Abstract
If K is an index of relative voting power for simple voting games, the bicameral postulate requires that the distribution of K -power within a voting assembly, as measured by the ratios of the powers of the voters, be independent of whether the assembly is viewed as a separate legislature or as one chamber of a bicameral system, provided that there are no voters common to both chambers. We argue that a reasonable index – if it is to be used as a tool for analysing abstract, ‘uninhabited’ decision rules – should satisfy this postulate. We show that, among known indices, only the Banzhaf measure does so. Moreover, the Shapley–Shubik, Deegan–Packel and Johnston indices sometimes witness a reversal under these circumstances, with voter x ‘less powerful’ than y when measured in the simple voting game G1 , but ‘more powerful’ than y when G1 is ‘bicamerally joined’ with a second chamber G2 . Thus these three indices violate a weaker, and correspondingly more compelling, form of the bicameral postulate. It is also shown that these indices are not always co-monotonic with the Banzhaf index and that as a result they infringe another intuitively plausible condition – the price monotonicity condition. We discuss implications of these findings, in light of recent work showing that only the Shapley–Shubik index, among known measures, satisfies another compelling principle known as the bloc postulate. We also propose a distinction between two separate aspects of voting power: power as share in a fixed purse (P-power) and power as influence (I-power).
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REFERENCES
Banzhaf, J.F.: 1965, ‘Weighted Voting Doesn't Work: A Mathematical Analysis', Rutgers Law Review 19, 317–343.
Brams, S.J., Affuso, P.J. and Kilgour, D.M.: 1989, ‘Presidential Power: A Game-Theoretic Analysis', in: Brace, P., Harrington, C.B. and King, G. (Eds.), The Presidency in American Politics, New York: New York University Press, pp. 55–74.
Coleman, J.S.: 1971, ‘Control of Collectivities and the Power of a Collectivity to Act', in: Lieberman, B. (Ed.), Social Choice, New York: Gordon and Breach, pp. 269–300.
Deegan, J. and Packel, E.W.: 1978, ‘A New Index of Power for Simple n-Person Games', International Journal of Game Theory 7, 113–123.
Deegan, J. and Packel, E.W.: 1982, ‘To the (Minimal Winning) Victors Go the (Equally Divided) Spoils: A New Index of Power for Simple n-Person Games', in: Brams, S.J., Lucas, W.F. and Straffin, P.D. (Eds.), Political and Related Models (Vol. 2 in series Models in Applied Mathematics, Ed. W.F. Lucas), New York: Springer, pp. 239–255.
Dubey, P.: 1975, ‘On the Uniqueness of the Shapley Value', International Journal of Game Theory 4, 131–140.
Dubey, P. and Shapley, L.S.: 1979, ‘Mathematical Properties of the Banzhaf Power Index', Mathematics of Operations Research 4, 99–131.
Feller, W.: 1957, An Introduction to Probability Theory and its Applications, Vol. I, 2nd. Edn., New York: Wiley.
Felsenthal, D.S. and Machover, M.: 1995, ‘Postulates and Paradoxes of Relative Voting Power — A Critical Re-Appraisal', Theory and Decision 38, 195–229.
Gul, F.: 1989, ‘Bargaining Foundations of Shapley Value', Econometrica 57, 81–95.
Johnston, R.J.: 1978, ‘On the Measurement of Power: Some Reactions to Laver', Environment and Planning A 10, 907–914.
Kreisel, G.: 1967, ‘Informal Rigour and Completeness Proofs', in: Lakatos, I. (Ed.) Problems in the Philosophy of Mathematics, pp. 138–171. Amsterdam: North-Holland.
Leech, D.: 1990, ‘Power Indices and Probabilistic Voting Assumptions', Public Choice 66, 293–299.
Lucas, W.F.: 1982, ‘Measuring Power in Weighted Voting Systems', in: Brams, S.J., Lucas, W.F. and Straffin, P.D. (Eds.), Political and Related Models (Vol. 2 in series Models in Applied Mathematics, Ed. W.F. Lucas), New York: Springer, pp. 183–238.
Luce, R.D. and Raiffa, H.: 1957, Games and Decisions: Introduction and Critical Survey, New York: Wiley.
Morriss, P.: 1987, Power — A Philosophical Analysis, Manchester: Manchester University Press.
Myerson, R.B.: 1991, Game Theory: Analysis of Conflict, Cambridge: Harvard University Press.
Owen, G.: 1972, ‘Multilinear Extensions of Games', Management Science, Series A, 18, 64–79.
Owen, G.: 1978a, ‘Characterization of the Banzhaf-Coleman Index', SIAM Journal of Applied Mathematics 35, 315–327.
Owen, G.: 1978b, ‘A Note on the Banzhaf-Coleman Axioms', in: Ordeshook, P. (Ed.), Game Theory and Political Science, New York: New York University Press, pp. 451–461.
Penrose, L.S.: 1946, ‘The Elementary Statistics of Majority Voting', Journal of the Royal Statistical Society 109, 53–57.
Quint, T.: 1994, ‘Measures of Powerlessness in Simple Games', Preprint Series, Report No. 28, University of Nevada at Reno.
Roth, A.E.: 1977, ‘Utility Functions for Simple Games', Journal of Economic Theory 16, 481–489.
Roth, A.E. (Ed.): 1988a, The Shapley Value, Cambridge: Cambridge University Press.
Roth, A.E.: 1988b, ‘Introduction to the Shapley Value', in: Roth, A.E. (Ed.), The Shapley Value, Cambridge: Cambridge University Press, pp. 1–27.
Shapley, L.S.: 1953, ‘A Value for n-Person Games', in: Kuhn, H.W. and Tucker, A.W. (Eds.), Contributions to the Theory of Games, II (Annals of Mathematics Studies 24), Princeton: Princeton University Press, pp. 307–317.
Shapley, L.S.: 1962, ‘Simple Games: An Outline of the Descriptive Theory', Behavioral Science 7, 59–66.
Shapley, L.S. and Shubik, M.: 1954, ‘A Method for Evaluating the Distribution of Power in a Committee System', American Political Science Review 48, 787–792.
Straffin, P.D.: 1982, ‘Power Indices in Politics', in: Brams, S.J., Lucas, W.F. and Straffin, P.D. (Eds.), Political and Related Models (Vol. 2 in series Models in Applied Mathematics, Ed. W.F. Lucas), New York: Springer, pp. 256–321.
Straffin, P.D.: 1988, ‘The Shapley-Shubik and Banzhaf Power Indices as Probabilities', in: Roth, A.E. (Ed.), The Shapley Value, Cambridge: Cambridge University Press, pp. 71–81.
Taylor, A. and Zwicker, W.: 1993, ‘Weighted Voting, Multicameral Representation, and Power', Games and Economic Behavior 5, 170–181.
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Felsenthal, D.S., Machover, M. & Zwicker, W. The Bicameral Postulates and Indices of a Priori Voting Power. Theory and Decision 44, 83–116 (1998). https://doi.org/10.1023/A:1004914608055
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DOI: https://doi.org/10.1023/A:1004914608055