work: a mathematical analysis, Rutgers Law Review, 19 (1965) 317-343.

Banzhaf, J., Multi-member electoral districts - do they violate the "one Man, one Vote' principle?, Yale Law J., 75 (1966) 1309-1338.

Banzhaf, J., One man, 3.312 votes: A mathematical analysis of the electoral college, Villanova Law Review, 13 (1968) 304-332.

Banzhaf, J., One man, ? votes: Mathematical analysis of political consequences and judicial choices, George Washington Law Rev., 36 (1968) 808-823.

Barua, R. and S. Chakravarty, S. Roy, P. Sarkar, A characterization and some properties of the Banzhaf-Coleman-Dubey-Shapley sensitivity index, Games and Economic Behavior, to appear.

Bilbao, J. and J. Fernandez, N. Jimenez, J. Lopez, Voting power in the European Union enlargement, European J. of Oper. Research 143 (2002) 181-196.

Bolger, E., Power indices for multicandidate voting games, Inter. J. of Game Theory, 14 (1986) 175-186.

Brams, S. and A. Doherty, M. Weidner, Game theory and multilateral negotiations: the Single European Act and the Uruguay round, In, I. Zartman (ed.), International and Multilateral Negotiation: Approaches to the Management of Complexity, Jossey-Bass, San Francisco, p. 95-112.

Carreras, F. and J. Freixas, Complete simple games, J. Math. Social Sciences, 32 (1996) 139-155.

Carreras, F. and J. Freixas, A power analysis of linear games with consensus, J. of Math. Social Sciences, 48 (2004) 207-221.

Chakravarty, N. and A. Goel, T. Sastry, Easy weighted majority games, Math. Social Sciences 40 (2000) 227-235.

Chvatal, V. and P. Hammer, Aggregation of inequalities in integer programming, Ann. Discrete Math., 1 (1977) 145-162.

Coleman, J., Control of collectivities and the power of a collectivity to act, In, B. Lieberman (ed.), Social Choice, Gordon and Breach, New York, 1971, p. 269-300.

Deegan, J. and E. Packel, A new index of power for simple n-person games, Int. J. Game Theory, 7 (1978) 113-123.

Diffo, L. and J. Moulen, Ordinal equivalence of power notions in voting games, Theory and Decision 53 (2002) 313-325.

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Dubey, P., and L. Shapley, Mathematical properties of the Banzhaf power index, Mathematics of Operations Research 4 (1979) 99-131.

Einy, E., The desirability relation of simple games, Math. Soc. Sci., 10 (1985) 155-158.

Felsenthal, D. and M. Machover, The weighted voting rule in the EU's council of ministers, 1958-1995: Intentions and Outcomes, Electoral Studies 16 (1997) 33-47.

Felsenthal, D. and M. Machover, W. Zwicker, The bicameral postulates and indices of a priori voting power, Theory and Decision 44 (1998) 83-116.

Felsenthal, D. and M. Machover, The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes, Edward Elgar Publishing, Cheltenham, 1998.

Freixas, J., The Banzhaf index for games with several levels of approval in the input and output, Annals of Operations Research, 137 (2005) 45-66.

Freixas, J., The dimension for the European Union Council under the Nice rules, European J. of Operations Research, 156 (2004) 415-419.

Freixas, J., The Shapley-Shubik power index for games with several levels of approval in the input and output, Decision Support Systems, 39 (2005)
185-195.

Freixas, J. and W. Zwicker, Weighted voting, abstention, and multiple levels of approval, Soc. Choice Welf., 21 (2003) 399-331.

Garrett, G. and I. McLean, M. Machover, Power, Power Indices and Blocking Power: A comment, British J. of Political Science, 25 (1995) 563-568.

Garrett, G. and G. Tsebelis, Why resist the temptation to apply power indices to the European Union?, J. of Theoretical Politics, 11 (1999) 291-308.

Garrett, G. and G. Tsebelis, More reasons to resist the temptation of power indices in the European Union, J. of Theoretical Politics 11 (1999) 331-338.

Gianaris, W., Weighted voting in the international monetary fund and the world bank, Fordham International Law Journal, 14 (1991-1992) 910-945.

Herne, K. and H. Nurmi, The distribution of a priori voting power in the EC Council of Ministers and the European Parliament, Scandinavian Political Studies, 16 (1993) 269-284.

Hilliard, M., Weighted voting theory and applications, Tech. Report No. 609, School of Operations Research and Industrial Engineering, Cornell University, 1983.

Hollwer, M. and M. Widgren, Why power indices for assessing European Union decision-making? J. of Theoretical Politics 11 (1999) 321-330.

Hosli, M., Admission of European Free Trade Association states to the European Community: effects on voting power in the European Community and Council of Ministers, International Organization, 47 (1993) 629-643.

Hosli, M., The balance between small and large: effects of a double-majority system on voting power in the European Union, International Studies Quart., 39 (1995) 351-370.

Hosli, M., Voting strength in the European Parliament: The influence of national and/or partisan actors, European J. of Political Research, 31 (1997) 351-356.

Imrie, R., The impact of weighted vote on representation in municipal governing bodies of New York State, In, L. Papayanopoulos, (ed.), Democratic Representation and Apportionment: Quantitative Methods, Measures, and Criteria, Annals of the New York Academy of Sciences, volume 219, 1973, p. 192-199.


Isbell, J., A class of majority games, Quart. J. Math. Oxford Series, 7 (1956) 183-187.

