Abstract
We analyze the propensity to approve a random proposal of a large committee that makes decisions by weighted voting. The approach is a generalized version of James Coleman’s “power of a collectivity to act”. Throughout the paper it is assumed that the voters are of two kinds: a fixed (possibly empty) set of “major” (big) voters with fixed weights, and an ever-increasing number of “minor” (small) voters, whose total weight is also fixed, but where each individual’s weight becomes negligible. As our main result, we obtain that asymptotically many minor voters act like a modification of the quota for the vote among major voters. The paper estimates the rate of convergence which turns out to be very high if the weight distribution among the small voters is not too skewed. The results obtained are illustrated by evaluating the decision rules for the Council of Ministers of the EU for various scenarios of EU enlargement.
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References
Baldwin R, Berglöf E, Giavazzi F, Widgren M (2000) EU reforms for tomorrow’s Europe. Centre for Economic Policy Research. Discussion Paper No. 2623, London
Banzhaf JF (1965) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19:317–343
Coleman JS (1971) Control of collectivities and the power of a collectivity to act. In: Lieberman B (eds). Social Choice, Gordon and Breach, New York, Reprinted in Coleman JS (1986), Individual interests and collective action. Cambridge University Press, Cambridge
Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Math Operat Res 4(2):99–131
Felsenthal DS, Machover M (1998) The measurement of voting power: theory and practise. Problems and paradoxes, Edward Elgar, Cheltenham
Felsenthal DS, Machover M (2000) Enlargement of the EU and weighted voting in its council of ministers [online], LSE research online, London. Available at: http://eprints.lse.ac.uk/archive/00000407
Felsenthal DS, Machover M (2001) The Treaty of Nice and qualified majority voting. Social Choice Welfare 18:431–464
Galloway D (2001) The Treaty of Nice and beyond: realities and illusions of power in the EU. Sheffield Academic Press, Sheffield
Hoeffding W (1963) Probability inequalities for sums of bounded random variables. J Am Stat Assoc 58:13–30
Kemperman JHB (1964) Probability methods in the theory of distribution modulo one. Compositio Math 16:106–137
Laruelle A, Valenciano F (2006) Bargaining in committees as an extension of Nash’s bargaining theory. Social Choice Welfare, in press
Leech D (2002a) Designing the voting system for the council of the European Union. Public Choice 113:437–464
Leech D (2002b) Voting power in the governance of the International Monetary Fund. Ann Oper Res 109:375–397
Mann I, Shapley LS (1960) Values of large games, IV: evaluating the Electoral College by Monte Carlo techniques, The RAND Corporation, Memorandum RM-2641 (ASTIA No. AD 246277), September 19
Petrov VV (1975) Sums of independent random variables. Springer, Heidelberg
Rae DW (1969) Decision rules and individual values in constitutional choice. Am Polit Sci Rev 63:40–56
Roth AE (1988) Introduction to the Shapley value. In: Roth AE (eds). The Shapley value. Cambridge University Press, Cambridge
Shapiro NZ, Shapley LS (1978) Values of large games, I: a limit theorem. Math Oper Res 3(1):1–9
Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW (eds, 1953), Contributions to the theory of games II, Annals of Mathematics Studies 28. Princeton University Press, Princeton Reprinted in Roth AE (ed. 1988), The Shapley value. Cambridge University Press, Cambridge
Taylor AD, Zwicker WS (1992) A characterization of weighted voting. Proc Am Math Soc 115:1089–1094
Taylor AD, Zwicker WS (1999) Simple games: desirability relations, trading, and pseudoweightings. Princeton University Press, New Jersey
Treaty of Nice (2001) Conference of the Representatives of the Governments of the Member States, Brussels, 28 February 2001. Treaty of Nice amending the Treaty on European Union, the Treaties establishing the European Communities and certain related Acts. EU document CONFER 4820/00
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I wish to thank Matthew Braham, Sidartha Gordon, Maurice Koster, Moshé Machover, Guillermo Owen and two anonymous referees for helpful comments.
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Lindner, I. The power of a collectivity to act in weighted voting games with many small voters. Soc Choice Welfare 30, 581–601 (2008). https://doi.org/10.1007/s00355-007-0256-x
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DOI: https://doi.org/10.1007/s00355-007-0256-x