Johnson, R., The conflict over qualified majority voting in the European Union Council of Ministers: An analysis of the UK negotiating stance using power indices, British J. of Political Science, 25 (1995) 245-254.

Johnson, R., Can power be reduced to a quantitative index - and if so, which one? A response to Garrett, McLean and Machover, British J. of Political Science, 25 (1995) 568-572.

Johnson, R. On keeping touch with reality and failing to be befuddled by mathematics, British J. of Political Science, 26 (1996) 598-599.

Kilgour, D., A formal analysis of the amending formula of Canada's Constitution Act,, Canadian J. of Political Science, 16 (1983) 771-777.

Krohn, I. and P. Sudhslter, Directed and weighted majority games, Mathematical Methods of Operations Research 42 (1995) 189-216.

Lane, J.-E., and R. Maeland, Voting power under the constitution, J. of Theoretical Politics, 7 (1995) 223-230.

Lapidot, E., The counting vector of a simple game, Proceedings of the Amer. Math. Soc., 31 (1972) 228-231.

Lane, J. and S. Berg, Relevance of voting power, J. of Theoretical Politics 11 (1999) 309-320.

Lucas, W., Measuring Power in Weighted Voting Games, Case Studies in Applied Mathematics, Mathematical Association of America, Washington, 1976, pp. 42-106.

Lucas, W., Measuring power in weighted voting games, Chapter 9, in Political and Related Models, S. Brams, W. Lucas, and P. Straffin, Jr. (eds.), Springer-Verlag, New York, 1983, p. 183-238.


Lucas, W., Fair Voting, Consortium for Mathematics and its Applications (COMAP), Lexington, 1993.

Maatsui, T. and Y. Matsui, A survey of algorithms for calculating power indices of weighted majority games, J. of the Operations Research Society of Japan 43 (2000) 71-86.

Morriss, P., Power: A Philosophical Analysis, Manchester U. Press, Manchester, 1987.

Nowak, A., On an axiomatization of the Banzhaf value without the additivity axiom, Int. J. Game Theory, 26 (1997) 137-141.

Owen, G., Political games, Naval Res. Logistics Quart., 18 (1971) 345-355.

Peleg, B., A theory of coalition formation in committees, J. Math. Econ., 7 (1980) 115-134.

Penrose, L., The elementary statistics of majority voting, J. of the Royal Statistical Society, 109 (1946) 53-57.

Saari, D., The ultimate of chaos resulting from weighted voting systems, Advances in Applied Math. 5 (1984) 286-308.

Shapley, L. and M. Shubik, A method for evaluating the distribution of power in a committee system, American Political Science Review, 48 (1954) 787-792.

Shapley, L., Simple games: An outline of the descriptive theory, Behav. Sci., 7 (1962) 59-66.

Shapley, L., A comparison of power indices and a nonsymmetric generalization, Rand Paper P-5872, The Rand Corporation, Santa Monica, 1977.

Steunenberg, B. D. Schmidtchen, C. Koboldt, Strategic power in the European Union: evaluating the distribution of power in policy games, J. of Theoretical Politics 11 (1999) 339-366.

Straffin, P., Homogeneity, independence and power indices, Public Choice, 30 (1977) 107-118.


Straffin, P., Topics in Voting, Birkhäuser, Boston, 1980.

Tannenbaum, P., Power in weighted voting systems, The Mathematica Journal 7 (1997) 59-63.

Taylor, A., Mathematics and Politics, Springer-Verlag, New York, l995.

Taylor, A. and W. Zwicker, A characterization of weighted voting, Pro. American Math. Soc. 115 (1992) 1089-1094.

Taylor, A. and W. Zwicker, Weighted voting, multicameral representation, and power, Games and Economic Behavior, 5 (1993) 170-181.

Taylor, A. and W. Zwicker, Simple Games and Magic Squares, J. Combinatorial Theory, ser. A., 71 (1995) 67-88.

Taylor, A. and W. Zwicker, Quasi-weightings, trading, and desirability relations in simple games, Games and Economic Behavior 16 (1996) 331-346.

Taylor, A. and W. Zwicker, Interval measures of power, Math. Social Sci., 33 (1997) 23-74.

Taylor, A. and W. Zwicker, Simple Games, Princeton U. Press, Princeton, 1999.

Tong, Z. and R. Kain, Vote assignments in weighted voting mechanisms, IEEE Transactions on Computers 40 (1991) 664-667.

Walther, E., An analysis of weighted voting systems using the Banzhaf value, Master of Science Thesis, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, 1977.


Widgren, M., Voting power in the EC decision making and the consequences of two different enlargements, European Economic Review, 38 (1994) 1153-1170.

Widgren, M., Probabilistic voting power in the EU Council: the cases of trade policy and social regulation, Scandinavian J. of Economics, 97 (1995) 345-346.

Winder, R., Threshold Logic, Doctoral Thesis, Princeton University, Princeton, 1962.

Young, S. and A. Taylor, W. Zwicker, Counting quota systems: a combinatorial question from social choice theory, Math. Mag., 68 (1995) 331-342.

(Unsigned) How the European Union Works, European Communities, 2003 (ISBN 92-894-5283-8)

Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal, which also provides bibliographic services.


  1. Introduction
  2. Voting systems and games
  3. Basic mathematical ideas
  4. Unintuitive behavior
  5. More voting games
  6. References

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